Volume of Composite Shapes worksheet featuring six 3D figures composed of basic geometric solids, designed for calculating total volume.

Let’s solve each problem step by step. We’ll find the volume of each composite shape by breaking it into simpler shapes (like rectangular prisms, triangular prisms, cylinders, cones, spheres), calculating their volumes separately, and then adding them up.

We’ll use these formulas:

- Volume of a rectangular prism = length × width × height
- Volume of a triangular prism = (½ × base × height of triangle) × length of prism
- Volume of a cylinder = π × r² × h
- Volume of a cone = (1/3) × π × r² × h
- Volume of a sphere = (4/3) × π × r³

We’ll round all final answers to two decimal places.

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Problem 1:

Shape is made of:
- A rectangular prism at the bottom: 8 ft long, 5 ft wide, 2 ft high → V = 8×5×2 = 80 ft³
- A triangular prism on top: base = 5 ft, height of triangle = 6 ft, length = 8 ft → V = ½ × 5 × 6 × 8 = 120 ft³

Total volume = 80 + 120 = 200.00 ft³

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Problem 2:

Shape has:
- Two identical triangular prisms on sides
- One rectangular prism in middle

Each triangular prism:
Base = 7 m, height of triangle = 9 m, length = 10 m
V = ½ × 7 × 9 × 10 = 315 m³ per prism → two of them = 630 m³

Rectangular prism in middle:
Length = 10 m, width = 5 m, height = 9 m? Wait — looking at diagram, the middle part connects the two triangles. Actually, the middle part is a rectangular prism with dimensions: length = 10 m, width = 5 m, height = ? Let’s check.

Actually, from the diagram, the entire base is 5m + 7m + 7m? No — wait, the total length along the bottom is labeled as 5m for the middle rectangle, and each triangle has base 7m? But that doesn’t add up visually.

Wait — let me re-read the labels.

Looking again: The middle rectangle is 5m wide (front-to-back?), and the triangles are attached to its sides. Each triangle has base 7m and height 9m, and the depth (length) of the whole thing is 10m.

Actually, the composite shape is like a house with two slanted roofs on sides and a flat center.

But more accurately: It's one big rectangular prism in the center: 5m (width) × 10m (depth) × 9m (height)? No — because the triangles go up to 9m, but the center might be lower? Wait, no — actually, the diagram shows the center part is also 9m tall? Or is it?

Wait — perhaps the center is a rectangular prism of size: length=10m, width=5m, height=9m? And then two triangular prisms attached to the left and right, each with base=7m, height=9m, length=10m.

But if you attach them to the sides, the total width would be 7+5+7=19m, which matches the label “19” at the bottom? Yes! The bottom says “19”, so yes.

So:

Center rectangular prism: 5m (width) × 10m (length) × 9m (height) → V = 5×10×9 = 450 m³

Left triangular prism: base=7m, height=9m, length=10m → V = ½ × 7 × 9 × 10 = 315 m³

Right triangular prism: same → 315 m³

Total = 450 + 315 + 315 = 1080.00 m³

Wait — but is the center really 9m tall? In the diagram, the triangles go up to 9m, and the center seems to be at the same height? Actually, looking closely, the center part might only be as tall as the base of the triangles? No — the vertical line from the peak goes down to the base, and the center block appears to fill the space between the two triangles. So yes, the center block is 9m tall.

Alternatively, maybe the center block is shorter? But there’s no indication of that. The diagram shows the center block going full height. So we’ll go with 9m.

Thus, total volume = 450 + 315 + 315 = 1080.00 m³

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Problem 3:

Three parts:
- Left cube: 4 cm × 4 cm × 4 cm → V = 64 cm³
- Middle connector: looks like a small rectangular prism: 2 cm × 2 cm × 2 cm? Wait, labeled as 2cm on each side? Actually, it says “2” on the connecting piece — probably 2cm × 2cm × 2cm? But let’s see: the connection is between two cubes. The left cube is 4x4x4, right cube is 4x4x4, and the middle piece is 2cm long (along the axis), and cross-section 2cm x 2cm? The diagram shows “2” on the middle piece — likely meaning 2cm in all dimensions? Or just length?

Actually, looking: the middle piece is drawn as a smaller cube or rectangular prism connecting the two larger cubes. The label “2” is on the side of the middle piece — probably indicating that the middle piece is 2cm × 2cm × 2cm? But that would make it a cube. However, the distance between the two large cubes is filled by this piece. Since the large cubes are 4cm each, and they’re separated by a gap filled by the middle piece.

The diagram shows the middle piece has dimension “2” — likely the length along the direction of connection. And since it’s connected face-to-face, the cross-section should match the face of the cube? But the cube is 4x4, so if the middle piece is only 2x2, that doesn’t make sense.

Wait — perhaps the “2” refers to the length of the connector, and the cross-section is 2cm x 2cm? That would mean the connector is a rectangular prism of size 2cm (length) × 2cm (width) × 2cm (height)? But then how is it attached to 4x4 faces?

This is ambiguous. Let me think differently.

Perhaps the entire assembly is aligned such that the middle piece is centered or something. But another way: maybe the “2” is the length of the connector, and the width and height are the same as the cubes? But that can’t be, because then it wouldn’t be labeled separately.

Looking back at the diagram description: it says “4” on the cubes, and “2” on the middle piece. Probably, the middle piece is a rectangular prism with dimensions: length = 2 cm, width = 2 cm, height = 2 cm? But that seems too small.

Wait — perhaps the middle piece is 2 cm long (in the direction of connection), and its cross-section is 2 cm by 2 cm, while the cubes are 4 cm by 4 cm. So when attached, it’s like a neck.

In that case:

Left cube: 4×4×4 = 64 cm³

Middle connector: 2×2×2 = 8 cm³? But is the length 2 cm? The label “2” is on the side, so likely all dimensions are 2 cm for the connector.

Right cube: 4×4×4 = 64 cm³

Total = 64 + 8 + 64 = 136 cm³

But let’s confirm: if the connector is 2x2x2, and attached to the center of the 4x4 face, that makes sense geometrically.

So total volume = 64 + 8 + 64 = 136.00 cm³

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Problem 4:

Shape is a cylinder with a hemisphere on top and a hemisphere on bottom? Wait, no — looking: it’s a cylinder with a cone on top and a hemisphere on bottom? Let’s see.

Diagram shows:
- Top: cone with radius 3 ft, height 4 ft
- Middle: cylinder with radius 3 ft, height 10 ft
- Bottom: hemisphere with radius 3 ft

Yes.

So:

Cone: V = (1/3)πr²h = (1/3)π(3)²(4) = (1/3)π(9)(4) = 12π ≈ 37.699 ft³

Cylinder: V = πr²h = π(9)(10) = 90π ≈ 282.743 ft³

Hemisphere: half of sphere → (1/2)*(4/3)πr³ = (2/3)π(27) = 18π ≈ 56.549 ft³

Total = 12π + 90π + 18π = 120π ≈ 376.99 ft³

Calculate numerically:

12π = 37.6991

90π = 282.7433

18π = 56.5487

Sum = 37.6991 + 282.7433 = 320.4424; +56.5487 = 376.9911 → 376.99 ft³

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Problem 5:

Shape consists of:
- Rectangular prism: 8 ft × 6 ft × 5 ft? Wait, labels: base is 8 ft by 6 ft? Height of prism is 5 ft? Then on top, a pyramid? Or a cone? Diagram shows a pyramid with square base? Wait, it says “pyramid” but let’s see.

Actually, it’s a rectangular prism with a pyramid on top. The pyramid has a rectangular base same as the prism: 8 ft by 6 ft, and height of pyramid is 7 ft? Label says “7” from apex to base.

Also, there’s a triangular prism attached to the side? Wait, no — looking: it’s a combination of a rectangular prism and a pyramid on top, and additionally, a triangular prism attached to one side? The diagram shows a shape that looks like a house with a pointed roof and an extension.

Wait, let’s parse:

From the diagram description: it has a rectangular base 8 ft by 6 ft, height 5 ft for the prism. On top of that, a pyramid with height 7 ft. Additionally, attached to one side (say, the 8-ft side), there is a triangular prism that extends out. The triangular prism has a triangular face with base 6 ft and height 4 ft? And length 8 ft? Wait, labels: “4” on the triangle, “6” on the base, “8” on the length.

Actually, the附加 part is a triangular prism attached to the side of the main structure. The main structure is the rectangular prism plus pyramid. The附加 triangular prism shares the same length (8 ft) and has a triangular cross-section with base 6 ft and height 4 ft.

But is it attached to the side? And does it overlap? Probably not — it’s an addition.

So let’s break it down:

Part A: Rectangular prism: 8 ft (length) × 6 ft (width) × 5 ft (height) → V = 8×6×5 = 240 ft³

Part B: Pyramid on top: base 8×6, height 7 ft → V = (1/3) × base area × height = (1/3)×(8×6)×7 = (1/3)×48×7 = 16×7 = 112 ft³

Part C: Triangular prism attached to the side: this is a bit tricky. The triangular prism has a triangular base with base=6 ft, height=4 ft, and the length of the prism is 8 ft (same as the main length). So V = (½ × 6 × 4) × 8 = (12) × 8 = 96 ft³

Now, is this triangular prism attached externally? Yes, so we add it.

Total volume = 240 + 112 + 96 = 448 ft³

But wait — is the triangular prism overlapping with the main structure? For example, if it’s attached to the side, and the main structure already has width 6 ft, attaching another prism with base 6 ft might mean it’s extending outward, so no overlap. So yes, add them.

Thus, total = 240 + 112 + 96 = 448.00 ft³

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Problem 6:

Three stacked shapes:
- Top: cylinder with radius 2 in, height 3 in
- Middle: sphere with radius 3 in? Wait, label says “3” for the sphere, but is that radius or diameter? Typically in such diagrams, if it’s a circle with a line through center labeled “3”, it’s diameter. But here, for the sphere, it’s drawn with a dimension “3” across, so likely diameter = 3 in, so radius = 1.5 in.

Similarly, bottom: cylinder with radius 3 in? Label “3” — probably radius? Or diameter? Let’s see consistency.

Top cylinder: labeled “2” — likely radius, since it’s common to label radius for cylinders. Height “3”.

Sphere: labeled “3” — if it’s the diameter, then radius=1.5. If radius, then 3. But looking at the drawing, the sphere is larger than the top cylinder’s radius 2, so if labeled “3”, it might be diameter. But let’s check the bottom cylinder: labeled “3” — if it’s radius, then it’s bigger.

Actually, in many textbooks, when a dimension is given for a circle without specification, it could be diameter. But to be safe, let’s assume that for circles, the number given is the radius unless specified otherwise. However, in this case, for the sphere, if it’s labeled “3” and it’s a full circle, it might be diameter.

Wait — look at the diagram description: for the sphere, it says “3” inside or across? Typically, if it’s written next to the sphere with a line, it’s diameter. But to resolve, let’s calculate both ways and see.

Standard interpretation in such problems: when a single number is given for a circular cross-section, it is often the radius. But for spheres, sometimes diameter is given.

Another clue: the bottom cylinder has “3” — if it’s radius, then volume is large. Top cylinder has “2” radius, height 3.

Sphere: if radius is 3, volume is huge. If diameter 3, radius 1.5.

Let me assume that all linear dimensions given are radii for circles, except where context suggests otherwise.

But for the sphere, the label “3” is likely the diameter, because if it were radius, it would be unusually large compared to others. Let’s see the relative sizes in the diagram — the sphere is drawn larger than the top cylinder but smaller than the bottom cylinder? Not clear.

To avoid mistake, let’s use the most logical approach.

In problem 6, the shapes are stacked vertically. The top cylinder has radius 2 in, height 3 in.

The sphere in the middle: the dimension “3” is probably the diameter, because if it were radius, the sphere would have diameter 6, which is larger than the bottom cylinder’s radius 3 (diameter 6), so possible. But let’s calculate with assumption.

I recall that in many similar problems, when a sphere is labeled with a single number, it is the diameter. For example, "a sphere of diameter 3".

Moreover, in the diagram, the sphere is shown with a horizontal line through center labeled “3”, which typically means diameter.

Similarly, for the bottom cylinder, “3” is likely the radius, as it’s common to label radius for cylinders.

Let’s assume:

- Top cylinder: radius r=2 in, height h=3 in → V = πr²h = π*4*3 = 12π

- Sphere: diameter d=3 in → radius r=1.5 in → V = (4/3)πr³ = (4/3)π(3.375) = (4/3)*(27/8)π wait, 1.5³ = 3.375 = 27/8? 1.5=3/2, (3/2)^3=27/8=3.375

V = (4/3)π*(27/8) = (4*27)/(3*8) π = (108)/24 π = 4.5π

- Bottom cylinder: radius r=3 in, height h=4 in? Label says “4” for height? In the diagram, it says “4” for the bottom cylinder’s height. Yes.

So V = π*9*4 = 36π

Total volume = 12π + 4.5π + 36π = 52.5π

Now calculate: 52.5 * π ≈ 52.5 * 3.1416 ≈ let's compute:

50*3.1416 = 157.08

2.5*3.1416 = 7.854

Total ≈ 164.934 → 164.93 in³

But is the sphere’s dimension diameter? Let me double-check.

If the sphere’s “3” is radius, then V = (4/3)π(27) = 36π, then total = 12π + 36π + 36π = 84π ≈ 263.89, which is larger.

But in the diagram, the sphere is between two cylinders; the top cylinder has radius 2, bottom has radius 3, so sphere with radius 1.5 (diameter 3) makes sense as intermediate size.

Moreover, in standard textbook problems, when a sphere is labeled with a number across, it's diameter.

So I'll go with diameter 3 for sphere.

Thus, total volume = 12π + 4.5π + 36π = 52.5π ≈ 164.93 in³

Confirm calculation:

52.5 * 3.1415926535 = ?

52.5 * 3 = 157.5

52.5 * 0.1415926535 ≈ 52.5 * 0.14 = 7.35, 52.5 * 0.0015926535 ≈ 0.0836, so total approx 7.4336

So 157.5 + 7.4336 = 164.9336 → rounds to 164.93 in³

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Now, summarizing all answers:

1. 200.00 ft³
2. 1080.00 m³
3. 136.00 cm³
4. 376.99 ft³
5. 448.00 ft³
6. 164.93 in³

Final Answer:
1. 200.00
2. 1080.00
3. 136.00
4. 376.99
5. 448.00
6. 164.93