13-2 Surface Area of Prisms and Pyramids Worksheet | PDF | Area ... - Free Printable

Let’s solve each problem step by step.

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Problem 1: Triangular Prism

We are given a triangular prism with:
- Triangle base = 8 ft
- Triangle height = 6 ft (this is the height of the triangle, not the prism)
- The two other sides of the triangle are both 10 ft? Wait — looking at the diagram description: it shows a right triangle? Actually, from the numbers: 6 ft and 8 ft are legs, and hypotenuse would be √(6² + 8²) = √(36+64)=√100=10 ft. So yes, it’s a right triangle with legs 6 ft and 8 ft, hypotenuse 10 ft.
- Length of prism (distance between triangles) = 5 ft

Surface area of a triangular prism = 2 × (area of triangular base) + (perimeter of triangle) × (length of prism)

Step 1: Area of one triangle = (1/2) × base × height = (1/2) × 8 × 6 = 24 ft²
→ Two triangles: 2 × 24 = 48 ft²

Step 2: Perimeter of triangle = 6 + 8 + 10 = 24 ft
→ Lateral surface area = perimeter × length = 24 × 5 = 120 ft²

Step 3: Total SA = 48 + 120 = 168 ft²

✔ Check: Yes, that matches standard formula.

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Problem 2: Cube

All sides = 9 in.

A cube has 6 identical square faces.

Area of one face = 9 × 9 = 81 in²
Total SA = 6 × 81 = 486 in²

✔ Easy check: 6 × 80 = 480, plus 6 × 1 = 6 → 486. Correct.

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Problem 3: Rectangular Prism — ℓ = 5.5 cm, w = 4.5 cm, h = 3.5 cm

Formula: SA = 2ℓw + 2ℓh + 2wh

Compute each term:

2ℓw = 2 × 5.5 × 4.5
First: 5.5 × 4.5
→ 5 × 4.5 = 22.5; 0.5 × 4.5 = 2.25 → total 24.75
Then ×2 = 49.5

2ℓh = 2 × 5.5 × 3.5
5.5 × 3.5 = (5×3.5) + (0.5×3.5) = 17.5 + 1.75 = 19.25
×2 = 38.5

2wh = 2 × 4.5 × 3.5
4.5 × 3.5 = (4×3.5) + (0.5×3.5) = 14 + 1.75 = 15.75
×2 = 31.5

Now add: 49.5 + 38.5 = 88; 88 + 31.5 = 119.5 cm²

✔ Double-check:
Alternatively: SA = 2(ℓw + ℓh + wh) = 2(24.75 + 19.25 + 15.75) = 2(59.75) = 119.5 → correct.

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Problem 4: ℓ = 15 in., w = 9 in., h = 3.8 in.

SA = 2(ℓw + ℓh + wh)

Compute inside:

ℓw = 15 × 9 = 135
ℓh = 15 × 3.8 = let's compute: 15×3 = 45, 15×0.8=12 → 57
wh = 9 × 3.8 = 9×3 = 27, 9×0.8=7.2 → 34.2

Sum: 135 + 57 = 192; 192 + 34.2 = 226.2
Multiply by 2: 2 × 226.2 = 452.4 in²

✔ Check:
2ℓw = 2×135=270
2ℓh=2×57=114
2wh=2×34.2=68.4
Total: 270+114=384; 384+68.4=452.4 → correct.

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Problem 5: ℓ = 2 yd, w = 6 yd, h = 1.7 yd

SA = 2(ℓw + ℓh + wh)

ℓw = 2×6 = 12
ℓh = 2×1.7 = 3.4
wh = 6×1.7 = 10.2

Sum: 12 + 3.4 = 15.4; 15.4 + 10.2 = 25.6
×2 = 51.2 yd²

✔ Check:
2ℓw=24, 2ℓh=6.8, 2wh=20.4 → 24+6.8=30.8; 30.8+20.4=51.2 → correct.

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Problem 6: Reasoning — Write dimensions of two different rectangular prisms with same surface area.

We need to pick two sets of (ℓ, w, h) that give same SA.

Let’s try small whole numbers.

Example 1: ℓ=1, w=1, h=1 → SA = 2(1+1+1)=6

Can we find another? Try ℓ=2, w=1, h=?

Set SA=6: 2(2×1 + 2×h + 1×h) = 6
→ 2(2 + 2h + h) = 6
→ 2(2 + 3h) = 6
→ 2 + 3h = 3
→ 3h = 1 → h=1/3 → not nice.

Try SA=22.

Prism A: ℓ=3, w=2, h=1
SA = 2(3×2 + 3×1 + 2×1) = 2(6+3+2)=2×11=22

Prism B: ℓ=4, w=1, h=1.5? Let’s test:

SA = 2(4×1 + 4×1.5 + 1×1.5) = 2(4 + 6 + 1.5)=2×11.5=23 → too big.

Try ℓ=5, w=1, h=0.6? Messy.

Better: Use known pair.

Actually, here’s a classic example:

Prism 1: 1×1×5 → SA = 2(1×1 + 1×5 + 1×5) = 2(1+5+5)=2×11=22

Prism 2: 2×2×1.5 → SA = 2(2×2 + 2×1.5 + 2×1.5) = 2(4 + 3 + 3)=2×10=20 → no.

Wait — let me calculate properly.

Try:

Prism A: ℓ=4, w=3, h=1 → SA=2(12 + 4 + 3)=2×19=38

Prism B: ℓ=5, w=2, h=1.8? Too messy.

Simple integer solution:

How about:

Prism 1: 2×3×4 → SA=2(6+8+12)=2×26=52

Prism 2: 1×4×5 → SA=2(4+5+20)=2×29=58 → no.

Wait — let’s use algebra or trial.

I recall: 1×2×3 → SA=2(2+3+6)=2×11=22

Another: 1×1×5 → SA=2(1+5+5)=22 → YES!

So:

Prism 1: length=1, width=2, height=3 → SA=22
Prism 2: length=1, width=1, height=5 → SA=22

Both have surface area 22 square units.

You can write them as:

First prism: 1 unit × 2 units × 3 units
Second prism: 1 unit × 1 unit × 5 units

They are different shapes but same surface area.

✔ Verified:
For 1×2×3: 2(1·2 + 1·3 + 2·3) = 2(2+3+6)=22
For 1×1×5: 2(1·1 + 1·5 + 1·5)=2(1+5+5)=22 → perfect.

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Final Answer:

1. 168 ft²
2. 486 in²
3. 119.5 cm²
4. 452.4 in²
5. 51.2 yd²
6. Example: One prism with dimensions 1 × 2 × 3 and another with 1 × 1 × 5 (both have surface area 22 square units). Other answers possible as long as they are different and yield same SA.