# Area Formulas

### Triangle

Three formulae can be used to calculate the area of a triangle:

\newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}} \begin{align*} \ctext{Area}& =\frac12\times\ctext{base}\times\ctext{perpendicular height}=\frac12 bh \cr \ctext{Area}&=\frac12 ab\sin C \cr \ctext{Area}&=\sqrt{S(S-a)(S-b)(S-c)}\ctext{, where }S=\frac{a+b+c}{2} \end{align*}

### Square

The area of a square can be found by multiplying its (equal) side lengths a.

$$\ctext{Area}=a^2$$

### Rectangle

The area of a rectangle is found by multiplying its long and short side lengths (a and b respectively).

$$\ctext{Area}=ab$$

### Trapezium

The area of a trapezium is found by multiplying it height h by the average of its two parallel side lengths (a and b)

$$\ctext{Area}=\frac{a+b}{2}\times h$$

### Circle

The area of a circle is found by multiplying the constant $\pi$ by the square of its radius r.

$$\ctext{Area}=\pi r^2$$

### Sector of a Circle

The area of a sector of a circle is found by halving the product of the circle's radius, r, and the sector arc length, s.

$$\ctext{Area}=\frac12 rs$$

### Segment of a Circle

The area of a segment of a circle is found by combining the circle's radius, r, segment arc length, s, chord length, c, and segment height, h.

$$\ctext{Area}=\frac12 rs-\frac12 c(r-h)=\frac12\left[rs-c(r-h)\right]$$

### Ellipse

The area of an ellipse is calculated by multiplying the constant $\pi$ by the product of the ellipse's semi-major axis (b) and semi-minor axis (a) lengths.

$$\ctext{Area}=\pi ab$$

### Parabolic Segment

The area of a parabolic segment is found by combining the segment's base length, b, and height, h.

$$\ctext{Area}=\frac23 b\times h$$