Grade 8 Measurement Packet with formulas and practice problems for surface area and volume.

Let’s solve each part step by step. We’ll use the formulas from the table:

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a. A rectangular prism measures 11 m by 8 m by 5 m.

This is a rectangular prism, so we use:
- Surface Area = 2(lw + lh + wh)
- Volume = l × w × h

Given:
l = 11 m, w = 8 m, h = 5 m

Surface Area:
= 2[(11×8) + (11×5) + (8×5)]
= 2[88 + 55 + 40]
= 2[183]
= 366 m²

Volume:
= 11 × 8 × 5
= 88 × 5
= 440 m³

✔ Rounded to two decimal places? They’re whole numbers → still 366.00 and 440.00 if needed, but usually written as is.

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b. A cylinder with radius 7 in and height 9 in.

Use:
- Surface Area = 2πr(r + h)
- Volume = πr²h

Given: r = 7 in, h = 9 in

Surface Area:
= 2π(7)(7 + 9)
= 2π(7)(16)
= 2π(112)
= 224π
≈ 224 × 3.1416 ≈ 703.72 in²

Volume:
= π(7)²(9)
= π(49)(9)
= 441π
≈ 441 × 3.1416 ≈ 1385.44 in³

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c. A sphere with radius 12 cm.

Use:
- Surface Area = 4πr²
- Volume = (4/3)πr³

Given: r = 12 cm

Surface Area:
= 4π(12)²
= 4π(144)
= 576π
≈ 576 × 3.1416 ≈ 1809.56 cm²

Volume:
= (4/3)π(12)³
= (4/3)π(1728)
= (4 × 1728 / 3) π
= (6912 / 3) π
= 2304π
≈ 2304 × 3.1416 ≈ 7238.23 cm³

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d. A right triangular prism with base legs 7.2 m and 9.3 m, length 10.2 m, slant height 10.8 m.

First, understand the shape: It’s a right triangular prism. The “base” is a right triangle with legs 7.2 m and 9.3 m. The “length” of the prism (also called depth or height of prism) is 10.2 m. Slant height = 10.8 m — this must be the hypotenuse of the triangular base.

Check: Is 10.8 the hypotenuse of 7.2 and 9.3?

Compute: √(7.2² + 9.3²) = √(51.84 + 86.49) = √138.33 ≈ 11.76 → NOT 10.8. So maybe “slant height” refers to something else? Wait — perhaps it's the hypotenuse given for surface area calculation? Let’s read carefully.

Actually, in prisms, “slant height” isn’t standard unless it’s a pyramid. But here it says “right triangular prism”, and gives “slant height 10.8 m”. Maybe they mean the third side of the triangular base? That would make sense for perimeter.

Wait — let’s assume the triangular base has sides: 7.2 m, 9.3 m, and 10.8 m (even though 7.2² + 9.3² ≠ 10.8²). Perhaps it’s not a right triangle at those legs? But problem says “right triangular prism with base 7.2 m, height 9.3 m” — that likely means the two legs of the right triangle are 7.2 and 9.3, so hypotenuse should be √(7.2² + 9.3²) ≈ 11.76 m. But they give “slant height 10.8 m” — contradiction?

Wait — re-read: “A right triangular prism with base 7.2 m, height 9.3 m, length 10.2 m and slant height 10.8 m.”

Perhaps “base” and “height” refer to the legs of the right triangle, and “slant height” is the hypotenuse — but then 10.8 doesn't match. Alternatively, maybe “slant height” is misnamed, and it’s actually the hypotenuse we should use for perimeter? Or perhaps it’s a typo? But since it’s given, we’ll use 10.8 as the third side of the triangle for surface area.

In many textbooks, for a triangular prism, surface area = 2 × (area of triangle) + (perimeter of triangle) × (length of prism)

So even if it’s not mathematically consistent, we’ll go with the given values.

So triangular base sides: 7.2 m, 9.3 m, 10.8 m
Length of prism (depth): 10.2 m

Area of triangular base (right triangle):
Since it’s a right triangle with legs 7.2 and 9.3, area = (1/2) × base × height = (1/2) × 7.2 × 9.3
= 3.6 × 9.3 = let’s compute:
3.6 × 9 = 32.4
3.6 × 0.3 = 1.08
Total = 33.48 m²

But wait — if it’s a right triangle with legs 7.2 and 9.3, then hypotenuse is √(7.2² + 9.3²) = √(51.84 + 86.49) = √138.33 ≈ 11.76 m — but they say slant height is 10.8 m. This is inconsistent.

Alternative interpretation: Maybe “base 7.2 m” and “height 9.3 m” are not the legs? But it says “right triangular prism”, so the base is a right triangle. Perhaps “slant height” is irrelevant? No — probably it’s meant to be the hypotenuse.

Wait — maybe “slant height” is a red herring? Or perhaps it’s the height of the prism? But no, it says “length 10.2 m” — that’s likely the prism’s length.

Another idea: In some contexts, “slant height” for a prism might mean the edge connecting corresponding vertices — but that’s just the length.

I think there’s confusion. Let me check standard formula for right triangular prism surface area:

SA = 2 × (area of triangle) + (sum of all three sides of triangle) × (length of prism)

We have area of triangle = (1/2)*7.2*9.3 = 33.48 m² (assuming right angle between them)

Perimeter of triangle = 7.2 + 9.3 + ?

If it’s a right triangle, third side = √(7.2² + 9.3²) = √(51.84 + 86.49) = √138.33 ≈ 11.76 m

But problem says “slant height 10.8 m” — perhaps they want us to use 10.8 as the third side? Even if it’s not accurate? Since it’s given, maybe we should use it.

Perhaps “slant height” is a mistake, and it’s supposed to be the hypotenuse. But 10.8 is given, so let’s proceed with the given numbers as is.

Assume the triangular base has sides: 7.2 m, 9.3 m, and 10.8 m (even if not right-angled, but problem says “right triangular prism”, so it must be right-angled). Contradiction.

Wait — perhaps “base 7.2 m” and “height 9.3 m” are the legs, and “slant height 10.8 m” is incorrect or for another purpose? But it’s listed.

Another thought: In some problems, “slant height” for a prism might refer to the lateral edge, but that’s the same as length.

I think the best approach is to ignore the inconsistency and use the given values as intended for calculation.

So, for surface area of prism:

SA = 2 × (area of triangular base) + (perimeter of triangular base) × (length of prism)

Area of triangle = (1/2) * 7.2 * 9.3 = 33.48 m² (since it’s right triangle with those legs)

Perimeter = 7.2 + 9.3 + 10.8 = 27.3 m (using given slant height as third side)

Length of prism = 10.2 m

So SA = 2*33.48 + 27.3 * 10.2
= 66.96 + let's compute 27.3 * 10.2

27.3 * 10 = 273
27.3 * 0.2 = 5.46
Total = 278.46

So SA = 66.96 + 278.46 = 345.42 m²

Volume = area of base * length = 33.48 * 10.2

Compute: 33.48 * 10 = 334.8
33.48 * 0.2 = 6.696
Total = 341.496 ≈ 341.50 m³

But is this correct? The triangle with sides 7.2, 9.3, 10.8 is not right-angled because 7.2² + 9.3² = 51.84 + 86.49 = 138.33, and 10.8² = 116.64 — not equal. So it can't be a right triangle with those legs and that hypotenuse.

Perhaps "slant height" is not the hypotenuse? Maybe it's the height of the prism? But it says "length 10.2 m".

Another possibility: "slant height" might be a distractor, and we should calculate the hypotenuse ourselves.

Let me try that.

If it's a right triangular prism with legs 7.2 m and 9.3 m, then hypotenuse c = √(7.2² + 9.3²) = √(51.84 + 86.49) = √138.33 ≈ 11.761 m

Then perimeter = 7.2 + 9.3 + 11.761 = 28.261 m

Area of triangle = (1/2)*7.2*9.3 = 33.48 m²

Length = 10.2 m

SA = 2*33.48 + 28.261*10.2 = 66.96 + 288.2622 = 355.2222 ≈ 355.22 m²

Volume = 33.48 * 10.2 = 341.496 ≈ 341.50 m³

But the problem gives "slant height 10.8 m", which is close to nothing. Perhaps it's a typo, and it's supposed to be 11.8 or something.

Maybe "slant height" is the height of the triangular face when viewed differently, but that doesn't make sense.

Another idea: in some contexts, for a right triangular prism, "slant height" might refer to the edge from the right angle vertex to the opposite vertex on the other base, but that would be sqrt(length^2 + hypotenuse^2), which is complicated.

I think the safest bet is to use the given values as provided, even if inconsistent, because the problem includes "slant height 10.8 m" specifically.

Perhaps "base 7.2 m" and "height 9.3 m" are not the legs of the right triangle, but one leg and the height of the prism? But it says "right triangular prism with base 7.2 m, height 9.3 m" — typically, for a triangular prism, "base" and "height" refer to the triangle's dimensions.

Let's look back at the worksheet image description — but I can't see it, so I have to rely on text.

The user said: "d. A right triangular prism with base 7.2 m, height 9.3 m, length 10.2 m and slant height 10.8 m."

In many educational materials, for a right triangular prism, they provide the two legs of the triangle and the length of the prism, and sometimes the hypotenuse is calculated or given. Here, they give "slant height" which is likely the hypotenuse.

But 7.2, 9.3, 10.8 do not form a right triangle. Let's check if 7.2^2 + 10.8^2 = 9.3^2? 51.84 + 116.64 = 168.48, 9.3^2=86.49 — no.

Or 9.3^2 + 10.8^2 = 86.49 + 116.64 = 203.13, 7.2^2=51.84 — no.

So none work. Therefore, perhaps "slant height" is not related to the triangle, but to the prism's lateral face? Unlikely.

Another possibility: "slant height" might be a mistake, and it's supposed to be the hypotenuse, but they gave 10.8 instead of 11.76. Or perhaps it's 11.8 rounded.

Maybe "base 7.2 m" is the hypotenuse? But it says "right triangular prism with base 7.2 m, height 9.3 m" — usually base and height are the legs.

I recall that in some problems, "slant height" for a prism is not used; perhaps it's extra information or for a different part.

Let's read the question again: "Determine the surface area and volume for each of the following."

For d, it's a right triangular prism, so we need the area of the triangular base and the perimeter for surface area.

Perhaps "slant height" is the length of the prism? But it says "length 10.2 m".

I think there's a error in the problem, but since it's given, and to match the expected answer, perhaps they intend for us to use the given "slant height" as the third side of the triangle.

So I'll go with that.

So for d:

Triangular base sides: 7.2 m, 9.3 m, 10.8 m

Area of triangle: since it's specified as "right triangular prism", but with these sides, it's not right-angled, so perhaps we should calculate area using Heron's formula? But that's advanced for grade 8.

The problem likely assumes that 7.2 and 9.3 are the legs, so area = (1/2)*7.2*9.3 = 33.48 m², and ignore the slant height for area, but use it for perimeter.

That seems reasonable.

So:

Area of base = 33.48 m²

Perimeter = 7.2 + 9.3 + 10.8 = 27.3 m

Length = 10.2 m

SA = 2*33.48 + 27.3*10.2 = 66.96 + 278.46 = 345.42 m²

Volume = 33.48 * 10.2 = 341.496 ≈ 341.50 m³

To confirm, let's calculate 27.3 * 10.2:

27.3 * 10 = 273

27.3 * 0.2 = 5.46

Sum 278.46 — yes.

33.48 * 10.2:

33.48 * 10 = 334.8

33.48 * 0.2 = 6.696

Sum 341.496 — rounds to 341.50

So I'll go with that.

Now, summarize all answers.

Final Answer:

a. Surface Area: 366 m², Volume: 440 m³
b. Surface Area: 703.72 in², Volume: 1385.44 in³
c. Surface Area: 1809.56 cm², Volume: 7238.23 cm³
d. Surface Area: 345.42 m², Volume: 341.50 m³

Note: For d, there is an inconsistency in the given dimensions for a right triangle, but we used the provided values as instructed.