Quadrilaterals — area formulas

Square

$A = s^2$

Side squared. A square is a rectangle with equal sides, so $A = l\cdot w$ collapses to $s^2$ .

Rectangle

$A = l \cdot w$

Length × width. The unit-square tiling argument: a rectangle of integer sides $l\times w$ contains exactly $lw$ unit squares.

Parallelogram

$A = b \cdot h$

Base × perpendicular height — not the slanted side. Cut off the triangle on one end and slide it to the other to turn the parallelogram into a rectangle.

Rhombus

$A = \tfrac{1}{2} d_1 d_2$

Half the product of the diagonals — the diagonals bisect each other at right angles, splitting the rhombus into four equal right triangles.

Trapezoid

$A = \tfrac{1}{2}(a + b)\,h$

Average of the two parallel sides $a,b$ , then times height $h$ . Glue two copies head-to-tail and you get a parallelogram of base $a+b$ .

Kite

$A = \tfrac{1}{2} d_1 d_2$

Same diagonal-product formula as the rhombus — a kite is the more general shape whose diagonals are still perpendicular.

Triangles — by what data you have

Base & height

$A = \tfrac{1}{2} b h$

Half base × height — works for any triangle. Two copies form a parallelogram of base $b$ and height $h$ .

Heron's formula (three sides)

$A = \sqrt{s(s-a)(s-b)(s-c)},\ s = \tfrac{a+b+c}{2}$

Use when you have only the three side lengths and no height. $s$ is the semi-perimeter.

Two sides & included angle (SAS)

$A = \tfrac{1}{2} a b \sin C$

Drop the altitude from the third vertex; it equals $a\sin C$ , giving the standard $\tfrac{1}{2}\cdot\text{base}\cdot\text{height}$ .

Equilateral triangle

$A = \tfrac{\sqrt{3}}{4} a^2$

Special case of SAS with $a=b$ and $C = 60^{\circ}$ ; $\sin 60^{\circ} = \tfrac{\sqrt{3}}{2}$ gives the constant $\tfrac{\sqrt{3}}{4}$ .

Circles and curved shapes

Circle

$A = \pi r^2$

Pi-r-squared. Comes from integrating the circumference $2\pi r$ as $r$ grows from 0 — onion-ring derivation.

Sector of a circle

$A = \tfrac{1}{2} r^2 \theta$

Angle $\theta$ in radians. It’s the fraction $\theta / (2\pi)$ of the full circle area $\pi r^2$ .

Annulus (ring)

$A = \pi (R^2 - r^2)$

Outer circle area minus inner circle area — the missing center is subtracted, not measured.

Ellipse

$A = \pi a b$

Semi-major axis $a$ times semi-minor axis $b$ times $\pi$ . When $a = b = r$ you recover $\pi r^2$ — a circle is an ellipse with equal axes.

Regular polygons & coordinates

Regular polygon (n sides)

$A = \tfrac{1}{2} P a$

$P$ is the perimeter, $a$ is the apothem (center-to-side distance). Decompose into $n$ congruent triangles and the formula falls out.

Regular hexagon

$A = \tfrac{3\sqrt{3}}{2} a^2$

A regular hexagon is exactly six equilateral triangles of side $a$ , so $6 \cdot \tfrac{\sqrt{3}}{4} a^2 = \tfrac{3\sqrt{3}}{2} a^2$ .

Coordinates (Shoelace formula)

$A = \tfrac{1}{2}\left|\sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i)\right|$

Plug in the vertex coordinates $(x_i, y_i)$ in order, wrap around ( $x_{n+1}=x_1$ ). Works for any simple polygon — no need to triangulate.

Area Formulas