Volumes of Solids - Kuta Software - Free Printable

Let’s solve each problem one by one. We’ll use the correct volume formulas for each shape and round to the nearest tenth as instructed.

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Problem 1: Triangular Prism

This is a triangular prism. The base is a triangle with base = 4 yd, height = 2 yd (the perpendicular height of the triangle). The length (or depth) of the prism is 5 yd.

Volume of a prism = Area of base × height (length)

Area of triangular base = (1/2) × base × height = (1/2) × 4 × 2 = 4 yd²

Volume = 4 × 5 = 20.0 yd³

Wait — let me double-check the diagram. There’s also a “1.5 yd” labeled inside the triangle. That might be the height? But it says “2 yd” on the side. Actually, looking again: the triangle has sides 4 yd (base), and the height from that base is marked as 2 yd? Or is 1.5 yd the height?

Actually, in the diagram, the red dashed line labeled “1.5 yd” is drawn from the top vertex perpendicular to the 4 yd base — so that’s the actual height of the triangle! The “2 yd” is probably the slant side or something else. Let me re-read.

Looking carefully: The triangle face has base 4 yd, and the perpendicular height (red dashed line) is 1.5 yd. The “2 yd” is likely the length of another edge, not the height for area calculation.

So:

Area of triangle = (1/2) × 4 × 1.5 = 3 yd²

Then volume = area × length = 3 × 5 = 15.0 yd³

But wait — the 5 yd is the length of the prism? Yes, it’s labeled along the top edge.

Also, there’s a “4 yd” at the bottom right — that might be the height of the rectangular face? No, this is confusing.

Alternative approach: Maybe it’s a trapezoidal prism? No, the front face is a triangle.

Actually, standard interpretation: For a triangular prism, if the triangular base has base 4 yd and height 1.5 yd (perpendicular), then area = 3 yd². The length of the prism (distance between triangles) is 5 yd. So volume = 3 × 5 = 15.0 yd³.

I think that’s correct.

✔ Final for #1: 15.0 yd³

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Problem 2: Square Pyramid

Base is a square with side 5 m. Height of pyramid is 8 m (from apex to center of base).

Volume of pyramid = (1/3) × base area × height

Base area = 5 × 5 = 25 m²

Volume = (1/3) × 25 × 8 = (200)/3 ≈ 66.666... → 66.7 m³

✔ Final for #2: 66.7 m³

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Problem 3: Square Pyramid

Base is square with side 3 yd. Height is 5 yd (given as vertical height).

Volume = (1/3) × base area × height = (1/3) × (3×3) × 5 = (1/3) × 9 × 5 = 15 yd³

✔ Final for #3: 15.0 yd³

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Problem 4: Cone

Radius = 2 km, height = 3 km

Volume of cone = (1/3)πr²h = (1/3) × π × (2)² × 3 = (1/3) × π × 4 × 3 = 4π ≈ 4 × 3.1416 = 12.5664 → 12.6 km³

✔ Final for #4: 12.6 km³

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Problem 5: Cylinder

Diameter = 4 in → radius = 2 in. Length (height) = 3 in.

Volume = πr²h = π × (2)² × 3 = π × 4 × 3 = 12π ≈ 37.699 → 37.7 in³

✔ Final for #5: 37.7 in³

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Problem 6: Cube

All sides = 2 m

Volume = side³ = 2³ = 8 m³

✔ Final for #6: 8.0 m³

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Problem 7: Triangular Prism

Front face is a triangle: base = 6 yd, height = 2.5 yd (red dashed line). The length of the prism (depth) is 5 yd? Wait, labeled “5 yd” on the top edge, but also “3 yd” on the side.

Actually, looking: The triangular base has base 6 yd and height 2.5 yd. The prism extends back 5 yd? But there’s also a “3 yd” label on the side — perhaps that’s the length?

Wait, the diagram shows:

- Triangle base: 6 yd
- Triangle height: 2.5 yd (perpendicular)
- Then the prism has a length (along the direction perpendicular to triangle) of 5 yd? But there’s also a “3 yd” on the side — maybe that’s the width?

Actually, in such diagrams, the “length” of the prism is usually the dimension going into the page. Here, the 5 yd is labeled on the top edge of the rectangle, which suggests it’s the length of the prism.

But let’s check: The solid has two triangular faces and three rectangular faces. The triangular face has base 6 yd and height 2.5 yd. The distance between the two triangular faces (the length of the prism) is 5 yd? But there’s also a “3 yd” labeled on the side — perhaps that’s irrelevant or mislabeled?

Wait, no — actually, looking again: The “3 yd” is labeled on the vertical edge of the rectangular face. That might mean the height of the rectangular face is 3 yd, but that doesn’t make sense because the triangle’s height is 2.5 yd.

Perhaps the 3 yd is the length of the prism? Let me think differently.

Standard formula: Volume = area of triangular base × length of prism.

Triangular base: base = 6 yd, height = 2.5 yd → area = (1/2)*6*2.5 = 7.5 yd²

Now, what is the length? The diagram shows “5 yd” on the top edge of the prism, which is parallel to the base of the triangle — that should be the length. Also, “3 yd” is on the side, which might be the height of the rectangular face, but since the triangle’s height is 2.5 yd, that doesn’t match.

Wait — perhaps the 3 yd is the length? Let me see the orientation.

Actually, in many textbooks, for a triangular prism like this, the “length” is the dimension perpendicular to the triangular face. In the diagram, the 5 yd is along the direction of the prism’s length, and the 3 yd might be a red herring or part of another measurement.

But let’s look at the labels:

- Bottom base of triangle: 6 yd
- Height of triangle (dashed): 2.5 yd
- Top edge of prism: 5 yd — this is likely the length
- Side edge: 3 yd — this might be the slant height or something else, but for volume, we don’t need it.

I think the length is 5 yd.

So volume = 7.5 × 5 = 37.5 yd³

But let me confirm with another way: If the prism has triangular ends with area 7.5 yd², and length 5 yd, yes.

Alternatively, if the 3 yd is the length, then volume = 7.5 × 3 = 22.5, but that seems less likely because the 5 yd is labeled on the long edge.

Looking at the diagram again mentally: The solid is lying on its rectangular face. The triangular face is at the end. The dimension along the length of the prism is 5 yd (top edge), and the 3 yd is the height of the rectangular side, which should equal the height of the triangle? But 3 ≠ 2.5, so inconsistency.

Wait — perhaps the 2.5 yd is not the height? No, it’s drawn as perpendicular to the 6 yd base.

Another possibility: The 3 yd is the length of the prism. Let me check online or standard problems.

Actually, I recall that in some diagrams, the number on the side indicates the length. But here, the 5 yd is clearly labeled on the edge that runs the length of the prism.

To resolve: Let’s calculate both ways, but I think 5 yd is intended as the length.

Moreover, in problem 1, similar labeling was used.

I’ll go with length = 5 yd.

Volume = (1/2)*6*2.5 * 5 = 7.5 * 5 = 37.5 yd³

✔ Final for #7: 37.5 yd³

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Problem 8: Square Pyramid

Base is square with side 2 in? Wait, labeled “2 in” and “1 in” — oh, it’s a rectangle? No, it says “square pyramid”, but base is 2 in by 1 in? That would be a rectangular pyramid.

Looking: Base has sides 2 in and 1 in, so it’s a rectangular base. Height is 1 in (vertical from apex to base).

Volume of pyramid = (1/3) × base area × height

Base area = 2 × 1 = 2 in²

Height = 1 in

Volume = (1/3) × 2 × 1 = 2/3 ≈ 0.666... → 0.7 in³

✔ Final for #8: 0.7 in³

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Now, compiling all answers:

1) 15.0
2) 66.7
3) 15.0
4) 12.6
5) 37.7
6) 8.0
7) 37.5
8) 0.7

All rounded to nearest tenth.

Final Answer:
1) 15.0 yd³
2) 66.7 m³
3) 15.0 yd³
4) 12.6 km³
5) 37.7 in³
6) 8.0 m³
7) 37.5 yd³
8) 0.7 in³