Pyramids

What makes a pyramid?

A pyramid has one polygonal base and triangular faces that meet at a single apex. A square-based pyramid, for example, has a square base and four triangular faces. Volume measures the space inside it, in cubic units such as cm³. Surface area measures the area covering its outside, in square units such as cm².

Two different heights matter. The perpendicular height runs from the apex at right angles to the base; it is used for volume. A slant height runs down a triangular face at right angles to that face’s base edge; it is used to find the area of that face. The sloping edge from the apex to a corner is neither of these heights.

Finding the volume

A pyramid has one-third the volume of a prism with the same base area and perpendicular height:

$V=\frac13\times\text{area of base}\times\text{perpendicular height}.$

The pyramid volume formula will be given in this OCR specification. You still need to identify the correct base area and height. For a square base of side $a$ and perpendicular height $h$, the base area is $a^2$, so $V=\frac13a^2h$.

Finding the surface area

Add the area of the base to the areas of all the triangular faces. Each triangular face has area $\frac12\times\text{base edge}\times\text{slant height of that face}$. If the faces are not all alike, find their areas separately before adding them.

For a regular square-based pyramid with base side $a$ and a common slant height $s$, the four triangular faces have combined area $4(\frac12as)=2as$. Its total surface area is therefore $a^2+2as$. This shortcut works because those four faces have the same base length and slant height.

A square-based pyramid whose apex is directly above the base centre has a 6 cm base side and a 4 cm perpendicular height. From the centre of the base to the midpoint of a side is 3 cm, making a right-angled triangle with the 4 cm perpendicular height and slant height $s$. Pythagoras’ theorem gives $s=\sqrt{4^2+3^2}=5\text{ cm}$. Its total surface area is $6^2+2(6)(5)=96\text{ cm}^2$. Its volume uses the 4 cm perpendicular height instead: $\frac13(6^2)(4)=48\text{ cm}^3$.

Pyramid calculations

Volume: $V=\frac13\times\text{base area}\times\text{perpendicular height}$. The volume formula will be given.

Total surface area: base area + the areas of every triangular face. For each face, use $\frac12\times\text{base edge}\times\text{face slant height}$. For a regular square-based pyramid, $A=a^2+2as$, where $a$ is the base side and $s$ is the slant height.

Keep the perpendicular height for volume and the face slant height for surface area. Use square units for area and cubic units for volume.