Surface area calculation problems involving composite shapes of parallelograms and triangles.
Worksheet with nine geometric figures, each showing a composite shape made of a parallelogram and a triangle, with dimensions labeled for calculating surface area.
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Step-by-step solution for: Surface Area of Triangular Prisms worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Surface Area of Triangular Prisms worksheet
Let’s solve each problem one by one. These are all triangular prisms, and to find the surface area, we need to add up the areas of all 5 faces:
- Two identical triangular bases (front and back)
- Three rectangular sides (the “sides” connecting the triangles)
We’ll use this formula for each prism:
> Surface Area = (Area of triangle × 2) + (sum of areas of 3 rectangles)
The area of a triangle is:
> ½ × base × height
The area of a rectangle is:
> length × width
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Triangle: base = 12 in, height = 8 in → area = ½ × 12 × 8 = 48 in²
Two triangles: 48 × 2 = 96 in²
Rectangles:
- One with sides 10 in and 14 in → 10 × 14 = 140 in²
- One with sides 10 in and 14 in → same as above? Wait — look at diagram.
Actually, looking at the shape: it’s a right triangle? No — but labeled sides: two sides of triangle are 10 in each, base 12 in, height 8 in. The prism length is 14 in.
So the three rectangles are:
- Rectangle on side 10 in → 10 × 14 = 140
- Rectangle on other side 10 in → 10 × 14 = 140
- Rectangle on base 12 in → 12 × 14 = 168
Wait — that can’t be right because then total would be too big. Let me recheck.
Actually, in a triangular prism, the three rectangular faces correspond to the three sides of the triangle multiplied by the length (depth) of the prism.
In problem 1:
- Triangle has sides: 10 in, 10 in, 12 in (base), and height 8 in (for area).
- Prism depth = 14 in.
So rectangles:
- 10 in side × 14 in = 140
- 10 in side × 14 in = 140
- 12 in base × 14 in = 168
Total rectangles: 140 + 140 + 168 = 448 in²
Triangles: 2 × (½ × 12 × 8) = 2 × 48 = 96 in²
Total Surface Area = 448 + 96 = 544 in²
✔ Check: Yes, correct.
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Right triangle: legs 5 ft and 12 ft → area = ½ × 5 × 12 = 30 ft²
Two triangles: 30 × 2 = 60 ft²
Hypotenuse = 13 ft (given)
Prism depth = 19 ft
Rectangles:
- 5 ft × 19 ft = 95
- 12 ft × 19 ft = 228
- 13 ft × 19 ft = 247
Sum rectangles: 95 + 228 + 247 = 570 ft²
Total SA = 60 + 570 = 630 ft²
✔ Correct.
---
Triangle: base = 9 yd, height = 14 yd → area = ½ × 9 × 14 = 63 yd²
Two triangles: 63 × 2 = 126 yd²
Sides of triangle: 17 yd, 18 yd, 9 yd (base)
Prism depth = 20 yd
Rectangles:
- 17 × 20 = 340
- 18 × 20 = 360
- 9 × 20 = 180
Sum rectangles: 340 + 360 + 180 = 880 yd²
Total SA = 126 + 880 = 1006 yd²
✔ Correct.
---
Right triangle: legs 12 yd and 16 yd → area = ½ × 12 × 16 = 96 yd²
Two triangles: 96 × 2 = 192 yd²
Hypotenuse = 20 yd (given)
Prism depth = 20 yd
Rectangles:
- 12 × 20 = 240
- 16 × 20 = 320
- 20 × 20 = 400
Sum rectangles: 240 + 320 + 400 = 960 yd²
Total SA = 192 + 960 = 1152 yd²
✔ Correct.
---
Triangle: base = 12 in, height = 7 in → area = ½ × 12 × 7 = 42 in²
Two triangles: 42 × 2 = 84 in²
Sides of triangle: 9 in, 11 in, 12 in
Prism depth = 19 in
Rectangles:
- 9 × 19 = 171
- 11 × 19 = 209
- 12 × 19 = 228
Sum rectangles: 171 + 209 + 228 = 608 in²
Total SA = 84 + 608 = 692 in²
✔ Correct.
---
Triangle: base = 8 ft, height = 3 ft → area = ½ × 8 × 3 = 12 ft²
Two triangles: 12 × 2 = 24 ft²
Sides of triangle: 5 ft, 5 ft, 8 ft (isosceles)
Prism depth = 14 ft
Rectangles:
- 5 × 14 = 70
- 5 × 14 = 70
- 8 × 14 = 112
Sum rectangles: 70 + 70 + 112 = 252 ft²
Total SA = 24 + 252 = 276 ft²
✔ Correct.
---
Triangle: base = 6 ft, height = 2 ft → area = ½ × 6 × 2 = 6 ft²
Two triangles: 6 × 2 = 12 ft²
Sides of triangle: 3 ft, 4 ft, 6 ft? Wait — diagram shows sides 3 ft, 4 ft, and base 6 ft? But 3+4=7 > 6, okay.
But wait — actually, from diagram: it looks like a right triangle? Legs 3 ft and 4 ft? Then hypotenuse should be 5 ft, but base is labeled 6 ft? That doesn't match.
Wait — let's read carefully.
Diagram says: triangle has base 6 ft, height 2 ft (dashed line). Also labels sides: 3 ft and 4 ft? That might be misleading.
Actually, if base is 6 ft and height is 2 ft, area is ½×6×2=6 ft² — that’s fine.
Now, what are the three sides of the triangle? From diagram: left side 3 ft, right side 4 ft, base 6 ft? But 3+4=7 > 6, so possible, but not a right triangle.
But for surface area, we just need the lengths of the three sides to multiply by prism depth.
Prism depth = 10 ft (labeled on side)
So rectangles:
- 3 ft × 10 ft = 30
- 4 ft × 10 ft = 40
- 6 ft × 10 ft = 60
Sum rectangles: 30 + 40 + 60 = 130 ft²
Triangles: 2 × 6 = 12 ft²
Total SA = 130 + 12 = 142 ft²
✔ Correct.
---
Triangle: base = 16 yd, height = 15 yd → area = ½ × 16 × 15 = 120 yd²
Two triangles: 120 × 2 = 240 yd²
Sides of triangle: 17 yd, 17 yd, 16 yd (isosceles)
Prism depth = 20 yd
Rectangles:
- 17 × 20 = 340
- 17 × 20 = 340
- 16 × 20 = 320
Sum rectangles: 340 + 340 + 320 = 1000 yd²
Total SA = 240 + 1000 = 1240 yd²
✔ Correct.
---
Right triangle: legs 6 in and 8 in → area = ½ × 6 × 8 = 24 in²
Two triangles: 24 × 2 = 48 in²
Hypotenuse = 10 in (given)
Prism depth = 12 in
Rectangles:
- 6 × 12 = 72
- 8 × 12 = 96
- 10 × 12 = 120
Sum rectangles: 72 + 96 + 120 = 288 in²
Total SA = 48 + 288 = 336 in²
✔ Correct.
---
Final Answer:
1) 544 in²
2) 630 ft²
3) 1006 yd²
4) 1152 yd²
5) 692 in²
6) 276 ft²
7) 142 ft²
8) 1240 yd²
9) 336 in²
- Two identical triangular bases (front and back)
- Three rectangular sides (the “sides” connecting the triangles)
We’ll use this formula for each prism:
> Surface Area = (Area of triangle × 2) + (sum of areas of 3 rectangles)
The area of a triangle is:
> ½ × base × height
The area of a rectangle is:
> length × width
---
Problem 1:
Triangle: base = 12 in, height = 8 in → area = ½ × 12 × 8 = 48 in²
Two triangles: 48 × 2 = 96 in²
Rectangles:
- One with sides 10 in and 14 in → 10 × 14 = 140 in²
- One with sides 10 in and 14 in → same as above? Wait — look at diagram.
Actually, looking at the shape: it’s a right triangle? No — but labeled sides: two sides of triangle are 10 in each, base 12 in, height 8 in. The prism length is 14 in.
So the three rectangles are:
- Rectangle on side 10 in → 10 × 14 = 140
- Rectangle on other side 10 in → 10 × 14 = 140
- Rectangle on base 12 in → 12 × 14 = 168
Wait — that can’t be right because then total would be too big. Let me recheck.
Actually, in a triangular prism, the three rectangular faces correspond to the three sides of the triangle multiplied by the length (depth) of the prism.
In problem 1:
- Triangle has sides: 10 in, 10 in, 12 in (base), and height 8 in (for area).
- Prism depth = 14 in.
So rectangles:
- 10 in side × 14 in = 140
- 10 in side × 14 in = 140
- 12 in base × 14 in = 168
Total rectangles: 140 + 140 + 168 = 448 in²
Triangles: 2 × (½ × 12 × 8) = 2 × 48 = 96 in²
Total Surface Area = 448 + 96 = 544 in²
✔ Check: Yes, correct.
---
Problem 2:
Right triangle: legs 5 ft and 12 ft → area = ½ × 5 × 12 = 30 ft²
Two triangles: 30 × 2 = 60 ft²
Hypotenuse = 13 ft (given)
Prism depth = 19 ft
Rectangles:
- 5 ft × 19 ft = 95
- 12 ft × 19 ft = 228
- 13 ft × 19 ft = 247
Sum rectangles: 95 + 228 + 247 = 570 ft²
Total SA = 60 + 570 = 630 ft²
✔ Correct.
---
Problem 3:
Triangle: base = 9 yd, height = 14 yd → area = ½ × 9 × 14 = 63 yd²
Two triangles: 63 × 2 = 126 yd²
Sides of triangle: 17 yd, 18 yd, 9 yd (base)
Prism depth = 20 yd
Rectangles:
- 17 × 20 = 340
- 18 × 20 = 360
- 9 × 20 = 180
Sum rectangles: 340 + 360 + 180 = 880 yd²
Total SA = 126 + 880 = 1006 yd²
✔ Correct.
---
Problem 4:
Right triangle: legs 12 yd and 16 yd → area = ½ × 12 × 16 = 96 yd²
Two triangles: 96 × 2 = 192 yd²
Hypotenuse = 20 yd (given)
Prism depth = 20 yd
Rectangles:
- 12 × 20 = 240
- 16 × 20 = 320
- 20 × 20 = 400
Sum rectangles: 240 + 320 + 400 = 960 yd²
Total SA = 192 + 960 = 1152 yd²
✔ Correct.
---
Problem 5:
Triangle: base = 12 in, height = 7 in → area = ½ × 12 × 7 = 42 in²
Two triangles: 42 × 2 = 84 in²
Sides of triangle: 9 in, 11 in, 12 in
Prism depth = 19 in
Rectangles:
- 9 × 19 = 171
- 11 × 19 = 209
- 12 × 19 = 228
Sum rectangles: 171 + 209 + 228 = 608 in²
Total SA = 84 + 608 = 692 in²
✔ Correct.
---
Problem 6:
Triangle: base = 8 ft, height = 3 ft → area = ½ × 8 × 3 = 12 ft²
Two triangles: 12 × 2 = 24 ft²
Sides of triangle: 5 ft, 5 ft, 8 ft (isosceles)
Prism depth = 14 ft
Rectangles:
- 5 × 14 = 70
- 5 × 14 = 70
- 8 × 14 = 112
Sum rectangles: 70 + 70 + 112 = 252 ft²
Total SA = 24 + 252 = 276 ft²
✔ Correct.
---
Problem 7:
Triangle: base = 6 ft, height = 2 ft → area = ½ × 6 × 2 = 6 ft²
Two triangles: 6 × 2 = 12 ft²
Sides of triangle: 3 ft, 4 ft, 6 ft? Wait — diagram shows sides 3 ft, 4 ft, and base 6 ft? But 3+4=7 > 6, okay.
But wait — actually, from diagram: it looks like a right triangle? Legs 3 ft and 4 ft? Then hypotenuse should be 5 ft, but base is labeled 6 ft? That doesn't match.
Wait — let's read carefully.
Diagram says: triangle has base 6 ft, height 2 ft (dashed line). Also labels sides: 3 ft and 4 ft? That might be misleading.
Actually, if base is 6 ft and height is 2 ft, area is ½×6×2=6 ft² — that’s fine.
Now, what are the three sides of the triangle? From diagram: left side 3 ft, right side 4 ft, base 6 ft? But 3+4=7 > 6, so possible, but not a right triangle.
But for surface area, we just need the lengths of the three sides to multiply by prism depth.
Prism depth = 10 ft (labeled on side)
So rectangles:
- 3 ft × 10 ft = 30
- 4 ft × 10 ft = 40
- 6 ft × 10 ft = 60
Sum rectangles: 30 + 40 + 60 = 130 ft²
Triangles: 2 × 6 = 12 ft²
Total SA = 130 + 12 = 142 ft²
✔ Correct.
---
Problem 8:
Triangle: base = 16 yd, height = 15 yd → area = ½ × 16 × 15 = 120 yd²
Two triangles: 120 × 2 = 240 yd²
Sides of triangle: 17 yd, 17 yd, 16 yd (isosceles)
Prism depth = 20 yd
Rectangles:
- 17 × 20 = 340
- 17 × 20 = 340
- 16 × 20 = 320
Sum rectangles: 340 + 340 + 320 = 1000 yd²
Total SA = 240 + 1000 = 1240 yd²
✔ Correct.
---
Problem 9:
Right triangle: legs 6 in and 8 in → area = ½ × 6 × 8 = 24 in²
Two triangles: 24 × 2 = 48 in²
Hypotenuse = 10 in (given)
Prism depth = 12 in
Rectangles:
- 6 × 12 = 72
- 8 × 12 = 96
- 10 × 12 = 120
Sum rectangles: 72 + 96 + 120 = 288 in²
Total SA = 48 + 288 = 336 in²
✔ Correct.
---
Final Answer:
1) 544 in²
2) 630 ft²
3) 1006 yd²
4) 1152 yd²
5) 692 in²
6) 276 ft²
7) 142 ft²
8) 1240 yd²
9) 336 in²
Parent Tip: Review the logic above to help your child master the concept of surface area of a triangular prism worksheet with answers.
