19.5 Surface area and volume of spheres, cones and pyramids — Learn

Prerequisites: use circle area; substitute into formulae and handle square and cube units.

Objectives: calculate surface area or volume of spheres, cones and pyramids.

Curved solids and pyramids

For a sphere, $V=\frac43\pi r^3$ and $A=4\pi r^2$. For a cone, $h$ is its perpendicular height, from the tip to the base at a right angle. The slant height $l$ follows the cone's side from the tip to the base edge. Cone volume is $V=\frac13\pi r^2h$ and curved area is $\pi rl$. Its total surface area includes the circular base: $\pi rl+\pi r^2$. A pyramid has $V=\frac13Ah$, where $A$ is its base area. To find its surface area, add the base area and the areas of its triangular faces.

Example 1. A sphere has radius 3 cm. Work out its volume and surface area.

Use the sphere volume formula: $V=\frac43\pi\times3^3=36\pi$.

Use the sphere surface area formula: $A=4\pi\times3^2=36\pi$.

So the volume is $36\pi\text{ cm}^3$ and the surface area is $36\pi\text{ cm}^2$.

Example 2. A cone has radius 3 cm and perpendicular height 10 cm. Work out its volume.

Substitute the radius and perpendicular height: $V=\frac13\pi\times3^2\times10$.

Calculate: $V=30\pi\text{ cm}^3$.

Example 3. A cone has radius 5 cm and slant height 13 cm. Work out its curved surface area.

Use the slant height in $A=\pi rl$: $A=\pi\times5\times13=65\pi$.

So its curved surface area is $65\pi\text{ cm}^2$.

Add the circular base area for total surface area: $65\pi+\pi\times5^2=90\pi\text{ cm}^2$.

Example 4. A square-based pyramid has base side 8 cm and perpendicular height 9 cm.

Find its base area: $A=8\times8=64\text{ cm}^2$.

Use one third of base area times height: $V=\frac13\times64\times9=192\text{ cm}^3$.

Example 5. A square-based pyramid has base side 6 cm. Each triangular face has perpendicular height 5 cm. Work out its surface area.

Add the base area and four triangular face areas: $6^2+4\times\frac12\times6\times5=96$.

So its surface area is $96\text{ cm}^2$.

Watch out. Cone surface area uses slant height; cone volume uses perpendicular height. Label the diagram before substituting.

Progress check

A conical cover over the pond has radius $4$ m and perpendicular height $3$ m. Work out its volume in terms of $\pi$. Then state which height the cone volume formula uses.

You should now be able to

• calculate sphere surface area and volume;
• calculate cone volume and surface area using the correct heights;
• calculate pyramid volume from base area and perpendicular height;
• find pyramid surface area by adding the base and triangular face areas;
• give square or cubic units as required.