Equable shape

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.

An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then it has an area of 45 square feet (4.2 m²) and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard). Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. However its common use as GCSE coursework has led to its being an accepted concept. 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 solids

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

References

• Bradley, Christopher J. (2005). Challenges in Geometry: For Mathematical Olympians Past and Present. Oxford University Press. p. 15. ISBN 0198566921.