Solid Geometry

A sphere is a perfectly symmetrical solid with all points on the surface at the same distance $r$ from the centre. It has no edges or corners.

Surface area of sphere $=4\pi r^2$

Volume of sphere $=\frac43 \pi r^3$

A cylinder has a flat circular base and top which both have the same radius, $r$. It has one curved side with height $h$.

Surface area of cylinder $=2\pi r(r+h)$

Volume of cylinder $=\pi r^2h$

A cone has a flat circular base with radius $r$. It has one curved side. The point at the top of the cone is called the apex. It has a height $h$ and side length $s$. Note that the side length is only defined if the cone is a right circular cone, that is the apex is directly above the centre of the base.

Surface area of cone $=\pi r(r+s)$

Volume of cone $=\frac13 \pi r^2h$

A triangular pyramid has four triangular faces with three side faces and one base. It has four corner points (vertices) and six edges.

A square pyramid has five faces. The four side faces are triangles while the base is a square. It has five vertices and eight edges.

The formula for the surface area and the volume is the same for both triangular and square pyramids, although the formula for the [Base area] will be different for each.

Surface area $=$ [Base area] $+\frac12\times$ perimeter $\times$ [slant height]

Volume $=\frac13\times$ [Base area] $\times$ height

Note: When side faces are different sizes we can calculate the area of the base and each triangular face separately and then add them up to find the total surface area.

• Common 3D Shapes - a useful pictorial reference of commonly occurring 3D shapes.
• Solid Geometry - the Khan Academy's collection of videos covering volumes of solids.
• Surface Areas of 3D Figures - a Udemy video showing how to calculate the surface areas of a variety of figures.