What are the area of the base and the volume when the height is 0? What do you notice?
What are the area of the base and the volume when the height is 1? What do you notice?
What are the area of the base and the volume when the height is 2? What do you notice?
What similarities do you notice with the volume of a prism?
Write a formula to represent the volume of the cylinder.

We have already seen how the volume of rectangular prisms can be calculated using the formula \text{Volume} = \text{Area of Base} \cdot \text{Height of Prism}

two Cylinders labeled 'Area A' on top and 'h' to indicate height. the first cylinder shows layers of circles. 'Volume= Base Area x Height = pi r^2 h is written below.

A cylinder is very similar to a prism, except that the base is a circle instead of a rectangle or other polygon, but the volume can be found using the same process.

\begin{aligned} \text{Volume of Cylinder} &= \text{Area of Base} \cdot \text{Height of Prism}\\ &=\pi \cdot r^2 \cdot h \\ &=\pi r^2 h \end{aligned}

Examples

Example 1

Find the volume of a cylinder correct to one decimal place if its radius is 5\text{ cm} and its height is 13\text{ cm.}

Worked Solution

Create a strategy

Use the formula of the volume of a cylinder.

Apply the idea

We have been given values for r=5 and h=13 into the formula.

\displaystyle V	\displaystyle =	\displaystyle \pi r^2 h	Use the formula
	\displaystyle =	\displaystyle \left(\pi \cdot 5^{2}\right) \cdot 13	Substitute the values
	\displaystyle =	\displaystyle \pi \cdot 25 \cdot 13	Evaluate the squares
	\displaystyle =	\displaystyle 1021.0\, \text{cm}^3	Evaluate

Reflect and check

Our answer may vary if we use an approixmation of 3.14 instead of \pi.

\displaystyle V	\displaystyle =	\displaystyle \pi r^{2} h	Use the formula
	\displaystyle =	\displaystyle \left(3.14 \cdot 5^{2}\right) \cdot 13	Substitute the values
	\displaystyle =	\displaystyle 3.14 \cdot 25 \cdot 13	Evaluate the squares
	\displaystyle =	\displaystyle 1020.5\, \text{cm}^3	Evaluate

Example 2

Consider the half-pipe with a diameter of 7\text{ cm} and a height of 16\text{ cm}.

Half of a cylinder with a diameter of 7 centimeters and height of 16 centimeters.

Find its volume, rounding to two decimal places.

Worked Solution

Create a strategy

Use the fact that the volume of the half-pipe is half of the volume of the cylinder, and the radius is half of the diameter.

Apply the idea

We have been given the values for diameter of 7\,\text{cm} and height of 16\,\text{cm}.

Since the radius is half of the diameter, then r=\dfrac{7}{2}.

\displaystyle V	\displaystyle =	\displaystyle \dfrac{1}{2} \cdot \pi r^{2} h	Multiply the cylinder formula by a half
	\displaystyle =	\displaystyle \dfrac{1}{2} \cdot \pi \cdot \left(\dfrac{7}{2}\right)^2\cdot 16	Substitute the values
	\displaystyle =	\displaystyle 307.88\, \text{cm}^3	Evaluate the product

Idea summary

The volume of the cylinder is given by:

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

\bm{r}

is the radius of the cylinder

\bm{h}

is the height of the cylinder

We can also think of the volume formula as \text{Volume} = \text{Area of Base} \cdot \text{Height}

Outcomes

7.MG.1

The student will investigate and determine the volume formulas for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.

7.MG.1a

Develop the formulas for determining the volume of right cylinders and solve problems, including those in contextual situations, using concrete objects, diagrams, and formulas.

7.04 Volume of right cylinders

Ideas

Volume of right cylinders

Exploration