Volume and Surface Area of a Regular Prism

A regular prism has matching regular polygons top and bottom, joined by vertical sides. The head of a hexagonal bolt, a pencil and a polygonal column are all this shape.

Both the volume and the surface area follow once the base area is known.

• $A$ is the base area
• $V$ is the volume
• $n$ is the number of sides of the base
• $s$ is the side length
• $h$ is the height of the prism

The sides are $n$ upright rectangles, so the lateral area is $n s h$. The total surface adds the top and bottom, giving $2A + n s h$.

Where the base formula comes from

Drawing lines from the centre of a regular polygon to each vertex cuts it into $n$ identical isosceles triangles. Each has the side $s$ as its base, and its height is the distance from the centre to that side.

That distance comes from halving the central angle. The central angle is $2\pi \div n$, and the tangent of half of it gives $\dfrac{s}{2\tan(\pi / n)}$. One triangle therefore has area $\dfrac{s}{2} \times \dfrac{s}{2\tan(\pi / n)}$, and multiplying by $n$ produces the formula above.

Example

The default input is a prism of height 10 on a regular hexagon of side 4.

The base area is $6 \times 16 \div (4\tan 30°)$, about 41.5692. The volume multiplies that by the height, about 415.6922. The lateral area is $6 \times 4 \times 10$, exactly 240, and the total surface is $2 \times 41.5692 + 240$, about 323.1384.

Setting the number of sides to 4 describes a prism on a square of side 4, in other words a cuboid, and the calculator duly returns a base area of 16 and a volume of 160.

Points to watch

The number of sides must be a whole number of at least 3. Fewer than that is not a polygon.

Adding sides moves the base towards a circle. Holding the side length fixed while adding sides makes the prism steadily fatter, so comparisons with a cylinder should keep the perimeter fixed instead.

A pyramid on the same base with the same height holds exactly one third of this volume, and there is a separate tool here for that.