Plane geometry formulas: triangles, circles and quadrilaterals

Why formulas are rarely enough on their own

Plane geometry questions almost never hand you every number a formula needs. A trapezoid comes with its slanted side instead of its height; a rhombus comes with a side and one diagonal; a square is described only through the circle around it. The formula is the last step. The real work is finding the one missing length, and that length nearly always comes from a right triangle hidden in the figure.

The formulas to know cold

Figure	Area	Perimeter or circumference
Triangle, base $b$, height $h$	$\frac{1}{2}bh$	sum of the three sides
Equilateral triangle, side $s$	$\frac{\sqrt{3}}{4}s^2$	$3s$
Rectangle, sides $a$, $b$	$ab$	$2(a+b)$
Square, side $s$, diagonal $d$	$s^2 = \frac{d^2}{2}$	$4s$
Rhombus, diagonals $d_1$, $d_2$, side $s$	$\frac{d_1 d_2}{2}$	$4s$
Trapezoid, parallel sides $a$, $b$, height $h$	$\frac{(a+b)h}{2}$	sum of the four sides
Circle, radius $r$	$\pi r^2$	$2\pi r$

The height of a triangle or trapezoid is always measured at a right angle to the base, never along a slanted side. Any side of a triangle can serve as the base, as long as the height is the one drawn to that side. In a right triangle the two legs are a base and its height, so its area is half the product of the legs.

The right triangle behind every missing length

The Pythagorean theorem says that in a right triangle with legs $a$, $b$ and hypotenuse $c$, $a^2 + b^2 = c^2$. Without a calculator it helps to recognise the common whole-number triples: $3, 4, 5$; $5, 12, 13$; $8, 15, 17$; $7, 24, 25$; and their multiples such as $6, 8, 10$.

Three hidden right triangles come up again and again:

• Equilateral triangle. The height splits it into two right triangles with hypotenuse $s$ and short leg $\frac{s}{2}$, so the height is $\frac{\sqrt{3}}{2}s$. Half the base times that height gives the area formula in the table.
• Rhombus. Its diagonals cross at right angles and cut each other in half. Each of the four small right triangles has legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$ and hypotenuse equal to the side $s$. So any two of side, first diagonal and second diagonal give the third.
• Isosceles trapezoid. Drop a perpendicular from each end of the shorter parallel side. The two right triangles at the ends each have a horizontal leg of $\frac{b - a}{2}$, where $b$ is the longer parallel side, and the slanted side as hypotenuse. The vertical leg is the height.

A square is both a rectangle and a rhombus: its diagonals are equal, so $\frac{d \cdot d}{2} = \frac{d^2}{2}$ gives its area from the diagonal, and its diagonal is $s\sqrt{2}$.

Circles, arcs and sectors

Turn whatever is given into the radius first. A circumference of $18\pi$ means $r = 9$ and an area of $81\pi$. A sector with central angle $\theta$ degrees is the fraction $\frac{\theta}{360}$ of the whole circle, so its area is $\frac{\theta}{360}\pi r^2$ and its arc is $\frac{\theta}{360} \cdot 2\pi r$. A quarter circle of radius 4 has area $4\pi$ and arc $2\pi$.

When a square sits inside a circle with its four corners on the circle, the square's diagonal is a diameter. When a circle sits inside a square touching all four sides, the circle's diameter equals the square's side. Read which one the question describes; the two give very different numbers.

Composite regions: add or subtract

A shaded region is usually a big figure minus a small one, or two simple figures joined along a side. Name the pieces before calculating. "The region inside the circle but outside the square" is circle area minus square area. An L-shaped floor plan is two rectangles. Leave $\pi$ and roots in exact form until the end: the answer options usually do too.

Scaling

If every length of a figure is multiplied by $k$, its perimeter is multiplied by $k$ and its area by $k^2$. Doubling the sides of a square multiplies its area by 4; a circle whose radius grows by 10% has an area $1.1^2 = 1.21$ times as large, 21% more. This is often the fastest route to an answer, and it is a good check on any area you compute.