Volume of cones, pyramids and spheres practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Measurement and similarity, Volume and capacity)

Volumes of cones, pyramids and spheres

A cone or pyramid is one third of its enclosing cylinder or prism. A sphere uses radius cubed, and a hemisphere is half of the corresponding sphere

V_{\text{cone or pyramid}}=\frac13Bh,\quad V_{\text{sphere}}=\frac43\pi r^3,\quad V_{\text{hemisphere}}=\frac12V_{\text{sphere}}

Find the base area For a square or rectangular base, calculate the base area before applying the one-third factor \(\displaystyle B=6^2=36\text{ cm}^2\)
Use one third Multiply the base area by the perpendicular height and divide by 3 for a cone or pyramid \(\displaystyle V=\frac13\times36\times9=108\text{ cm}^3\)
Cube the radius Sphere volume uses the radius cubed, rather than the radius squared or the radius alone \(\displaystyle V=\frac43\pi(3)^3=36\pi\text{ cm}^3\)
Halve a sphere A hemisphere has half the volume of a full sphere with the same radius \(\displaystyle V_{\text{hemisphere}}=\frac12\times36\pi=18\pi\text{ cm}^3\)

Worked example

A hemispherical dome has radius 6 m. Find its volume to 1 decimal place

r=6\text{ m}

\(\displaystyle r^3=6^3=216\)Cube the radius for the sphere-volume formula
\(\displaystyle V_{\text{sphere}}=\frac43\pi(6)^3=288\pi\text{ m}^3\)Calculate the volume of the full sphere with radius 6 m
\(\displaystyle V_{\text{hemisphere}}=\frac12\times288\pi=144\pi\text{ m}^3\)A hemisphere has half the volume of the corresponding sphere
\(\displaystyle 144\pi\approx452.4\text{ m}^3\)Round the exact hemisphere volume once at the end

Common mistakes

Dropping the one-third factor

✕\(\displaystyle V=\pi(3)^2(12)=108\pi\text{ cm}^3\)✓\(\displaystyle V=\frac13\pi(3)^2(12)=36\pi\text{ cm}^3\)

A cone occupies one third of the enclosing cylinder with the same circular base and perpendicular height

Squaring instead of cubing the radius

✕\(\displaystyle V=\frac43\pi(4)^2=\frac{64}{3}\pi\text{ cm}^3\)✓\(\displaystyle V=\frac43\pi(4)^3=\frac{256}{3}\pi\text{ cm}^3\)

Sphere volume uses the radius cubed because the solid extends in three dimensions

Not halving a hemisphere

✕\(\displaystyle V=\frac43\pi(5)^3=\frac{500}{3}\pi\text{ cm}^3\)✓\(\displaystyle V=\frac12\left(\frac43\pi(5)^3\right)=\frac{250}{3}\pi\text{ cm}^3\)

A hemisphere is exactly half of a full sphere with the same radius

Difficulty levels

introFind the exact volume of a square-based pyramid from its base dimensions and perpendicular height

standardChoose and apply a cone or sphere formula, giving a volume to 1 decimal place

challengeHandle a hemisphere, a pyramid with a stated base area or a composite solid

examChoose a sphere or hemisphere volume from options containing fraction and radius-power errors

Questions about this topic

Why is a cone or pyramid multiplied by one third?+

A cone is one third of the cylinder with the same circular base and height, and a pyramid is one third of the prism with the same base and height

Why is the radius cubed for a sphere?+

Volume measures three-dimensional space. Multiply four-thirds of pi by the radius cubed to find the sphere volume

How is a hemisphere volume found?+

Calculate the volume of the full sphere with the same radius and divide it by 2

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity

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Volume of cones, pyramids and spheres

Volumes of cones, pyramids and spheres

Volumes of cones, pyramids and spheres

Worked example

Common mistakes

Difficulty levels

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