Volume of Cylinders

A **cylinder** is a 3D shape with two identical, parallel **circular bases** joined by one curved surface. Think of a soda can, a candle or a water tank. You already know that the volume of a prism is **base area times height**. A cylinder works the same way: stack thin circles on top of each other until you reach the height. $$V = B \times h$$ The base is a circle, and the area of a circle is $\pi r^2$. So: $$V = \pi r^2 h$$ Where: - $r$ = **radius** of the base (from the center to the edge) - $h$ = **height** (the distance between the two bases) - $\pi \approx 3.14$ **Radius or diameter?** The formula needs the radius. If you are given the **diameter** $d$ (all the way across), halve it first: $r = d \div 2$. **Two ways to give the answer:** - **In terms of $\pi$** (exact): keep $\pi$ as a symbol, for example $90\pi$ cm³. - **As a number**: use $\pi \approx 3.14$ and round to the place the problem asks for, for example $90 \times 3.14 = 282.6$ cm³. **Liters:** $1 \text{ cm}^3 = 1 \text{ mL}$, so $1000 \text{ cm}^3 = 1 \text{ L}$. For big tanks: $1 \text{ m}^3 = 1000 \text{ L}$.

Volume from the Radius

A cylinder has radius $3$ cm and height $10$ cm. Find its volume in terms of $\pi$, then use $\pi \approx 3.14$.

Write the formula: $V = \pi r^2 h$ = Formula ready

Square the radius: $r^2 = 3^2 = 9$ = $r^2 = 9$

Multiply by the height: $V = \pi \times 9 \times 10 = 90\pi$ = $V = 90\pi$ cm³ (exact)

Use $\pi \approx 3.14$: $90 \times 3.14 = 282.6$ = $V \approx 282.6$ cm³

Answer: $V = 90\pi$ cm³, which is about $282.6$ cm³

Watch Out: the Diameter Is Given

A can has a diameter of $8$ cm and a height of $12$ cm. Find its volume using $\pi \approx 3.14$, rounded to the nearest whole cm³.

Find the radius from the diameter: $r = 8 \div 2 = 4$ = $r = 4$ cm

Square the radius: $4^2 = 16$ = $r^2 = 16$

Multiply by the height: $16 \times 12 = 192$ = $V = 192\pi$ cm³

Multiply by $3.14$ and round: $192 \times 3.14 = 602.88 \approx 603$ = $V \approx 603$ cm³

Answer: The can holds about $603$ cm³, which is about $603$ mL

How Many Liters in a Water Tank?

A cylindrical water tank has radius $50$ cm and height $120$ cm. How many liters does it hold? Use $\pi \approx 3.14$.

Square the radius: $50^2 = 2500$ = $r^2 = 2500$

Multiply by the height: $2500 \times 120 = 300000$ = $V = 300000\pi$ cm³

Use $\pi \approx 3.14$: $300000 \times 3.14 = 942000$ = $V \approx 942000$ cm³

Convert to liters ($1000$ cm³ $= 1$ L): $942000 \div 1000 = 942$ = $942$ L

Answer: The tank holds about $942$ liters

Finding the Height

A cylinder has a volume of $471$ cm³ and a radius of $5$ cm. How tall is it? Use $\pi \approx 3.14$.

Find the base area: $B = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$ = $B = 78.5$ cm²

Use $V = B \times h$: $471 = 78.5 \times h$ = Equation set up

Divide the volume by the base area: $h = 471 \div 78.5 = 6$ = $h = 6$ cm

Check: $78.5 \times 6 = 471$ = Correct

Answer: The height is $6$ cm

Finding the Radius

A cylinder is $20$ cm tall and has a volume of $1004.8$ cm³. What is its radius? Use $\pi \approx 3.14$.

Write the formula with what you know: $1004.8 = 3.14 \times r^2 \times 20$ = Equation set up

Multiply $\pi$ by the height: $3.14 \times 20 = 62.8$ = $1004.8 = 62.8 \times r^2$

Divide to find $r^2$: $r^2 = 1004.8 \div 62.8 = 16$ = $r^2 = 16$

Take the square root: $r = \sqrt{16} = 4$ = $r = 4$ cm

Answer: The radius is $4$ cm (so the diameter is $8$ cm)

Cylinders are everywhere, and their volume tells you how much they hold: - **Drinks and food**: how many milliliters fit in a can, a bottle or a glass? - **Water tanks and rain barrels**: how many liters can you store in the garden? - **Pools**: how much water do you need to fill a round paddling pool? - **Engineering**: engine cylinders, pipes and silos are all measured with $V = \pi r^2 h$. - **Cooking and baking**: a round cake pan holds a different amount than a square one. Once you know one formula, you can answer all of these questions, and even work backwards to find how tall or how wide a container must be.

Mistake: Using the diameter instead of the radius

Why: The formula squares the radius. If you square the diameter, the answer is 4 times too big. For $d = 10$ cm and $h = 5$ cm: $\pi \times 10^2 \times 5 = 500\pi$ is wrong.

Correct: Halve the diameter first: $r = 5$, so $V = \pi \times 5^2 \times 5 = 125\pi$ cm³.

Mistake: Doubling the radius instead of squaring it

Why: $r^2$ means $r \times r$, not $r \times 2$. For $r = 3$: $3^2 = 9$, not $6$.

Correct: Always multiply the radius by itself: $r^2 = 3 \times 3 = 9$.

Mistake: Dividing by 100 instead of 1000 when converting to liters

Why: A liter is a cube $10$ cm on each side, so it holds $10 \times 10 \times 10 = 1000$ cm³, not $100$.

Correct: $942000$ cm³ $\div 1000 = 942$ L. Remember: $1$ cm³ $= 1$ mL.

Mistake: Rounding too early

Why: If you round the base area before multiplying by the height, the error grows. For $r = 2.5$ cm and $h = 10$ cm, rounding $B = 19.625$ to $20$ gives $200$ cm³.

Correct: Keep all digits until the end: $19.625 \times 10 = 196.25$ cm³, then round only the final answer.

Drink Cans

Drink companies design cans so they hold a set amount, like 330 mL. The volume formula tells them how tall and how wide the can must be.

A can with radius $3.3$ cm and height $10$ cm holds about $3.14 \times 3.3^2 \times 10 \approx 342$ cm³, enough for $330$ mL plus some air.

Rain Barrels

Gardeners collect rainwater in cylindrical barrels. Knowing the volume tells them how many liters they can save for dry weeks.

A barrel with radius $25$ cm and height $80$ cm holds $3.14 \times 625 \times 80 = 157000$ cm³, which is $157$ L.

Paddling Pools

Before filling a round paddling pool, it helps to know how much water you need. Here the height is the depth of the water, not the height of the pool wall.

A pool with radius $1.5$ m filled to $0.2$ m holds $3.14 \times 2.25 \times 0.2 = 1.413$ m³, which is $1413$ L.

A cylinder has two identical circular bases, and its volume is base area times height: $V = B \times h$

Because the base is a circle, $V = \pi r^2 h$

The formula needs the radius: if you are given the diameter, divide it by $2$ first

Give the answer in terms of $\pi$ (exact, like $90\pi$ cm³) or use $\pi \approx 3.14$ and round only at the end

To find the height, divide the volume by the base area: $h = V \div (\pi r^2)$; to find the radius, find $r^2$ first and then take the square root

$1$ cm³ $= 1$ mL, $1000$ cm³ $= 1$ L and $1$ m³ $= 1000$ L

Q: Why is there a $\pi r^2$ in the formula?

A: Because $\pi r^2$ is the area of the circular base. The volume of any prism-like shape is base area times height, and a cylinder is like a prism with a circle as its base.

Q: When should I leave the answer in terms of $\pi$?

A: When the problem asks for an exact answer or says "in terms of $\pi$". Otherwise use $\pi \approx 3.14$ (or the $\pi$ key on a calculator) and round to the place the problem asks for.

Q: What if the cylinder is lying on its side?

A: Nothing changes. The height is always the distance between the two circular bases, so for a pipe or a rolling pin lying down, the height is its length.

Q: What happens to the volume if I double the radius?

A: The volume becomes 4 times bigger, because the radius is squared: $(2r)^2 = 4r^2$. Doubling the height only doubles the volume.

Exit Presentation

Volume of Cylinders

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Measurement / Volume & Capacity

Volume of Cylinders

Learn how to find the volume of a cylinder with V = πr²h, use the radius instead of the diameter, and convert cm³ to liters for cans, tanks and glasses.

Intermediate25 min

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