Calculating Volumes of Composite Solids

Hello! Last lesson focused on choosing the correct formula and identifying the measurements that belong in it. Now you will use those formulas to calculate volumes accurately, including solids built by joining simpler shapes together.

The main rule remains:

$$V=\text{area of cross-section}\times \text{length}$$

For a single solid, this usually means one calculation. For a composite solid, you split it into non-overlapping simpler solids, calculate each volume, then add them.

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Volume answers need cubic units

Volume measures three-dimensional space. If every length is measured in centimetres, the calculation multiplies three centimetres:

$$\text{cm}\times \text{cm}\times \text{cm}={\text{cm}}^3$$

So:

• area of a face or cross-section uses square units, such as ${\text{cm}}^2$;
• volume of a solid uses cubic units, such as ${\text{cm}}^3$.

This is an important test check. An answer such as $150\text{ cm}$ or $150{\text{ cm}}^2$ cannot be a volume.

Also make sure all dimensions use the same length unit before calculating. Converting between units is the focus of the next module; for now, do not multiply metres and centimetres together as though they match.

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Calculating volumes of single solids

A clear working layout reduces mistakes:

1. Write the formula.
2. Substitute the numbers.
3. Calculate.
4. Include cubic units.

Rectangular prism

For a rectangular prism:

$$V=lwh$$

Suppose a storage box is $12\text{ cm}$ long, $5\text{ cm}$ wide, and $8\text{ cm}$ high.

$$V=12\times 5\times 8$$ $$V=480{\text{ cm}}^3$$

There is no need to identify one special “base” for a rectangular prism: multiplying any three perpendicular edge lengths gives the same result.

Triangular prism

For a triangular prism, first calculate the area of the triangular cross-section:

$$A_{△}=\frac{1}{2}bh$$

Then multiply it by the prism length $L$:

$$V=\left(\frac{1}{2}bh\right)L$$

Suppose the triangular end has base $9\text{ cm}$, perpendicular height $4\text{ cm}$, and the prism is $10\text{ cm}$ long.

First find the triangular area:

$$A_{△}=\frac{1}{2}\times 9\times 4$$ $$A_{△}=18{\text{ cm}}^2$$

Now extend that same triangular area along the length of the prism:

$$V=18\times 10$$ $$V=180{\text{ cm}}^3$$

Keeping the triangle-area calculation on its own line makes it much less likely that you will forget the $\frac{1}{2}$.

Cylinder

For a cylinder:

$$V=\pi r^{2}h$$

Suppose a cylindrical container has diameter $10\text{ cm}$ and height $12\text{ cm}$. Because the formula needs a radius, halve the diameter first:

$$r=10÷2=5\text{ cm}$$

Then substitute:

$$V=\pi \times 5^2\times 12$$ $$V=300\pi {\text{ cm}}^3$$

If the question asks for an exact answer, stop there. If it requests a decimal approximation:

$$V\approx 942.5{\text{ cm}}^3$$

Use the $\pi$ button on your calculator unless the question specifically tells you to use $3.14$.

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What makes a solid composite?

A composite solid is made by joining two or more simple solids. It might look complicated as a whole, but its volume is still just the amount of space inside it.

The most reliable method is:

1. Split the object into simple, non-overlapping pieces.
2. Find the volume of every piece.
3. Add the volumes.

You may split an object in more than one correct way. The best split is usually the one where each part has dimensions you can clearly identify.

Two rules keep the method safe:

• Do not leave out a part of the shape.
• Do not count the same region twice.

Faces where two solids touch do not add any extra volume. They are only boundaries between the pieces.

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A composite prism with an L-shaped cross-section

Sometimes it is easier not to split the entire 3D solid. Instead, find the area of its complicated cross-section, then multiply by the length of the prism.

The L-shaped cross-section can be split into two rectangles:

$$A_A=2\times 6=12{\text{ cm}}^2$$ $$A_B=3\times 3=9{\text{ cm}}^2$$

So the total cross-sectional area is:

$$A=12+9=21{\text{ cm}}^2$$

The prism has length $13\text{ cm}$, so:

$$V=21\times 13$$ $$V=273{\text{ cm}}^3$$

Notice a small but important unit issue in the diagram: the area of the cross-section should be $21{\text{ cm}}^2$, not $21{\text{ cm}}^3$. Its final volume of $273{\text{ cm}}^3$ is correct.

A rectangular prism with a triangular-prism roof

A useful composite shape is a “shed” cross-section: a rectangle with a triangle on top. Suppose the cross-section has:

• a rectangular part $6\text{ m}$ wide and $3\text{ m}$ high;
• a triangular roof with base $6\text{ m}$ and perpendicular height $2\text{ m}$;
• a constant length of $10\text{ m}$.

Find the rectangle area:

$$A_{\text{rectangle}}=6\times 3=18{\text{ m}}^2$$

Find the triangle area:

$$A_{△}=\frac{1}{2}\times 6\times 2=6{\text{ m}}^2$$

Add them to get the whole cross-sectional area:

$$A=18+6=24{\text{ m}}^2$$

Finally, multiply by the length:

$$V=24\times 10=240{\text{ m}}^3$$

You could also calculate the rectangular-prism volume and triangular-prism volume separately, then add them. Both methods describe the same space.

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Composite cylinders

Cylinders can be joined too. Calculate the volume of each cylinder separately unless the joined cylinders have the same radius and form one longer cylinder.

In this diagram, all three cylinders have diameter $7$, so:

$$r=7÷2=3.5$$

The combined length is:

$$4+3+4=11$$

Since the radius stays the same throughout, treat it as one cylinder of length $11$:

$$V=\pi \times 3.5^2\times 11$$ $$V=134.75\pi \text{ cubic units}$$ $$V\approx 423.3\text{ cubic units}$$

If the cylinders had different radii, you could not merge their lengths. Instead, calculate each cylinder’s volume using its own radius and add the answers.

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A test-ready checking routine

Before writing your final answer, scan your work with these checks:

Check	What to look for
Correct shape formula	$lwh$, $\left(\frac{1}{2}bh\right)L$, or $\pi r^{2}h$
Cylinder radius	Halve the diameter before squaring
Triangle height	It must be perpendicular to the chosen base
Composite pieces	Every piece counted once; no overlap
Units	Final answer is in ${\text{mm}}^3$, ${\text{cm}}^3$, ${\text{m}}^3$, or cubic units
Reasonableness	A composite solid should have more volume than any one of its positive-volume parts

A quick size check can catch calculator slips. For example, a cylinder with radius about $5\text{ cm}$ and height about $12\text{ cm}$ should have volume a little under $1000{\text{ cm}}^3$, not $100{\text{ cm}}^3$ and not $10000{\text{ cm}}^3$.

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Key takeaways

For a single solid, calculate using the appropriate formula:

$$V_{\text{rectangular prism}}=lwh$$ $$V_{\text{triangular prism}}=\left(\frac{1}{2}bh\right)L$$ $$V_{\text{cylinder}}=\pi r^{2}h$$

For composite solids, either split the whole solid into simple volumes or split its repeated cross-section into simple areas. Add only non-overlapping parts, and always finish with cubic units.

Next, you will move from calculating a volume in one unit to converting between units such as ${\text{cm}}^3$, millilitres, and litres.