Equable shapes

A two-dimensional equable shape (or perfect shape) is one whose area is numerically equal to its perimeter. For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both equal to 30 units.

Equable shapes are not regarded as a true mathematical concept, as it is incorrect to state that an area can be equal to a length, however its common use as GCSE coursework has led to it being an accepted term. For any shape, there is a similar equable shape: if a shape S has perimeter p and area A, then scaling S by a factor of p/A leads to an equable shape. Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. In the case of the square, for instance, this equation is

$displaystyle\; x^2\; =\; 4x.$

Solving this yields that x = 4, so a 4 × 4 square is equable.

In three dimensions, a shape is equable when its surface area is numerically equal to its volume.

Equable regular polygons

The perimeter of an n-sided regular polygon with side length x is nx. The area is

$A=frac\{nt^2\}\{4tanfrac\{pi\}\{n\}\}$

equating these two values and solving for x, one finds that

$x=frac\{4\}\{tan(frac\{pi\}\{2\}-frac\{pi\}\{n\})\}$

This formula can then be used to find the perfect pentagon, hexagon, heptagon etc.

Equable solids

As with equable shapes in two dimensions, one may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor.