There is no picture. How do I imagine the shape?

Read the description slowly and make a quick sketch. Write each part with its own numbers, for example: cylinder $r = 2$, $h = 6$; hemisphere $r = 2$. Once the parts are listed, the problem becomes two small problems.

Should I use $\pi \approx 3.14$ or the $\pi$ button?

Use what the question says. This lesson uses $3.14$. The $\pi$ button gives a slightly bigger, more exact answer (for example $102\pi$ is $320.28$ with $3.14$ and $320.44$ with the button). If a question says leave the answer in terms of $\pi$, write something like $102\pi$.

Do I count the flat face where the two parts touch?

For volume, no. The face where two parts meet has no thickness, so it adds no volume. It only matters for surface area, which is a different lesson (Surface Area of Composite Shapes).

Teacher Resources

Volume of Composite Shapes

Learn how to find the volume of solids built from cubes, prisms, cylinders, cones, spheres and pyramids by adding or subtracting the volumes of their parts.

Advanced30 minLesson

Printable worksheet

Definition

A composite solid (or composite shape) is a 3D object made of two or more simple solids joined together, or a simple solid with a part taken out. A pencil is a cylinder with a cone on the end, a grain silo is a cylinder with a half sphere on top, and a pipe is a cylinder with a smaller cylinder removed.

There is no new formula to learn. You split the solid into parts you already know, find the volume of each part, then combine them:

V_{\text{total}} = V_1 + V_2 \quad \text{(parts joined together)}

V = V_{\text{outer}} - V_{\text{removed}} \quad \text{(a hole or hollow part)}

Formula recap

Cube: $V = s^3$
Rectangular prism: $V = l \times w \times h$
Cylinder: $V = \pi r^2 h$
Cone: $V = \frac{1}{3}\pi r^2 h$
Sphere: $V = \frac{4}{3}\pi r^3$ , so a hemisphere (half sphere) has $V = \frac{2}{3}\pi r^3$
Pyramid: $V = \frac{1}{3}Bh$ , where $B$ is the area of the base

A method that always works

1.Describe the solid in words (or sketch it): which simple solids, how they are joined, which dimensions belong to which part.
2.Find the dimensions of each part. Parts often share a radius, but each part has its own height.
3.Find each volume. Keeping answers in terms of $\pi$ (like $90\pi$ ) makes adding easy.
4.Add or subtract.
5.**Use $\pi \approx 3.14$ and round only at the end**, to the place the question asks for. Write cubic units (cm³, m³). Remember $1$ L $= 1000$ cm³ and $1$ m³ $= 1000$ L.

Try it now

A solid is a cylinder with a cone attached to its top, so the cone's circular base sits exactly on the top of the cylinder. How do you find the volume of the whole solid?

Worked Examples

A garden shed has a box-shaped base that is $4$ m long, $3$ m wide and $2.5$ m high. On top sits a roof shaped like a pyramid with the same $4$ m by $3$ m rectangular base and a height of $1.5$ m. What is the total volume of the shed?

Name the parts

Rectangular prism ( $4 \times 3 \times 2.5$ ) + pyramid (base $4 \times 3$ , height $1.5$ ) → Two parts, joined

Volume of the prism

$V_1 = 4 \times 3 \times 2.5 = 30$ → $30$ m³

Base area of the pyramid

$B = 4 \times 3 = 12$ → $12$ m²

Volume of the pyramid

$V_2 = \frac{1}{3} \times 12 \times 1.5 = 6$ → $6$ m³

Add the parts

$V = 30 + 6 = 36$ → $36$ m³

Answer: The shed has a volume of $36$ m³.

Common Mistakes

Using the total height for every part (a pencil $14$ cm long treated as a cylinder of height $14$ plus a cone of height $4$ )

Why it's wrong: Each part has its own height. The total height belongs to the whole object, not to any single part, so using it counts the cone's part twice.

Correct: Split the height: cylinder $h = 10$ , cone $h = 4$ . Check that the heights of the parts add up to the total: $10 + 4 = 14$ .

Using the full sphere formula for a dome or scoop

Why it's wrong: A dome, a scoop or the round end of a silo is only half a sphere. Using $\frac{4}{3}\pi r^3$ doubles that part.

Correct: A hemisphere is $\frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$ . Two hemispheres (the two ends of a capsule) make one whole sphere.

Subtracting the radii of a pipe first: $\pi (5 - 4)^2 \times 100 = 100\pi$

Why it's wrong: Volume depends on $r^2$ , so you cannot subtract the radii before squaring. The difference of the volumes is not the volume of the difference.

Correct: Subtract whole volumes: $\pi \times 5^2 \times 100 - \pi \times 4^2 \times 100 = \pi (25 - 16) \times 100 = 900\pi$ .

Rounding each part before adding

Why it's wrong: Small rounding errors in each part add up, and the final answer can be off in the last digit.

Correct: Keep each part in terms of $\pi$ (or keep all decimals), add or subtract, and round once at the end to the place the question asks for.

Interactive Visual

Composite solid

Rotate X (20°)

Rotate Y (30°)

Volume

Rectangular Prism: V = lwh = 30 m³
+ Rectangular Pyramid: V = ⅓lwh = 6 m³

Total volume: 36 m³

The volume of a composite solid is the sum of the volumes of its parts. A hole is subtracted.

Surface Area

Rectangular Prism: 59 − 12 (touching faces) = 47 m²
+ Rectangular Pyramid: 27.99 − 12 (touching faces) ≈ 15.99 m²

Total surface area: 62.99 m²

Surface area counts only the outside: where two parts touch, that face is hidden and is taken away from both parts. A hole takes away its openings and adds its inner wall.

Calculated with π ≈ 3.14.

Interactive Sandbox

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Practice Problems

18 problems

Problem 1 of 18

Easy

A solid is a cylinder with a cone attached to its top, so the cone's circular base sits exactly on the top of the cylinder. How do you find the volume of the whole solid?

Practice Problems: Volume of Composite Shapes

1. A solid is a cylinder with a cone attached to its top, so the cone's circular base sits exactly on the top of the cylinder. How do you find the volume of the whole solid?
2. A cube with edges of 4 cm is glued on top of a bigger cube with edges of 6 cm. What is the total volume of the solid in cm³?
3. A step block is made of two boxes. The bottom box is 8 cm long, 4 cm wide and 2 cm high. A second box, 4 cm long, 4 cm wide and 2 cm high, sits on top of it. Find the total volume in cm³.
4. Which expression gives the volume of a hemisphere (half a sphere) with radius r?
5. A small shed is a cube with edges of 2 m. Its roof is a pyramid with a square base of 2 m by 2 m (the top of the cube) and a height of 1.5 m. What is the total volume of the shed in m³?
6. A model house is a box 6 m long, 4 m wide and 3 m high, with a pyramid roof on top. The roof has the same 6 m by 4 m rectangular base and a height of 2 m. Find the total volume in m³.
7. A wooden cube has edges of 10 cm. A small cube with edges of 4 cm is cut out of one corner. What volume of wood is left, in cm³?
8. A rocket model is 15 cm tall: a cylinder with radius 2 cm and height 9 cm, with a cone of radius 2 cm and height 6 cm on top. Which expression gives its volume?
9. A crayon is a cylinder with radius 1 cm and length 9 cm, with a cone-shaped tip of radius 1 cm and height 3 cm. Use π ≈ 3.14 to find its volume in cm³.
10. An ice cream cone has radius 3 cm and height 8 cm and is completely filled. A hemisphere-shaped scoop of radius 3 cm sits on top. The total volume of ice cream is kπ cm³. What is k?
11. A capsule-shaped container is a cylinder with radius 3 cm and length 12 cm, with a hemisphere of radius 3 cm on each end. Use π ≈ 3.14 and round the total volume to the nearest whole cm³.
12. A wooden block is 10 cm long, 10 cm wide and 6 cm high. A cylindrical hole with radius 2 cm is drilled straight down through its whole 6 cm height. Using π ≈ 3.14, what volume of wood is left, rounded to the nearest tenth?
13. A silo is a cylinder with radius 2 m and height 8 m, with a hemisphere-shaped roof of radius 2 m. Use π ≈ 3.14. What is the volume of the silo in m³, rounded to the nearest whole number?
14. A solid cylinder has radius 6 cm and height 10 cm. A cone with the same base (radius 6 cm) and the same height (10 cm) is hollowed out of it, with its tip pointing down inside the cylinder. Use π ≈ 3.14 to find the volume of material that is left, in cm³.
15. A metal pipe is 50 cm long. Its outer radius is 6 cm and its inner radius (the hole) is 5 cm. Which expression gives the volume of metal in the pipe?
16. A concrete drain pipe is 200 cm long. Its outer radius is 10 cm and its inner radius is 8 cm. Use π ≈ 3.14 to find how many liters of concrete it contains (1 L = 1000 cm³).
17. A fish tank is a box 30 cm long, 20 cm wide and 20 cm high. A solid decoration shaped like a cylinder, with radius 5 cm and height 20 cm, stands upright inside it. Using π ≈ 3.14, how many liters of water are needed to fill the rest of the tank? (1 L = 1000 cm³)
18. A solid is a cylinder with a hemisphere on top, both with radius 3 cm. Its total volume is 90π cm³. How tall is the cylinder part? (Work in terms of π.)

Why It Matters

Real objects are rarely a single perfect shape:

Engineering: Gas tanks and medicine capsules are cylinders with rounded ends (two hemispheres).
Farming: A silo is a cylinder with a dome or cone on top, and farmers need its volume to know how much grain fits.
Construction: Builders order concrete for blocks and pipes that have holes in them, so they must subtract the empty space.
Food: An ice cream cone with a scoop on top, a cake with a hollow center, a pencil-shaped candy.
Architecture: A house is a box with a pyramid or prism roof, and heating depends on the volume of air inside.

If you can split a shape into simple parts, you can find the volume of almost anything.

Real World Applications

Gas Tanks and Capsules

Tanks for gas, water or milk are often cylinders with a hemisphere on each end, because round ends handle pressure better than flat ones. The two hemispheres together make one whole sphere.

Example:

A pill capsule with radius $3$ mm and a $4$ mm long cylinder in the middle has volume $\pi \times 3^2 \times 4 + \frac{4}{3}\pi \times 3^3 = 36\pi + 36\pi = 72\pi \approx 226.08$ mm³.

1Try It Yourself

A water tank is a cylinder with radius $30$ cm and length $100$ cm, with a hemisphere of radius $30$ cm on each end. Use $\pi \approx 3.14$ .

How many liters of water does the tank hold?

Step 1: Write the mathematical expression

Cylinder plus one whole sphere:

Concrete Planters

Garden planters are often cast from concrete as a block with a hole for the soil. The concrete needed is the volume of the block minus the volume of the hole.

Example:

A block $8$ cm by $8$ cm by $5$ cm with a cylindrical hole of radius $1$ cm through its height uses $320 - \pi \times 1^2 \times 5 \approx 304.3$ cm³ of material.

2Try It Yourself

A planter is a concrete cube with edges of $50$ cm. A cylinder-shaped hole with radius $20$ cm and depth $40$ cm is left in the top for soil. Use $\pi \approx 3.14$ .

How many cm³ of concrete does the planter need?

Step 1: Write the mathematical expression

Cube minus the cylinder hole:

Model Rockets

A model rocket body is a cylinder with a cone-shaped nose. Designers use the volume to plan how much space is inside for parts and how heavy a solid model will be.

Example:

A rocket with radius $4$ cm, a $15$ cm cylinder and a $6$ cm nose cone has volume $240\pi + 32\pi = 272\pi \approx 854.08$ cm³.

3Try It Yourself

A toy rocket is a cylinder with radius $2$ cm and height $12$ cm, topped by a cone with radius $2$ cm and height $6$ cm. Use $\pi \approx 3.14$ .

What is the volume of the rocket?

Step 1: Write the mathematical expression

Cylinder plus cone:

Key Takeaways

1A composite solid is made of simple solids joined together, or a simple solid with a part removed
2Split the solid into parts you know: prisms, cubes, cylinders, cones, spheres, hemispheres, pyramids
3Joined parts: add the volumes. Holes and hollow parts: subtract the volume that is removed
4Each part has its own height; parts that touch often share a radius
5A hemisphere has volume $\frac{2}{3}\pi r^3$ , and two hemispheres make one sphere
6Keep parts in terms of $\pi$ , use $\pi \approx 3.14$ and round only at the end; give cubic units or liters ( $1$ L $= 1000$ cm³)

Frequently Asked Questions

Ask what the solid is made of. If two pieces are stuck together (a cone on a cylinder), add. If something is cut out or empty (a hole, the inside of a pipe), subtract the empty part from the bigger solid.

Glossary

Composite solid: A 3D shape made of two or more simple solids, or a simple solid with a part removed
Hemisphere: Half of a sphere, cut through its center; volume $\frac{2}{3}\pi r^3$
Decompose: To split a shape into simpler parts whose volumes you can find
Hollow cylinder: A cylinder with a smaller cylinder removed from its middle, like a pipe or a tube
Capacity: How much a container can hold, often given in liters or milliliters ( $1$ L $= 1000$ cm³)

Formula Card

Joined parts

V_{\text{total}} = V_1 + V_2 + \dots

Add the volumes of the parts

Part removed

V = V_{\text{outer}} - V_{\text{removed}}

Subtract the volume of the hole or hollow part

Cylinder and cone

V = \pi r^2 h

\quad V = \frac{1}{3}\pi r^2 h

A cone is one third of the cylinder with the same base and height

Sphere and hemisphere

V = \frac{4}{3}\pi r^3

\quad V = \frac{2}{3}\pi r^3

A hemisphere is half a sphere

Prism and pyramid

V = Bh

\quad V = \frac{1}{3}Bh

$B$ is the base area; a pyramid is one third of the prism with the same base and height

Hollow cylinder (pipe)

V = \pi (R^2 - r^2) h

$R$ is the outer radius, $r$ the inner radius

Volume of Composite Shapes

Definition

Formula recap

A method that always works

Worked Examples

Common Mistakes

Interactive Visual

Composite solid

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Gas Tanks and Capsules

Concrete Planters

Model Rockets

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Joined parts

Part removed

Cylinder and cone

Sphere and hemisphere

Prism and pyramid

Hollow cylinder (pipe)

More in This Topic

Introduction to Volume

Metric Volume Units (mL, L)

Volume of Rectangular Prisms

Imperial Volume Units (cups, pints, gallons)