Area Formulas

It will be assumed that you are familiar with the results from Euclidean and coordinate geometry listed below.

Area of a parallelogram:

(C.110)

Area of a triangle:

(C.111)

Area of a trapezoid:

(C.112)

We will always assume that angles are measured in radians unless otherwise specified. If an angle $\theta$ is inscribed in a circle of radius $r$ and $s$ is the length of the subtended arc, then

(C.113)

A right angle is $\pi/2$ and the sum of the angles of a triangle is $\pi$. When $\theta$ is four right angles in (C.113) we get

$$\mbox{circumference(circle)\index{circle circumference of}} = 2 \pi r.$$ (C.114)

Area of a circular sector:

$$\mbox{Area} = \frac{1}{2} \cdot \mbox{radius} \cdot \mbox{subtended arc}$$

$$= \frac{1}{2} r s$$

$$= \frac{1}{2} \cdot \mbox{central angle} \cdot \mbox{radius}^2$$

$$= \frac{1}{2} \theta r^2.$$ (C.115)

In particular when $\theta$ is four right angles

$$\mbox{Area(circle)} = \pi r^2.\index{circle area of}$$ (C.116)