9.08 Volume of prisms and cylinders

We can find the volume of any prism by multiplying the base area by the height of the prism, where the height of the prism is the distance between the two base faces.

The volume of a prism is given by V=Ah where A is the base area and h is the height of the prism.

Examples

Example 1

Find the volume of the triangular prism shown.

Worked Solution

Create a strategy

Use the volume formula V=Ah, where the area of the base is given by A=\dfrac{1}{2} b H.

Apply the idea

We are given base length b=5, height of the triangular base H=5, and height of prism h=10.

\displaystyle V	\displaystyle =	\displaystyle \left(\dfrac{1}{2} b H\right) h	Write the formula
	\displaystyle =	\displaystyle \left(\dfrac{1}{2} \times 5 \times 5 \right)\times 10	Subsitute the values
	\displaystyle =	\displaystyle 125 \text{ cm}^3	Evaluate

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Example 2

A prism has a volume of 1080 \text{ cm}^3.

If it has a base area of 120 \text{ cm}^2, find the height of the prism.

Worked Solution

Create a strategy

Use the volume of a prism formula: V=A h.

Apply the idea

We are given V=1080 and A=120.

\displaystyle V	\displaystyle =	\displaystyle A h	Write the formula
\displaystyle 1080	\displaystyle =	\displaystyle 120\times h	Subsitute the values
\displaystyle \dfrac{1080}{120}	\displaystyle =	\displaystyle \dfrac{120\times h}{120}	Divide both sides by 120
\displaystyle h	\displaystyle =	\displaystyle 9 \text{ cm}	Evaluate

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We can find the volume of a cylinder using the same method, by multiplying the area of the circular base by the height of the cylinder.

Since the area of a circle is given by the formula A=\pi r^2, the formula for the volume of a cylinder is: V=\pi r^2h, where r is the radius and h is the height of the cylinder.

Examples

Example 3

Calculate the volume of the solid. Assume that the solid is a quarter of a cylinder.

Round your answer to one decimal place.

Worked Solution

Create a strategy

Multiply the volume of a cylinder formula by a quarter.

Apply the idea

We are given r=5 and h=14.

\displaystyle V	\displaystyle =	\displaystyle \dfrac{1}{4}\pi r^2 h	Multiply the formula by \dfrac{1}{4}
	\displaystyle =	\displaystyle \dfrac{1}{4}\pi \times 5^2 \times 14	Subsitute the values
	\displaystyle =	\displaystyle 274.9 \text{ cm}^3	Evaluate

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Example 4

A wedding cake with three tiers rests on a table. The layers have radii of 50 cm, 54 cm and 58 cm as shown in the figure. If each layer is 21 cm high, calculate the total volume of the cake in cubic metres. Round your answer to two decimal places.

Worked Solution

Create a strategy

Add the three volumes and convert the units to cubic metres.

Apply the idea

\displaystyle \text{Total volume}	\displaystyle =	\displaystyle \pi r_1^2 h +\pi r_2^2 h +\pi r_3^2 h	Add the volumes
	\displaystyle =	\displaystyle \pi \times 50^2 \times 21+\pi \times 54^2 \times 21+\pi \times 58^2 \times 21	Substitute the radii and heights
	\displaystyle \approx	\displaystyle 579\,247\text{ cm}^3	Evaluate

To convert to metres cubed we need to divide by 100^3.

\displaystyle V	\displaystyle =	\displaystyle \dfrac{579\,247}{1\,000\,000}	Convert to \text{m}^3
	\displaystyle =	\displaystyle 0.58 \text{ m}^3	Evaluate and round

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