Surface Area of Prisms and Cylinders worksheet with nine problems requiring calculation of surface area for various geometric shapes.
A worksheet titled "Surface Area of Prisms and Cylinders" with nine problems, each showing a different 3D shape (prisms and a cylinder) with labeled dimensions. Students are asked to calculate the surface area and round to the nearest hundredth if necessary. The website Math-Aids.com is visible at the bottom right.
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Show Answer Key & Explanations
Step-by-step solution for: Solved Surface Area of Prisms and Cylinders Find the surface ...
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Show Answer Key & Explanations
Step-by-step solution for: Solved Surface Area of Prisms and Cylinders Find the surface ...
Let’s solve each problem one by one. We’ll find the surface area of each prism or cylinder using the correct formula for each shape.
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1) Cube with side 5 ft
A cube has 6 identical square faces.
Surface Area = 6 × (side)²
= 6 × (5)²
= 6 × 25
= 150 ft²
✔ Correct as written.
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2) Cube with side 3 in
Same as above:
Surface Area = 6 × (3)²
= 6 × 9
= 54 in²
✔ Correct as written.
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3) Rectangular prism: 3 yd × 3 yd × 6 yd
This is a rectangular prism with dimensions: length=3, width=3, height=6.
Formula: SA = 2(lw + lh + wh)
SA = 2[(3×3) + (3×6) + (3×6)]
= 2[9 + 18 + 18]
= 2[45]
= 90 yd²
Wait — the student wrote “64 yd²” and did some fractions? That’s wrong.
Let me double-check:
Front/back: 3×6 = 18 → two of them = 36
Left/right: 3×6 = 18 → two of them = 36
Top/bottom: 3×3 = 9 → two of them = 18
Total = 36 + 36 + 18 = 90 yd²
So correct answer is 90 yd²
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4) Rectangular prism: 13 mm × 7 mm × 6 mm
Dimensions: l=13, w=7, h=6
SA = 2(lw + lh + wh)
= 2[(13×7) + (13×6) + (7×6)]
= 2[91 + 78 + 42]
= 2[211]
= 422 mm²
Student wrote “422 mm²” — ✔ Correct!
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5) Triangular prism: base triangle sides 4 cm, 4 cm, 5 cm; height of prism = 8 cm
Wait — this is a triangular prism. The bases are triangles, and there are 3 rectangular sides.
First, we need to know if the triangle is equilateral? No — it says 4 cm, 4 cm, 5 cm → so it’s an isosceles triangle.
But to find surface area, we need:
- Area of two triangular bases
- Plus areas of three rectangular lateral faces
We don’t have the height of the triangle yet. Let’s calculate it.
Base = 5 cm, equal sides = 4 cm each.
Drop a perpendicular from top vertex to base → splits base into 2.5 cm each.
Height of triangle = √(4² - 2.5²) = √(16 - 6.25) = √9.75 ≈ 3.122 cm
Area of one triangle = (base × height)/2 = (5 × 3.122)/2 ≈ 7.805 cm²
Two triangles = 2 × 7.805 ≈ 15.61 cm²
Now rectangles:
- Two rectangles with sides 4 cm and 8 cm → 2 × (4×8) = 64 cm²
- One rectangle with side 5 cm and 8 cm → 5×8 = 40 cm²
Total SA = 15.61 + 64 + 40 = 119.61 cm²
Rounded to nearest hundredth: 119.61 cm²
But wait — maybe the diagram shows the triangle has height given? Looking back at image description — no, only sides 4,4,5 and prism height 8.
Alternatively, perhaps they expect us to use Heron’s formula?
Heron’s formula: s = (4+4+5)/2 = 6.5
Area = √[s(s-a)(s-b)(s-c)] = √[6.5(2.5)(2.5)(1.5)]
= √[6.5 × 2.5 × 2.5 × 1.5]
Calculate step by step:
6.5 × 1.5 = 9.75
2.5 × 2.5 = 6.25
Then 9.75 × 6.25 = let's compute:
9.75 × 6 = 58.5
9.75 × 0.25 = 2.4375
Total = 60.9375
√60.9375 ≈ 7.806 cm² per triangle → same as before.
So total SA ≈ 2×7.806 + 4×8 + 4×8 + 5×8? Wait no — the three rectangles are:
Actually, the three rectangular faces correspond to the three sides of the triangle times the prism height.
So:
Rectangle 1: 4 cm × 8 cm = 32
Rectangle 2: 4 cm × 8 cm = 32
Rectangle 3: 5 cm × 8 cm = 40
Sum = 32+32+40 = 104
Plus two triangles: 2 × 7.806 = 15.612
Total = 104 + 15.612 = 119.612 → rounded to 119.61 cm²
So final answer: 119.61 cm²
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6) Rectangular prism: 10 ft × 7 ft × 3 ft
l=10, w=7, h=3
SA = 2(lw + lh + wh)
= 2[(10×7) + (10×3) + (7×3)]
= 2[70 + 30 + 21]
= 2[121]
= 242 ft²
---
7) Hexagonal prism: regular hexagon base with side 8 yd, height 14 yd
For a regular hexagonal prism:
Surface Area = 2 × (area of hexagon) + (perimeter of hexagon × height)
Area of regular hexagon = (3√3 / 2) × s²
s = 8
Area = (3√3 / 2) × 64 = (3 × 1.732 / 2) × 64 ≈ (5.196 / 2) × 64 ≈ 2.598 × 64 ≈ 166.272 yd²
Two bases: 2 × 166.272 ≈ 332.544 yd²
Perimeter = 6 × 8 = 48 yd
Lateral area = 48 × 14 = 672 yd²
Total SA = 332.544 + 672 = 1004.544 → rounded to 1004.54 yd²
But let’s be more precise.
Exact value:
Area of hexagon = (3√3 / 2) × s² = (3√3 / 2) × 64 = 96√3
√3 ≈ 1.7320508
96 × 1.7320508 ≈ 166.2768768
Two bases: 332.5537536
Lateral: 48 × 14 = 672
Total: 332.5537536 + 672 = 1004.5537536 → 1004.55 yd² when rounded to nearest hundredth.
Wait — but sometimes textbooks use approximation √3 ≈ 1.732
Let me recalculate with that:
96 × 1.732 = 166.272
×2 = 332.544
+672 = 1004.544 → rounds to 1004.54 yd²
Depending on rounding convention. But since 0.544 is closer to 0.54 than 0.55? Actually, 0.544 rounds to 0.54 (since third digit is 4 < 5).
Standard rounding: look at thousandths place.
1004.544 → thousandths digit is 4 → round down → 1004.54
But let’s check: 1004.544 is exactly halfway between 1004.54 and 1004.55? No — 1004.544 is less than 1004.545, so rounds to 1004.54.
However, many calculators would give:
Using exact: 96 * sqrt(3) = 96 * 1.73205080757 = 166.27687752672
*2 = 332.55375505344
+672 = 1004.55375505344 → which rounds to 1004.55 because the third decimal is 3? Wait:
1004.553755... → to nearest hundredth: look at thousandths digit = 3 → which is less than 5 → so round down → 1004.55? Wait no:
Number is 1004.553755...
Breakdown:
Units: 1004
Tenths: 5
Hundredths: 5
Thousandths: 3
To round to hundredths, look at thousandths digit: 3 < 5 → so leave hundredths digit as is → 1004.55
Wait — 1004.553... means:
It's 1004 and 553 thousandths.
So to two decimal places: the digit in hundredths place is 5, next digit is 3 → so we do not round up → still 1004.55? No:
Actually, 1004.553 is greater than 1004.55, but when rounding to two decimals, we look at the third decimal.
Rule: if third decimal ≥5, round up second decimal; else leave it.
Here, third decimal is 3 → so we keep second decimal as 5 → so 1004.55
But 1004.553 is actually 1004.55 when rounded to two decimals? Let me think numerically:
The number is approximately 1004.553755
Position:
- Integer part: 1004
- Decimal: .553755
First decimal: 5 (tenths)
Second decimal: 5 (hundredths)
Third decimal: 3 (thousandths)
Since third decimal is 3 < 5, we do not round up the hundredths place.
So it remains 1004.55? But 1004.55 is already two decimals — yes.
Wait, confusion: 1004.553755 rounded to two decimal places:
Look at the third digit after decimal: it's 3, which is less than 5, so we truncate after second digit → so 1004.55
But 1004.55 is the same as writing it — yes.
Actually, 1004.553755 is closer to 1004.55 than to 1004.56? Yes, because 1004.555 would be midpoint.
1004.553755 < 1004.555, so rounds down to 1004.55.
But earlier calculation with approximate √3 gave 1004.544, which rounds to 1004.54.
This discrepancy is due to approximation of √3.
In most school contexts, they use √3 ≈ 1.732
So let's stick with that for consistency.
Area of hexagon = (3√3 / 2) * s² = (3 * 1.732 / 2) * 64 = (5.196 / 2) * 64 = 2.598 * 64
2.598 * 64:
2 * 64 = 128
0.598 * 64 = let's compute: 0.6*64=38.4, minus 0.002*64=0.128 → 38.4 - 0.128 = 38.272? No:
Better: 0.598 * 64 = 598/1000 * 64 = (598*64)/1000
598*64:
600*64 = 38400
minus 2*64=128 → 38400-128=38272
so 38.272
Thus total area = 128 + 38.272 = 166.272 per hexagon? No:
Earlier: 2.598 * 64
2.598 * 60 = 155.88
2.598 * 4 = 10.392
Total = 166.272 — yes.
So two bases: 332.544
Lateral: 48 * 14 = 672
Total: 332.544 + 672 = 1004.544 → now, to nearest hundredth: look at third decimal = 4 → which is less than 5 → so round down → 1004.54 yd²
I think for consistency with typical textbook problems, we'll go with 1004.54 yd²
---
8) Triangular prism: base triangle with base 4 in, height 3 in, and other sides? Diagram shows 3 in, 4 in, and hypotenuse? Also says 3.61 in — probably the slant height or something.
Looking at description: "3 in" vertical, "4 in" base, and "3.61 in" might be the other leg? Or perhaps it's a right triangle with legs 3 and 4, so hypotenuse 5? But it says 3.61 — that doesn't match.
Wait — perhaps the triangle has sides: base 4 in, height 3 in, and the other two sides are not given, but 3.61 in is the length of the non-base side? This is confusing.
Perhaps it's a right triangle with legs 3 in and 4 in, so hypotenuse should be 5 in, but it says 3.61 in — that doesn't make sense.
Another possibility: the 3.61 in is the height of the triangular face? But it's labeled on the side.
Looking back: "3 in" is vertical, "4 in" is base, and "3.61 in" is the slanted side — but in a right triangle with legs 3 and 4, hypotenuse is 5, not 3.61.
Unless it's not a right triangle.
Perhaps the 3.61 in is the height of the triangle corresponding to the base of 4 in? But then area would be (4*3.61)/2 = 7.22, but why label it on the side?
I think there might be a misinterpretation.
Another idea: perhaps the triangular base has sides 3 in, 4 in, and 3.61 in? But 3.61 is approximately √13, not standard.
Let me calculate: if base is 4 in, and height is 3 in, then area of triangle is (4*3)/2 = 6 in², regardless of other sides, as long as height is perpendicular to base.
And the prism height is 11 in (given as "11 in" along the length).
Also, the three rectangular faces:
- One with width 4 in (base) and height 11 in → area 44 in²
- One with width 3 in (height of triangle) and height 11 in → but is that a face? Only if it's a right triangle.
Assume it's a right triangle with legs 3 in and 4 in, so hypotenuse 5 in.
Then the three rectangular faces are:
- 3 in × 11 in = 33 in²
- 4 in × 11 in = 44 in²
- 5 in × 11 in = 55 in²
Sum of rectangles = 33+44+55 = 132 in²
Two triangular bases: each (3*4)/2 = 6 in², so 12 in²
Total SA = 132 + 12 = 144 in²
But the diagram mentions "3.61 in" — perhaps that's a red herring or for another purpose.
Maybe the 3.61 in is the actual length of the side, and it's not a right triangle.
Suppose the triangle has base 4 in, and two other sides: one is 3 in, and the other is 3.61 in.
Then we can find area using Heron's formula.
Sides a=3, b=4, c=3.61
s = (3+4+3.61)/2 = 10.61/2 = 5.305
Area = √[s(s-a)(s-b)(s-c)] = √[5.305(5.305-3)(5.305-4)(5.305-3.61)]
= √[5.305 * 2.305 * 1.305 * 1.695]
Calculate step by step:
First, 5.305 * 2.305 ≈ let's compute:
5.3 * 2.3 = 12.19, more precisely:
5.305 * 2 = 10.61
5.305 * 0.305 = approximately 5.305*0.3=1.5915, 5.305*0.005=0.026525, total 1.618025
So total 10.61 + 1.618025 = 12.228025
Better use calculator in mind or approximate.
Note that 3.61 is close to √13 ≈ 3.6056, so perhaps it's intended to be √13.
Assume sides 3, 4, √13.
But 3^2 + (√13)^2 = 9 + 13 = 22 ≠ 16, not right triangle.
Perhaps the height is 3.61 in for base 4 in.
If base is 4 in, and height is 3.61 in, then area of triangle = (4 * 3.61)/2 = 7.22 in²
Then two bases = 14.44 in²
Now for rectangles: the three sides of the triangle are needed.
If it's not specified, perhaps we assume the triangle has sides corresponding to the labels.
From the diagram description: "3 in" is vertical, "4 in" is base, "3.61 in" is the slanted side, and "11 in" is the length of the prism.
So likely, the triangular base has sides: 3 in, 4 in, and 3.61 in.
But 3.61 is approximately the length if it were a right triangle with legs 3 and 2.5 or something — let's calculate what it should be.
Perhaps the 3.61 in is the height, and the base is 4 in, and the other sides are not needed for area, but for lateral surface area, we need the perimeter.
This is ambiguous.
Another thought: in some diagrams, the 3.61 in might be the apothem or something, but for a triangle, unlikely.
Perhaps it's a typo, and it's supposed to be 5 in for the hypotenuse.
Given that 3-4-5 is a common right triangle, and 3.61 is close to nothing obvious, I think it's safe to assume it's a right triangle with legs 3 in and 4 in, so hypotenuse 5 in, and the 3.61 is either a mistake or for another purpose.
Moreover, in the student's work, they might have used 3.61, but let's see the context.
Perhaps the 3.61 in is the length of the edge, and we should use it as is.
Let me try with sides 3, 4, 3.61.
s = (3+4+3.61)/2 = 5.305
s-a = 5.305-3 = 2.305
s-b = 5.305-4 = 1.305
s-c = 5.305-3.61 = 1.695
Product = 5.305 * 2.305 * 1.305 * 1.695
First, 5.305 * 1.695 ≈ 5.3*1.7 = 9.01, more accurately:
5.305 * 1.6 = 8.488
5.305 * 0.095 = approximately 0.503975
Total ≈ 8.991975
Then 2.305 * 1.305 ≈ 2.3*1.3 = 2.99, more accurately:
2.305 * 1.3 = 2.9965
2.305 * 0.005 = 0.011525
Total ≈ 3.008025
Then product ≈ 8.991975 * 3.008025 ≈ let's say 9*3 = 27, but more accurately:
8.992 * 3.008 ≈ 8.992*3 = 26.976, 8.992*0.008=0.071936, total 27.047936
So area = √27.047936 ≈ 5.2008 in² (since 5.2^2=27.04)
So approximately 5.20 in² per triangle.
Two bases = 10.40 in²
Now lateral faces: three rectangles with widths 3, 4, 3.61 and height 11 in.
Areas: 3*11=33, 4*11=44, 3.61*11=39.71
Sum = 33+44+39.71 = 116.71 in²
Total SA = 10.40 + 116.71 = 127.11 in²
But this seems messy, and 3.61 is likely meant to be the height.
Perhaps the 3.61 in is the height of the triangle, and the base is 4 in, so area = (4*3.61)/2 = 7.22 in², two bases = 14.44 in²
Then for lateral surface area, we need the perimeter of the base triangle.
But we don't have the other two sides.
Unless the triangle is isosceles or something.
Another idea: perhaps the "3 in" is not a side, but the height, and "4 in" is the base, and "3.61 in" is the length of the equal sides if it's isosceles.
Assume the triangular base is isosceles with base 4 in, and height 3 in, then the equal sides can be calculated.
Height to base 4 in, so half-base is 2 in, so side = √(2^2 + 3^2) = √(4+9) = √13 ≈ 3.6056 in, which matches "3.61 in"!
Yes! So the triangle has base 4 in, height 3 in, and two equal sides of √13 ≈ 3.61 in.
Perfect.
So area of triangle = (base * height)/2 = (4 * 3)/2 = 6 in²
Two bases = 12 in²
Now lateral surface area: three rectangles:
- Base rectangle: 4 in × 11 in = 44 in²
- Two side rectangles: each 3.61 in × 11 in = 39.71 in² each, so 2 × 39.71 = 79.42 in²
Sum of rectangles = 44 + 79.42 = 123.42 in²
Total SA = 12 + 123.42 = 135.42 in²
Rounded to nearest hundredth: 135.42 in²
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9) Trapezoidal prism: trapezoid base with parallel sides 6 cm and 13 cm, height of trapezoid 7 cm, and prism height 8 cm? Diagram shows "5 cm", "8 cm", etc.
From description: "5 cm", "8 cm", "7 cm", "6 cm", "13 cm"
Likely, the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm (the distance between them), and the non-parallel sides are 5 cm and 8 cm? But 5 and 8 may not fit.
Prism height is probably the length, which might be 8 cm or something.
Looking: "5 cm" might be one leg, "8 cm" might be the other leg or the prism height.
Typically, for a trapezoidal prism, we need:
- Area of two trapezoidal bases
- Plus areas of four rectangular lateral faces (since trapezoid has 4 sides)
First, area of trapezoid = (sum of parallel sides)/2 × height = (6 + 13)/2 × 7 = (19/2) × 7 = 9.5 × 7 = 66.5 cm²
Two bases = 133 cm²
Now, the four lateral faces: each is a rectangle with width equal to a side of the trapezoid and height equal to the prism's length.
What is the prism's length? The diagram likely shows it as 8 cm or 5 cm.
From the labels: "5 cm", "8 cm", "7 cm", "6 cm", "13 cm"
Probably, the trapezoid has sides: bottom 13 cm, top 6 cm, left side 5 cm, right side 8 cm, and height 7 cm (which is the perpendicular distance).
But is that possible? Let's verify if such a trapezoid exists.
The difference in bases is 13 - 6 = 7 cm. This extra length is distributed on both sides.
If it's not isosceles, the overhang on left and right may be different.
Let the overhang on left be x, on right be y, so x + y = 7 cm.
Then, for the left side: if it's 5 cm, and height 7 cm, then by Pythagoras, x = √(5^2 - 7^2) = √(25-49) = √(-24) — impossible!
Oh no! 5 cm side with height 7 cm — the side must be longer than the height, but 5 < 7, so impossible for a right triangle.
Perhaps the 7 cm is not the height of the trapezoid, but something else.
Another possibility: the "7 cm" is the length of the prism, and the height of the trapezoid is given by other means.
Let's read the diagram description again: "7 cm" is labeled on the side, "6 cm" and "13 cm" are the parallel sides, "5 cm" and "8 cm" are the non-parallel sides, and "8 cm" might be the prism height.
Perhaps the prism height is 8 cm, and the trapezoid has parallel sides 6 cm and 13 cm, and the legs are 5 cm and 8 cm, but then the height of the trapezoid needs to be calculated.
But with legs 5 and 8, and bases 6 and 13, the difference is 7 cm, so if we drop perpendiculars, the sum of the projections is 7 cm.
Let the projection of the 5 cm side be a, of the 8 cm side be b, so a + b = 7.
Then, for the 5 cm side: a^2 + h^2 = 5^2 = 25
For the 8 cm side: b^2 + h^2 = 64
Subtract the first equation from the second: (b^2 + h^2) - (a^2 + h^2) = 64 - 25 => b^2 - a^2 = 39
But b = 7 - a, so (7-a)^2 - a^2 = 39
49 - 14a + a^2 - a^2 = 39 => 49 - 14a = 39 => 14a = 10 => a = 10/14 = 5/7 ≈ 0.7143 cm
Then b = 7 - 5/7 = 44/7 ≈ 6.2857 cm
Then h^2 = 25 - a^2 = 25 - (25/49) = (1225 - 25)/49 = 1200/49
h = √(1200/49) = (√1200)/7 = (20√3)/7 ≈ (20*1.732)/7 ≈ 34.64/7 ≈ 4.9486 cm
Then area of trapezoid = (6+13)/2 * h = 9.5 * 4.9486 ≈ 47.0117 cm²
Two bases = 94.0234 cm²
Now lateral faces: four rectangles:
- Bottom: 13 cm × prism_height
- Top: 6 cm × prism_height
- Left: 5 cm × prism_height
- Right: 8 cm × prism_height
What is the prism height? The diagram likely shows it as 8 cm or 7 cm. In the list, "8 cm" is mentioned, and "7 cm" is also there.
Perhaps the "8 cm" is the prism height, and "7 cm" is the height of the trapezoid, but we saw that's impossible with sides 5 and 8.
Another possibility: the "7 cm" is the prism height, and the trapezoid height is to be calculated, but then we have the same issue.
Perhaps the trapezoid is right-angled or something.
Let's look back at the user's input: "7 cm" is labeled, and "5 cm", "8 cm", "6 cm", "13 cm"
Perhaps the 7 cm is the height of the trapezoid, and the 5 cm and 8 cm are the lengths of the non-parallel sides, but as above, with height 7, the sides must be at least 7, but 5<7, so impossible.
Unless the 5 cm is not a side, but something else.
Perhaps "5 cm" is the length of the top or bottom, but it's already given as 6 and 13.
Another idea: perhaps the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm, and the non-parallel sides are not 5 and 8, but the 5 cm and 8 cm are the lengths of the lateral edges or something.
I think there might be a mislabeling.
Perhaps the "5 cm" is the difference or something.
Let's assume that the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm, and the non-parallel sides are equal or something, but not specified.
But in the diagram, it shows "5 cm" and "8 cm" on the sides, so likely those are the lengths of the legs.
But as calculated, with height 7, the leg cannot be 5 cm.
Unless the height is not 7 cm for the trapezoid.
Perhaps the "7 cm" is the length of the prism, and the height of the trapezoid is given by the 5 cm or 8 cm, but that doesn't make sense.
Let's try to interpret differently.
Suppose the trapezoid has bases 6 cm and 13 cm, and the legs are 5 cm and 8 cm, and the prism height is 7 cm.
Then, as above, we can calculate the height of the trapezoid.
From earlier:
a + b = 7 (difference in bases)
a^2 + h^2 = 25
b^2 + h^2 = 64
Then b^2 - a^2 = 39
(b-a)(b+a) = 39
b+ a = 7, so (b-a)*7 = 39 => b-a = 39/7 ≈ 5.5714
Then b + a = 7
b - a = 39/7
Add: 2b = 7 + 39/7 = (49 + 39)/7 = 88/7 => b = 44/7 ≈ 6.2857 cm
a = 7 - b = 7 - 44/7 = (49-44)/7 = 5/7 ≈ 0.7143 cm
Then h^2 = 25 - a^2 = 25 - (25/49) = (1225 - 25)/49 = 1200/49
h = √(1200/49) = (√1200)/7 = (20√3)/7 ≈ 20*1.73205/7 ≈ 34.641/7 ≈ 4.9487 cm
Area of trapezoid = (6+13)/2 * h = 9.5 * 4.9487 ≈ 47.01265 cm²
Two bases = 94.0253 cm²
Now lateral surface area: the four rectangles have heights equal to the prism height, which is likely 7 cm (since "7 cm" is listed, and not used yet).
So:
- Rectangle for bottom base: 13 cm × 7 cm = 91 cm²
- Rectangle for top base: 6 cm × 7 cm = 42 cm²
- Rectangle for left leg: 5 cm × 7 cm = 35 cm²
- Rectangle for right leg: 8 cm × 7 cm = 56 cm²
Sum of lateral areas = 91 + 42 + 35 + 56 = 224 cm²
Total SA = 94.0253 + 224 = 318.0253 cm² → rounded to nearest hundredth: 318.03 cm²
But this is approximate.
To be precise, let's keep exact values.
h = √(1200/49) = (√1200)/7 = (20√3)/7
Area of one trapezoid = (19/2) * (20√3)/7 = (19 * 20 * √3) / (2*7) = (380 √3) / 14 = (190 √3) / 7
Two bases = 2 * (190 √3) / 7 = 380 √3 / 7
√3 ≈ 1.7320508, so 380 * 1.7320508 = 658.179304, divided by 7 = 94.025614857 cm²
Lateral: 13*7 = 91, 6*7=42, 5*7=35, 8*7=56, sum 91+42=133, +35=168, +56=224 cm²
Total = 94.025614857 + 224 = 318.025614857 → rounds to 318.03 cm²
So we'll go with that.
---
Now, let's compile all answers:
1) 150 ft²
2) 54 in²
3) 90 yd² (corrected from 64)
4) 422 mm²
5) 119.61 cm²
6) 242 ft²
7) 1004.54 yd²
8) 135.42 in²
9) 318.03 cm²
For problem 3, the student had 64, but it should be 90.
For others, we have calculated.
Final Answer:
1) 150 ft²
2) 54 in²
3) 90 yd²
4) 422 mm²
5) 119.61 cm²
6) 242 ft²
7) 1004.54 yd²
8) 135.42 in²
9) 318.03 cm²
---
1) Cube with side 5 ft
A cube has 6 identical square faces.
Surface Area = 6 × (side)²
= 6 × (5)²
= 6 × 25
= 150 ft²
✔ Correct as written.
---
2) Cube with side 3 in
Same as above:
Surface Area = 6 × (3)²
= 6 × 9
= 54 in²
✔ Correct as written.
---
3) Rectangular prism: 3 yd × 3 yd × 6 yd
This is a rectangular prism with dimensions: length=3, width=3, height=6.
Formula: SA = 2(lw + lh + wh)
SA = 2[(3×3) + (3×6) + (3×6)]
= 2[9 + 18 + 18]
= 2[45]
= 90 yd²
Wait — the student wrote “64 yd²” and did some fractions? That’s wrong.
Let me double-check:
Front/back: 3×6 = 18 → two of them = 36
Left/right: 3×6 = 18 → two of them = 36
Top/bottom: 3×3 = 9 → two of them = 18
Total = 36 + 36 + 18 = 90 yd²
So correct answer is 90 yd²
---
4) Rectangular prism: 13 mm × 7 mm × 6 mm
Dimensions: l=13, w=7, h=6
SA = 2(lw + lh + wh)
= 2[(13×7) + (13×6) + (7×6)]
= 2[91 + 78 + 42]
= 2[211]
= 422 mm²
Student wrote “422 mm²” — ✔ Correct!
---
5) Triangular prism: base triangle sides 4 cm, 4 cm, 5 cm; height of prism = 8 cm
Wait — this is a triangular prism. The bases are triangles, and there are 3 rectangular sides.
First, we need to know if the triangle is equilateral? No — it says 4 cm, 4 cm, 5 cm → so it’s an isosceles triangle.
But to find surface area, we need:
- Area of two triangular bases
- Plus areas of three rectangular lateral faces
We don’t have the height of the triangle yet. Let’s calculate it.
Base = 5 cm, equal sides = 4 cm each.
Drop a perpendicular from top vertex to base → splits base into 2.5 cm each.
Height of triangle = √(4² - 2.5²) = √(16 - 6.25) = √9.75 ≈ 3.122 cm
Area of one triangle = (base × height)/2 = (5 × 3.122)/2 ≈ 7.805 cm²
Two triangles = 2 × 7.805 ≈ 15.61 cm²
Now rectangles:
- Two rectangles with sides 4 cm and 8 cm → 2 × (4×8) = 64 cm²
- One rectangle with side 5 cm and 8 cm → 5×8 = 40 cm²
Total SA = 15.61 + 64 + 40 = 119.61 cm²
Rounded to nearest hundredth: 119.61 cm²
But wait — maybe the diagram shows the triangle has height given? Looking back at image description — no, only sides 4,4,5 and prism height 8.
Alternatively, perhaps they expect us to use Heron’s formula?
Heron’s formula: s = (4+4+5)/2 = 6.5
Area = √[s(s-a)(s-b)(s-c)] = √[6.5(2.5)(2.5)(1.5)]
= √[6.5 × 2.5 × 2.5 × 1.5]
Calculate step by step:
6.5 × 1.5 = 9.75
2.5 × 2.5 = 6.25
Then 9.75 × 6.25 = let's compute:
9.75 × 6 = 58.5
9.75 × 0.25 = 2.4375
Total = 60.9375
√60.9375 ≈ 7.806 cm² per triangle → same as before.
So total SA ≈ 2×7.806 + 4×8 + 4×8 + 5×8? Wait no — the three rectangles are:
Actually, the three rectangular faces correspond to the three sides of the triangle times the prism height.
So:
Rectangle 1: 4 cm × 8 cm = 32
Rectangle 2: 4 cm × 8 cm = 32
Rectangle 3: 5 cm × 8 cm = 40
Sum = 32+32+40 = 104
Plus two triangles: 2 × 7.806 = 15.612
Total = 104 + 15.612 = 119.612 → rounded to 119.61 cm²
So final answer: 119.61 cm²
---
6) Rectangular prism: 10 ft × 7 ft × 3 ft
l=10, w=7, h=3
SA = 2(lw + lh + wh)
= 2[(10×7) + (10×3) + (7×3)]
= 2[70 + 30 + 21]
= 2[121]
= 242 ft²
---
7) Hexagonal prism: regular hexagon base with side 8 yd, height 14 yd
For a regular hexagonal prism:
Surface Area = 2 × (area of hexagon) + (perimeter of hexagon × height)
Area of regular hexagon = (3√3 / 2) × s²
s = 8
Area = (3√3 / 2) × 64 = (3 × 1.732 / 2) × 64 ≈ (5.196 / 2) × 64 ≈ 2.598 × 64 ≈ 166.272 yd²
Two bases: 2 × 166.272 ≈ 332.544 yd²
Perimeter = 6 × 8 = 48 yd
Lateral area = 48 × 14 = 672 yd²
Total SA = 332.544 + 672 = 1004.544 → rounded to 1004.54 yd²
But let’s be more precise.
Exact value:
Area of hexagon = (3√3 / 2) × s² = (3√3 / 2) × 64 = 96√3
√3 ≈ 1.7320508
96 × 1.7320508 ≈ 166.2768768
Two bases: 332.5537536
Lateral: 48 × 14 = 672
Total: 332.5537536 + 672 = 1004.5537536 → 1004.55 yd² when rounded to nearest hundredth.
Wait — but sometimes textbooks use approximation √3 ≈ 1.732
Let me recalculate with that:
96 × 1.732 = 166.272
×2 = 332.544
+672 = 1004.544 → rounds to 1004.54 yd²
Depending on rounding convention. But since 0.544 is closer to 0.54 than 0.55? Actually, 0.544 rounds to 0.54 (since third digit is 4 < 5).
Standard rounding: look at thousandths place.
1004.544 → thousandths digit is 4 → round down → 1004.54
But let’s check: 1004.544 is exactly halfway between 1004.54 and 1004.55? No — 1004.544 is less than 1004.545, so rounds to 1004.54.
However, many calculators would give:
Using exact: 96 * sqrt(3) = 96 * 1.73205080757 = 166.27687752672
*2 = 332.55375505344
+672 = 1004.55375505344 → which rounds to 1004.55 because the third decimal is 3? Wait:
1004.553755... → to nearest hundredth: look at thousandths digit = 3 → which is less than 5 → so round down → 1004.55? Wait no:
Number is 1004.553755...
Breakdown:
Units: 1004
Tenths: 5
Hundredths: 5
Thousandths: 3
To round to hundredths, look at thousandths digit: 3 < 5 → so leave hundredths digit as is → 1004.55
Wait — 1004.553... means:
It's 1004 and 553 thousandths.
So to two decimal places: the digit in hundredths place is 5, next digit is 3 → so we do not round up → still 1004.55? No:
Actually, 1004.553 is greater than 1004.55, but when rounding to two decimals, we look at the third decimal.
Rule: if third decimal ≥5, round up second decimal; else leave it.
Here, third decimal is 3 → so we keep second decimal as 5 → so 1004.55
But 1004.553 is actually 1004.55 when rounded to two decimals? Let me think numerically:
The number is approximately 1004.553755
Position:
- Integer part: 1004
- Decimal: .553755
First decimal: 5 (tenths)
Second decimal: 5 (hundredths)
Third decimal: 3 (thousandths)
Since third decimal is 3 < 5, we do not round up the hundredths place.
So it remains 1004.55? But 1004.55 is already two decimals — yes.
Wait, confusion: 1004.553755 rounded to two decimal places:
Look at the third digit after decimal: it's 3, which is less than 5, so we truncate after second digit → so 1004.55
But 1004.55 is the same as writing it — yes.
Actually, 1004.553755 is closer to 1004.55 than to 1004.56? Yes, because 1004.555 would be midpoint.
1004.553755 < 1004.555, so rounds down to 1004.55.
But earlier calculation with approximate √3 gave 1004.544, which rounds to 1004.54.
This discrepancy is due to approximation of √3.
In most school contexts, they use √3 ≈ 1.732
So let's stick with that for consistency.
Area of hexagon = (3√3 / 2) * s² = (3 * 1.732 / 2) * 64 = (5.196 / 2) * 64 = 2.598 * 64
2.598 * 64:
2 * 64 = 128
0.598 * 64 = let's compute: 0.6*64=38.4, minus 0.002*64=0.128 → 38.4 - 0.128 = 38.272? No:
Better: 0.598 * 64 = 598/1000 * 64 = (598*64)/1000
598*64:
600*64 = 38400
minus 2*64=128 → 38400-128=38272
so 38.272
Thus total area = 128 + 38.272 = 166.272 per hexagon? No:
Earlier: 2.598 * 64
2.598 * 60 = 155.88
2.598 * 4 = 10.392
Total = 166.272 — yes.
So two bases: 332.544
Lateral: 48 * 14 = 672
Total: 332.544 + 672 = 1004.544 → now, to nearest hundredth: look at third decimal = 4 → which is less than 5 → so round down → 1004.54 yd²
I think for consistency with typical textbook problems, we'll go with 1004.54 yd²
---
8) Triangular prism: base triangle with base 4 in, height 3 in, and other sides? Diagram shows 3 in, 4 in, and hypotenuse? Also says 3.61 in — probably the slant height or something.
Looking at description: "3 in" vertical, "4 in" base, and "3.61 in" might be the other leg? Or perhaps it's a right triangle with legs 3 and 4, so hypotenuse 5? But it says 3.61 — that doesn't match.
Wait — perhaps the triangle has sides: base 4 in, height 3 in, and the other two sides are not given, but 3.61 in is the length of the non-base side? This is confusing.
Perhaps it's a right triangle with legs 3 in and 4 in, so hypotenuse should be 5 in, but it says 3.61 in — that doesn't make sense.
Another possibility: the 3.61 in is the height of the triangular face? But it's labeled on the side.
Looking back: "3 in" is vertical, "4 in" is base, and "3.61 in" is the slanted side — but in a right triangle with legs 3 and 4, hypotenuse is 5, not 3.61.
Unless it's not a right triangle.
Perhaps the 3.61 in is the height of the triangle corresponding to the base of 4 in? But then area would be (4*3.61)/2 = 7.22, but why label it on the side?
I think there might be a misinterpretation.
Another idea: perhaps the triangular base has sides 3 in, 4 in, and 3.61 in? But 3.61 is approximately √13, not standard.
Let me calculate: if base is 4 in, and height is 3 in, then area of triangle is (4*3)/2 = 6 in², regardless of other sides, as long as height is perpendicular to base.
And the prism height is 11 in (given as "11 in" along the length).
Also, the three rectangular faces:
- One with width 4 in (base) and height 11 in → area 44 in²
- One with width 3 in (height of triangle) and height 11 in → but is that a face? Only if it's a right triangle.
Assume it's a right triangle with legs 3 in and 4 in, so hypotenuse 5 in.
Then the three rectangular faces are:
- 3 in × 11 in = 33 in²
- 4 in × 11 in = 44 in²
- 5 in × 11 in = 55 in²
Sum of rectangles = 33+44+55 = 132 in²
Two triangular bases: each (3*4)/2 = 6 in², so 12 in²
Total SA = 132 + 12 = 144 in²
But the diagram mentions "3.61 in" — perhaps that's a red herring or for another purpose.
Maybe the 3.61 in is the actual length of the side, and it's not a right triangle.
Suppose the triangle has base 4 in, and two other sides: one is 3 in, and the other is 3.61 in.
Then we can find area using Heron's formula.
Sides a=3, b=4, c=3.61
s = (3+4+3.61)/2 = 10.61/2 = 5.305
Area = √[s(s-a)(s-b)(s-c)] = √[5.305(5.305-3)(5.305-4)(5.305-3.61)]
= √[5.305 * 2.305 * 1.305 * 1.695]
Calculate step by step:
First, 5.305 * 2.305 ≈ let's compute:
5.3 * 2.3 = 12.19, more precisely:
5.305 * 2 = 10.61
5.305 * 0.305 = approximately 5.305*0.3=1.5915, 5.305*0.005=0.026525, total 1.618025
So total 10.61 + 1.618025 = 12.228025
Better use calculator in mind or approximate.
Note that 3.61 is close to √13 ≈ 3.6056, so perhaps it's intended to be √13.
Assume sides 3, 4, √13.
But 3^2 + (√13)^2 = 9 + 13 = 22 ≠ 16, not right triangle.
Perhaps the height is 3.61 in for base 4 in.
If base is 4 in, and height is 3.61 in, then area of triangle = (4 * 3.61)/2 = 7.22 in²
Then two bases = 14.44 in²
Now for rectangles: the three sides of the triangle are needed.
If it's not specified, perhaps we assume the triangle has sides corresponding to the labels.
From the diagram description: "3 in" is vertical, "4 in" is base, "3.61 in" is the slanted side, and "11 in" is the length of the prism.
So likely, the triangular base has sides: 3 in, 4 in, and 3.61 in.
But 3.61 is approximately the length if it were a right triangle with legs 3 and 2.5 or something — let's calculate what it should be.
Perhaps the 3.61 in is the height, and the base is 4 in, and the other sides are not needed for area, but for lateral surface area, we need the perimeter.
This is ambiguous.
Another thought: in some diagrams, the 3.61 in might be the apothem or something, but for a triangle, unlikely.
Perhaps it's a typo, and it's supposed to be 5 in for the hypotenuse.
Given that 3-4-5 is a common right triangle, and 3.61 is close to nothing obvious, I think it's safe to assume it's a right triangle with legs 3 in and 4 in, so hypotenuse 5 in, and the 3.61 is either a mistake or for another purpose.
Moreover, in the student's work, they might have used 3.61, but let's see the context.
Perhaps the 3.61 in is the length of the edge, and we should use it as is.
Let me try with sides 3, 4, 3.61.
s = (3+4+3.61)/2 = 5.305
s-a = 5.305-3 = 2.305
s-b = 5.305-4 = 1.305
s-c = 5.305-3.61 = 1.695
Product = 5.305 * 2.305 * 1.305 * 1.695
First, 5.305 * 1.695 ≈ 5.3*1.7 = 9.01, more accurately:
5.305 * 1.6 = 8.488
5.305 * 0.095 = approximately 0.503975
Total ≈ 8.991975
Then 2.305 * 1.305 ≈ 2.3*1.3 = 2.99, more accurately:
2.305 * 1.3 = 2.9965
2.305 * 0.005 = 0.011525
Total ≈ 3.008025
Then product ≈ 8.991975 * 3.008025 ≈ let's say 9*3 = 27, but more accurately:
8.992 * 3.008 ≈ 8.992*3 = 26.976, 8.992*0.008=0.071936, total 27.047936
So area = √27.047936 ≈ 5.2008 in² (since 5.2^2=27.04)
So approximately 5.20 in² per triangle.
Two bases = 10.40 in²
Now lateral faces: three rectangles with widths 3, 4, 3.61 and height 11 in.
Areas: 3*11=33, 4*11=44, 3.61*11=39.71
Sum = 33+44+39.71 = 116.71 in²
Total SA = 10.40 + 116.71 = 127.11 in²
But this seems messy, and 3.61 is likely meant to be the height.
Perhaps the 3.61 in is the height of the triangle, and the base is 4 in, so area = (4*3.61)/2 = 7.22 in², two bases = 14.44 in²
Then for lateral surface area, we need the perimeter of the base triangle.
But we don't have the other two sides.
Unless the triangle is isosceles or something.
Another idea: perhaps the "3 in" is not a side, but the height, and "4 in" is the base, and "3.61 in" is the length of the equal sides if it's isosceles.
Assume the triangular base is isosceles with base 4 in, and height 3 in, then the equal sides can be calculated.
Height to base 4 in, so half-base is 2 in, so side = √(2^2 + 3^2) = √(4+9) = √13 ≈ 3.6056 in, which matches "3.61 in"!
Yes! So the triangle has base 4 in, height 3 in, and two equal sides of √13 ≈ 3.61 in.
Perfect.
So area of triangle = (base * height)/2 = (4 * 3)/2 = 6 in²
Two bases = 12 in²
Now lateral surface area: three rectangles:
- Base rectangle: 4 in × 11 in = 44 in²
- Two side rectangles: each 3.61 in × 11 in = 39.71 in² each, so 2 × 39.71 = 79.42 in²
Sum of rectangles = 44 + 79.42 = 123.42 in²
Total SA = 12 + 123.42 = 135.42 in²
Rounded to nearest hundredth: 135.42 in²
---
9) Trapezoidal prism: trapezoid base with parallel sides 6 cm and 13 cm, height of trapezoid 7 cm, and prism height 8 cm? Diagram shows "5 cm", "8 cm", etc.
From description: "5 cm", "8 cm", "7 cm", "6 cm", "13 cm"
Likely, the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm (the distance between them), and the non-parallel sides are 5 cm and 8 cm? But 5 and 8 may not fit.
Prism height is probably the length, which might be 8 cm or something.
Looking: "5 cm" might be one leg, "8 cm" might be the other leg or the prism height.
Typically, for a trapezoidal prism, we need:
- Area of two trapezoidal bases
- Plus areas of four rectangular lateral faces (since trapezoid has 4 sides)
First, area of trapezoid = (sum of parallel sides)/2 × height = (6 + 13)/2 × 7 = (19/2) × 7 = 9.5 × 7 = 66.5 cm²
Two bases = 133 cm²
Now, the four lateral faces: each is a rectangle with width equal to a side of the trapezoid and height equal to the prism's length.
What is the prism's length? The diagram likely shows it as 8 cm or 5 cm.
From the labels: "5 cm", "8 cm", "7 cm", "6 cm", "13 cm"
Probably, the trapezoid has sides: bottom 13 cm, top 6 cm, left side 5 cm, right side 8 cm, and height 7 cm (which is the perpendicular distance).
But is that possible? Let's verify if such a trapezoid exists.
The difference in bases is 13 - 6 = 7 cm. This extra length is distributed on both sides.
If it's not isosceles, the overhang on left and right may be different.
Let the overhang on left be x, on right be y, so x + y = 7 cm.
Then, for the left side: if it's 5 cm, and height 7 cm, then by Pythagoras, x = √(5^2 - 7^2) = √(25-49) = √(-24) — impossible!
Oh no! 5 cm side with height 7 cm — the side must be longer than the height, but 5 < 7, so impossible for a right triangle.
Perhaps the 7 cm is not the height of the trapezoid, but something else.
Another possibility: the "7 cm" is the length of the prism, and the height of the trapezoid is given by other means.
Let's read the diagram description again: "7 cm" is labeled on the side, "6 cm" and "13 cm" are the parallel sides, "5 cm" and "8 cm" are the non-parallel sides, and "8 cm" might be the prism height.
Perhaps the prism height is 8 cm, and the trapezoid has parallel sides 6 cm and 13 cm, and the legs are 5 cm and 8 cm, but then the height of the trapezoid needs to be calculated.
But with legs 5 and 8, and bases 6 and 13, the difference is 7 cm, so if we drop perpendiculars, the sum of the projections is 7 cm.
Let the projection of the 5 cm side be a, of the 8 cm side be b, so a + b = 7.
Then, for the 5 cm side: a^2 + h^2 = 5^2 = 25
For the 8 cm side: b^2 + h^2 = 64
Subtract the first equation from the second: (b^2 + h^2) - (a^2 + h^2) = 64 - 25 => b^2 - a^2 = 39
But b = 7 - a, so (7-a)^2 - a^2 = 39
49 - 14a + a^2 - a^2 = 39 => 49 - 14a = 39 => 14a = 10 => a = 10/14 = 5/7 ≈ 0.7143 cm
Then b = 7 - 5/7 = 44/7 ≈ 6.2857 cm
Then h^2 = 25 - a^2 = 25 - (25/49) = (1225 - 25)/49 = 1200/49
h = √(1200/49) = (√1200)/7 = (20√3)/7 ≈ (20*1.732)/7 ≈ 34.64/7 ≈ 4.9486 cm
Then area of trapezoid = (6+13)/2 * h = 9.5 * 4.9486 ≈ 47.0117 cm²
Two bases = 94.0234 cm²
Now lateral faces: four rectangles:
- Bottom: 13 cm × prism_height
- Top: 6 cm × prism_height
- Left: 5 cm × prism_height
- Right: 8 cm × prism_height
What is the prism height? The diagram likely shows it as 8 cm or 7 cm. In the list, "8 cm" is mentioned, and "7 cm" is also there.
Perhaps the "8 cm" is the prism height, and "7 cm" is the height of the trapezoid, but we saw that's impossible with sides 5 and 8.
Another possibility: the "7 cm" is the prism height, and the trapezoid height is to be calculated, but then we have the same issue.
Perhaps the trapezoid is right-angled or something.
Let's look back at the user's input: "7 cm" is labeled, and "5 cm", "8 cm", "6 cm", "13 cm"
Perhaps the 7 cm is the height of the trapezoid, and the 5 cm and 8 cm are the lengths of the non-parallel sides, but as above, with height 7, the sides must be at least 7, but 5<7, so impossible.
Unless the 5 cm is not a side, but something else.
Perhaps "5 cm" is the length of the top or bottom, but it's already given as 6 and 13.
Another idea: perhaps the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm, and the non-parallel sides are not 5 and 8, but the 5 cm and 8 cm are the lengths of the lateral edges or something.
I think there might be a mislabeling.
Perhaps the "5 cm" is the difference or something.
Let's assume that the trapezoid has parallel sides 6 cm and 13 cm, height 7 cm, and the non-parallel sides are equal or something, but not specified.
But in the diagram, it shows "5 cm" and "8 cm" on the sides, so likely those are the lengths of the legs.
But as calculated, with height 7, the leg cannot be 5 cm.
Unless the height is not 7 cm for the trapezoid.
Perhaps the "7 cm" is the length of the prism, and the height of the trapezoid is given by the 5 cm or 8 cm, but that doesn't make sense.
Let's try to interpret differently.
Suppose the trapezoid has bases 6 cm and 13 cm, and the legs are 5 cm and 8 cm, and the prism height is 7 cm.
Then, as above, we can calculate the height of the trapezoid.
From earlier:
a + b = 7 (difference in bases)
a^2 + h^2 = 25
b^2 + h^2 = 64
Then b^2 - a^2 = 39
(b-a)(b+a) = 39
b+ a = 7, so (b-a)*7 = 39 => b-a = 39/7 ≈ 5.5714
Then b + a = 7
b - a = 39/7
Add: 2b = 7 + 39/7 = (49 + 39)/7 = 88/7 => b = 44/7 ≈ 6.2857 cm
a = 7 - b = 7 - 44/7 = (49-44)/7 = 5/7 ≈ 0.7143 cm
Then h^2 = 25 - a^2 = 25 - (25/49) = (1225 - 25)/49 = 1200/49
h = √(1200/49) = (√1200)/7 = (20√3)/7 ≈ 20*1.73205/7 ≈ 34.641/7 ≈ 4.9487 cm
Area of trapezoid = (6+13)/2 * h = 9.5 * 4.9487 ≈ 47.01265 cm²
Two bases = 94.0253 cm²
Now lateral surface area: the four rectangles have heights equal to the prism height, which is likely 7 cm (since "7 cm" is listed, and not used yet).
So:
- Rectangle for bottom base: 13 cm × 7 cm = 91 cm²
- Rectangle for top base: 6 cm × 7 cm = 42 cm²
- Rectangle for left leg: 5 cm × 7 cm = 35 cm²
- Rectangle for right leg: 8 cm × 7 cm = 56 cm²
Sum of lateral areas = 91 + 42 + 35 + 56 = 224 cm²
Total SA = 94.0253 + 224 = 318.0253 cm² → rounded to nearest hundredth: 318.03 cm²
But this is approximate.
To be precise, let's keep exact values.
h = √(1200/49) = (√1200)/7 = (20√3)/7
Area of one trapezoid = (19/2) * (20√3)/7 = (19 * 20 * √3) / (2*7) = (380 √3) / 14 = (190 √3) / 7
Two bases = 2 * (190 √3) / 7 = 380 √3 / 7
√3 ≈ 1.7320508, so 380 * 1.7320508 = 658.179304, divided by 7 = 94.025614857 cm²
Lateral: 13*7 = 91, 6*7=42, 5*7=35, 8*7=56, sum 91+42=133, +35=168, +56=224 cm²
Total = 94.025614857 + 224 = 318.025614857 → rounds to 318.03 cm²
So we'll go with that.
---
Now, let's compile all answers:
1) 150 ft²
2) 54 in²
3) 90 yd² (corrected from 64)
4) 422 mm²
5) 119.61 cm²
6) 242 ft²
7) 1004.54 yd²
8) 135.42 in²
9) 318.03 cm²
For problem 3, the student had 64, but it should be 90.
For others, we have calculated.
Final Answer:
1) 150 ft²
2) 54 in²
3) 90 yd²
4) 422 mm²
5) 119.61 cm²
6) 242 ft²
7) 1004.54 yd²
8) 135.42 in²
9) 318.03 cm²
Parent Tip: Review the logic above to help your child master the concept of surface area of prisms and cylinders worksheet answers.