# 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 of figures of which to derive the area. It makes sense that the area of a rectangle with lenght 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

Some popular notations for area include:

• A
• When there are multiple regions involved in a problem, subscripts can be added to the A, such as $A_1, A_2,\ldots$ or $A_{ABC}, A_{BCD},\ldots$.
• Brackets around the name of the region, e.g. $[ABC]$.

## Area Formulas

In the following formulas, assume we have $\triangle ABC$ with $a = BC, b= CA, c = AB, s=\frac 12(a+b+c)$, base b, height h, circumradius R and inradius r. Here are important formulas for the area of a triangle.

• [ABC]=
• $\frac 12 bh$
• $\sqrt{s(s-a)(s-b)(s-c)}$
• $\frac 12 ab\sin C$
• $\displaystyle rs$
• $\frac{abc}{4R}$
• For an equilateral triangle with side length s:
• $[ABC] = \frac{s^2\sqrt{3}}4$

For more formulas for the area of a figure, go to Area of common geometric figures.