Area

Introduction

In mathematics, area refers to the size of the region that a two-dimensional figure occupies.

Generally, contest problems are only concerned with finding the area of regions bounded by straight line segments, circles, or sometimes even ellipses.

One can find the area of even more complex regions via the use of calculus.

Rectangles are the most basic figures whose area we can study. It makes sense that the area of a rectangle with length l and width w is simply $l\cdot w$ .

Once we know the area of a rectangle, we can easily find the area of a triangle by just noting that if our triangle has base b and height h, then the rectangle with length b and width h has exactly twice as much area as the original triangle. Thus, the area of a triangle is

$A=\frac 12 bh.$

We can now find the area of any polygon by breaking it up into triangles.

Notation

The letters A and K are frequently used to stand for area. When there are multiple regions under consideration, subscripts are often employed, like $A_1, K_2,\ldots$ or $A_{ABC}, K_{BCD},\ldots$ . For example, $K_{ABCDEF}$ would mean the area of hexagon ABCDEF. An alternative notation is to use square brackets around the region to denote its area, e.g. $[ABC]$ for the area of triangle $\triangle ABC$ .

Area of Regular Geometric Figures

The area of any regular geometric figure can be found as follows:

Inscribe the figure, with n sides of length s, in a circle and draw a line from two adjacent vertices to the circumcenter. This creates a triangle that is $\frac{1}{n},$ of the total area (consider the regular octagon below as an example).

Drawing the altitude in creates two right triangles, with an angle of $\frac{180}{n}^{\circ}$ at the top vertex. If the polygon has side length s, the height of the triangle can be found using trigonometry to be of length $\displaystyle \frac s2 \cot \frac{180}{n}^{\circ}$ .

The area of each triangle is $\frac12$ the base times the height, so the area of each triangle is $\displaystyle \frac{s^2}{4} \cot\frac{180}{n}^{\circ}$ and the area of the entire polygon is

$\displaystyle \frac{n\cdot s^2}{4} \cot\frac{180}{n}^{\circ}$ .

Area of Triangle

There are many ways to find the area of a triangle. In all of these formulae, ${K}$ will be used to indicate area.

$K=\frac{bh}{2}$ where b is a base, and h is the altitude of the triangle to that base.
Heron's formula: $K=\sqrt{s(s-a)(s-b)(s-c)}$ , with semi-perimeter $s=\frac{a+b+c}{2}$ .
$\displaystyle K=rs$ , where r is the radius of the incircle, and s is the semi-perimeter.
$K=\frac{ab\sin{\theta}}{2}$ where a and b are adjacent sides of the triangle, and $\theta$ is the angle between them.
$K=\frac{abc}{4R}$ , where $\displaystyle a,b,c$ are the sides of the triangle and $\displaystyle R$ is the circumradius.

Area of a Quadrilateral

To find the area of most quadrilaterals, you must divide the quadrilateral up into smaller triangles and find the area of each triangle. However, some quadrilaterals have special formulas to find their areas. Again, $K$ is the area.