Area of polygons

Area is measured in squares

The area of a figure is the number of unit squares it takes to cover it. It's measured in square units: \(\text{in}^2\), \(\text{ft}^2\), \(\text{cm}^2\) and so on. The perimeter, by contrast, is the distance around the figure and is measured in plain units.

Rectangles and parallelograms

Why does the parallelogram work? Cut off the triangle on one end and slide it to the other: you get a rectangle with the same base and height.

Triangles

A diagonal splits a parallelogram into two congruent triangles, so a triangle is half a parallelogram.

A triangle with base \(14\) and height \(9\) has area \(\frac{1}{2} \cdot 14 \cdot 9 = 63\). In a right triangle, the legs are each other's height: \(A = \frac{1}{2} \cdot \text{leg}_1 \cdot \text{leg}_2\).

Trapezoids, rhombuses and kites

Regular polygons

Connecting the center of a regular polygon to its vertices divides it into \(n\) congruent triangles. Each has a side \(s\) of the polygon as its base and the apothem \(a\) as its height. Their total area is \(n \cdot \frac{1}{2}sa = \frac{1}{2}a(ns)\), and \(ns\) is the perimeter \(P\).

Area and scale

If every length of a figure is multiplied by \(k\), the area is multiplied by \(k^2\). Double the sides of a square and you get four copies of the original inside it, not two.

Practice