Missing Dimensions & Composite Solids

Q: What do I do after I get r² by itself?

Take the square root of both sides. If $r^2 = 16$, then $r = 4$. The squaring and the square root are opposite operations, so the square root is the finishing move whenever the radius is missing.

Sometimes a problem hands you the volume and asks for a dimension instead — the height of a cylinder, or its radius. Nothing about the formula changes. You still write $V = \pi r^2 h$ ; you just substitute what you know and solve for the letter that's left. The volume formula becomes an equation, and you solve it like any other.

The same unit also covers composite solids — shapes built from two simple solids stacked together, like a grain silo (a cylinder with a cone on top). The strategy there is even simpler: find each piece's volume with its own formula, then add.

Working the formula backward

Plug in everything you know, then solve for the missing letter. If a cylinder has volume $226.08$ and radius $3$ , substitution gives $226.08 = 3.14 \cdot 3^2 \cdot h$ . Simplify the right side to $28.26h$ , then divide both sides by $28.26$ to get $h = 8$ .

Always simplify the known numbers into a single coefficient before dividing. The equation $226.08 = 28.26h$ is a one-step equation — territory you already know.

When the radius is missing

A missing radius takes one extra step, because the formula squares it. Dividing both sides leaves you with something like $r^2 = 16$ — and $r^2$ is not the answer. Take the square root of both sides: $r = 4$ .

A quick self-check: a radius is a length, so it should be a sensible positive number. If your answer looks like the square of what you'd expect, you skipped the square root.

Composite solids

A composite solid is two (or more) simple solids joined together. To find the total volume, find each solid's volume with its own formula, then add the volumes.

Keep the formulas separate: the cylinder part uses $V = \pi r^2 h$ and the cone part uses $\displaystyle V = \frac{1}{3} \pi r^2 h$ . Forgetting the $\displaystyle \frac{1}{3}$ on the cone is the classic mistake — the cone on top of a silo holds a third of what the same-size cylinder would.

Worked examples

Example 1: missing height

A cylinder has volume $100.48$ and radius $2$ . Find its height. Use $\pi \approx 3.14$ .

Write the formula

V = \pi r^2 h

Substitute what you know

100.48 = 3.14 \cdot 2^2 \cdot h

Simplify the right side

100.48 = 12.56h

Divide both sides by

12.56

h = 8

Answer: $h = 8$

Example 2: missing radius

A cylinder has volume $251.2$ and height $5$ . Find its radius. Use $\pi \approx 3.14$ .

Write the formula

V = \pi r^2 h

Substitute what you know

251.2 = 3.14 \cdot r^2 \cdot 5

Simplify the right side

251.2 = 15.7r^2

Divide both sides by

15.7

r^2 = 16

Take the square root

r = 4

Answer: $r = 4$

Example 3: a composite solid

A grain silo is a cylinder with a cone on top. The cylinder has radius $2$ and height $3$ ; the cone has radius $2$ and height $3$ . Find the total volume. Use $\pi \approx 3.14$ .

Cylinder volume

V = 3.14 \cdot 2^2 \cdot 3 = 37.68

Cone volume — keep the one third

V = \dfrac{1}{3} \cdot 3.14 \cdot 2^2 \cdot 3 = 12.56

Add the volumes

37.68 + 12.56 = 50.24

Answer: $V = 50.24$

Try one yourself

Common questions

How do I know which dimension is missing?

Substitute everything the problem gives you into the formula. Whatever letter is left is what you're solving for. If the equation still has $h$ in it, you're finding the height; if it has $r^2$ , you're finding the radius.

What do I do after I get r² by itself?

Take the square root of both sides. If $r^2 = 16$ , then $r = 4$ . The squaring and the square root are opposite operations, so the square root is the finishing move whenever the radius is missing.

Do I ever subtract volumes instead of adding?

Yes — when a piece is removed, like a hole drilled through a block, subtract the missing piece's volume from the whole. For stacked solids like a silo, the pieces sit together, so you add.

Why do the answer choices often include my number times 9 or divided by 3?

Those are the trap answers: skipping the square on the radius, squaring the whole product, or dropping the $\displaystyle \frac{1}{3}$ on a cone. Write the formula first and substitute line by line, and the traps have nowhere to hide.

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Missing Dimensions & Composite Solids

Working the formula backward

When the radius is missing

Composite solids

Worked examples

Example 1: missing height

Example 2: missing radius

Example 3: a composite solid

Try one yourself

Common questions

Keep learning

Volume of Cylinders

Volume of Cones & Spheres

Surface Area of Prisms & Cylinders

Two-Step Equations