4-20-20 Grade 7 Volume of a Triangular Prism Worksheet online ... - Free Printable

Explanation:
We are asked to find the surface area of each triangular prism or pyramid shown. Let’s go one by one, using the correct formulas.

General ideas:

- A triangular prism has:
- 2 triangular bases (identical)
- 3 rectangular lateral faces
- Surface Area = 2 × (area of triangle base) + (perimeter of triangle base) × height of prism

- A triangular pyramid (tetrahedron with triangular base) has:
- 1 triangular base
- 3 triangular lateral faces
- But in this worksheet, figures like #3 and #4 look like triangular prisms laid on their side, where the “height” is the length of the prism, and the triangle is the cross-section.

Actually, looking carefully at all figures, all 9 are triangular prisms, just drawn differently — some standing upright, some lying down. The key is identifying:
- The triangular base (with base and height given)
- The length (or depth) of the prism — the distance between the two triangular faces.

Surface Area =
2 × (area of triangular base) +
(sum of the three side lengths of triangle) × (length of prism)

Let’s compute each:

---

1) Triangular base: base = 2 ft + 2 ft = 4 ft? Wait — no. Look: the base is a triangle with base = 2 ft + 2 ft = 4 ft? Actually, the bottom shows two segments of 2 ft each, so the full base of the triangle is 4 ft, and the height of the triangle is not given directly — but since it's a right triangle? No, the diagram shows a triangle with base split into two 2 ft parts, and vertical height is not labeled. Hmm — wait! In figure 1, it's a right triangular prism with triangular base that is a right triangle with legs 2 ft and 2 ft? Actually, the front face is a rectangle 4 ft tall, and the base is a triangle with two sides labeled 2 ft each — likely an isosceles triangle with base = 4 ft (2+2) and height = ? Not given.

Wait — re-examining: In many such worksheets, when a triangular prism is drawn like this (a rectangle with a triangle on top), the triangle’s base is the bottom edge (2 ft + 2 ft = 4 ft), and the height of the triangle is implied by the vertical edge? But no height is labeled.

Hold on — maybe these are right triangular prisms where the triangular base is a right triangle with legs given.

Look again at figure 1:
- It shows a triangular prism.
- The triangular face has two sides labeled “2 ft” on the base (left and right), and the vertical edge of the prism is labeled “4 ft” — that’s the length (height) of the prism.
- The triangle itself appears to be a right triangle with legs 2 ft and 2 ft? But then the hypoten’t is not needed unless we need perimeter.

Actually, standard interpretation for these diagrams (common in Math-Aids worksheets):
The triangular base is a right triangle with legs equal to the two numbers on the base (e.g., 2 ft and 2 ft), and the third side (hypotenuse) can be found via Pythagoras if needed.

But in figure 1: bottom edge is split into two 2 ft segments → total base = 4 ft. The triangle is likely isosceles, with two equal sides of unknown length, but height not given. That can’t be.

Alternative: Maybe the triangle is right, with legs 2 ft (horizontal) and ? — no.

Let me check figure 6: same shape as 1, but labeled: base 2 ft and 2 ft, height of prism = 4 ft. In many answer keys, for such a prism with triangular base being a right triangle with legs 2 ft and 2 ft, area = (1/2)(2)(2) = 2 ft². Then perimeter = 2 + 2 + √(2²+2²) = 2 + 2 + √8 ≈ 2 + 2 + 2.828 = 6.828. Then SA = 2×2 + 6.828×4 = 4 + 27.313 = 31.31 ft².

But let’s verify with figure 3, which gives more info:
- Triangle base: base = 8 ft, height = 6 ft (vertical dashed line), so area = (1/2)(8)(6) = 24 ft².
- The slant edge is labeled 7.21 ft — that’s likely the hypotenuse of the triangle: √(6² + 4²) = √(36+16)=√52≈7.21 — yes! So the triangle has base 8 ft, height 6 ft, and the two equal sides? Wait, if height is 6 ft to midpoint of base (8 ft), then half-base = 4 ft, so each slant side = √(4²+6²)=√52≈7.21 ft. So triangle sides: 8 ft, 7.21 ft, 7.21 ft.
- Prism length = 14 ft (the long edge).
- So surface area = 2×(area of triangle) + (perimeter of triangle)×length
= 2×24 + (8 + 7.21 + 7.21)×14
= 48 + (22.42)×14
= 48 + 313.88 = 361.88 ft² → round to nearest hundredth: 361.88

That seems correct.

Now apply same logic to others.

Let’s define for each:

Figure 1:

- Triangular base: looks identical to figure 6.
- Base = 2 + 2 = 4 ft. Height of triangle? Not labeled. But in figure 6, same shape, and no height labeled — yet in many such worksheets, when only the two halves of base are given (2 and 2) and no height, it implies the triangle is right with legs 2 ft and ? — wait, maybe the vertical edge of the prism (4 ft) is *not* the prism length, but the triangle height?

Actually, look at orientation:
- In fig 1: the prism stands on the triangular base. The vertical dimension labeled “4 ft” is the height of the prism (distance between triangles). The triangle itself lies horizontally, with base 2+2=4 ft, and its height is not labeled — but perhaps it's a right triangle with legs 2 ft and 2 ft, i.e., the two 2 ft segments are the legs, meeting at right angle. That would make base = 2 ft, height = 2 ft, and hypotenuse = √8 ≈ 2.828 ft.

Yes — that’s standard: when a triangle is drawn with two perpendicular sides labeled 2 ft and 2 ft, it’s a right triangle. The “2 ft” labels are on the two legs, not halves of base.

So reinterpret:
- Figure 1: triangular base is right triangle with legs 2 ft and 2 ft.
- Area = ½ × 2 × 2 = 2 ft²
- Sides: 2, 2, √8 ≈ 2.8284
- Perimeter = 2 + 2 + 2.8284 = 6.8284 ft
- Prism length (height) = 4 ft
- SA = 2×2 + 6.8284×4 = 4 + 27.3136 = 31.31 ft²

Figure 2:

- Triangle: legs 3 ft and 3 ft (right triangle)
- Area = ½ × 3 × 3 = 4.5 ft²
- Hypotenuse = √(3²+3²) = √18 ≈ 4.2426
- Perimeter = 3 + 3 + 4.2426 = 10.2426
- Prism length = 6 ft
- SA = 2×4.5 + 10.2426×6 = 9 + 61.4556 = 70.46 ft²

Figure 3:

We did:
- Triangle: base 8 ft, height 6 ft → area = 24
- Half-base = 4 ft, so equal sides = √(4²+6²)=√52≈7.2111
- Perimeter = 8 + 7.2111 + 7.2111 = 22.4222
- Length = 14 ft
- SA = 2×24 + 22.4222×14 = 48 + 313.9108 = 361.91 ft² (round to nearest hundredth)

Wait earlier I got 313.88, but √52 = 7.21110255, ×2 = 14.4222051, +8 = 22.4222051, ×14 = 313.9108714, +48 = 361.9108714 → 361.91

Figure 4:

Same triangle as fig 3: base 8 ft, height 6 ft, sides 7.21 ft each, area 24
Prism length = 14 ft (same as fig 3)
→ Same SA = 361.91 ft²

Wait, but figure 4 drawing shows the 14 ft as the slant edge? No — label says “14 ft” along the length (the prism direction), same as fig 3. Yes, identical dimensions → same answer.

Figure 5:

Triangle: base = 4 ft, height = 3 ft (vertical dashed line)
→ area = ½ × 4 × 3 = 6 ft²
Half-base = 2 ft, so slant side = √(2² + 3²) = √13 ≈ 3.6056 ft (matches label 3.61 ft)
Other side = same = 3.6056 ft
Perimeter = 4 + 3.6056 + 3.6056 = 11.2112 ft
Prism length = 11 ft
SA = 2×6 + 11.2112×11 = 12 + 123.3232 = 135.32 ft²

Figure 6:

Same as fig 1: right triangle legs 2 ft and 2 ft, prism height = 4 ft
→ SA = 31.31 ft²

Figure 7:

Triangle: base = 6 ft, height = 4 ft (vertical dashed), area = ½×6×4 = 12 ft²
Slant side given as 5 ft — check: half-base = 3 ft, height 4 ft → hypotenuse = √(3²+4²)=5 ft ✔
Other side = also 5 ft (isosceles)
Perimeter = 6 + 5 + 5 = 16 ft
Prism length = 12 ft
SA = 2×12 + 16×12 = 24 + 192 = 216.00 ft²

Figure 8:

Right triangle base: legs 2 ft and 2 ft (like fig 1), prism length = 4
→ area = 2, perimeter = 2+2+√8 = 6.8284
SA = 4 + 6.8284×4 = 4 + 27.3136 = 31.31 ft²

Figure 9:

Same as 8 and 6 and 1: legs 2 and 2, length 4 → 31.31 ft²

Wait — figure 9 is identical to 8 and 6.

Now list all:

1) 31.31
2) 70.46
3) 361.91
4) 361.91
5) 135.32
6) 31.31
7) 216.00
8) 31.31
9) 31.31

Let me double-check figure 2:
- legs 3,3 → area = 4.5
- hyp = √18 = 4.242640687
- perimeter = 3+3+4.24264 = 10.24264
- ×6 = 61.45585
- +9 = 70.45585 → round to hundredth = 70.46 ✔

Figure 5:
- base 4, height 3 → area 6
- sides: 3.605551275 each
- perimeter = 4 + 2×3.605551275 = 11.21110255
- ×11 = 123.322128
- +12 = 135.322128 → 135.32 ✔

Figure 7:
- base 6, height 4 → area 12
- sides: 5 and 5 (since 3-4-5 triangle)
- perimeter = 16
- ×12 = 192
- +24 = 216.00 ✔

All good.

Final Answer:
1) 31.31
2) 70.46
3) 361.91
4) 361.91
5) 135.32
6) 31.31
7) 216.00
8) 31.31
9) 31.31