Special right triangles
Two right triangles appear so often that their side ratios are worth memorising. They give exact answers without a calculator.
The 45-45-90 triangle
An isosceles right triangle has angles 45, 45, and 90 degrees. Its legs are equal, and by Pythagoras the hypotenuse is \(\sqrt{2}\) times a leg. Legs of 5 give a hypotenuse of \(5\sqrt{2} \approx 7.07\). It is half of a square cut along its diagonal.
Going backwards in 45-45-90
If the hypotenuse is known, divide by \(\sqrt{2}\) to get each leg: a hypotenuse of 10 gives legs of \(\frac{10}{\sqrt{2}} = 5\sqrt{2} \approx 7.07\). A square with a diagonal of 10 has sides of about 7.07.
The 30-60-90 triangle
A 30-60-90 triangle is half of an equilateral triangle. If the shortest side, opposite 30 degrees, is s, then the hypotenuse is 2s and the longer leg, opposite 60 degrees, is \(s\sqrt{3}\). With s = 4, the sides are 4, \(4\sqrt{3} \approx 6.93\), and 8.
Working from any side
In a 30-60-90 triangle, always find the short side first. If the hypotenuse is 14, the short side is 7 and the long leg \(7\sqrt{3}\). If the long leg is 9, the short side is \(\frac{9}{\sqrt{3}} = 3\sqrt{3}\) and the hypotenuse \(6\sqrt{3}\).
Equilateral triangle height
Cutting an equilateral triangle of side s in half gives two 30-60-90 triangles, so its height is \(\frac{\sqrt{3}}{2}s\) and its area is \(\frac{\sqrt{3}}{4}s^2\). An equilateral triangle with side 6 has height \(3\sqrt{3} \approx 5.2\) and area \(9\sqrt{3} \approx 15.6\).
Why the ratios matter
These triangles connect to the trigonometric values of 30, 45, and 60 degrees, which are used throughout trigonometry, physics, and engineering. Knowing the side ratios means you can write exact values, such as the sine of 30 degrees being one half, without a calculator.
In design and construction
Set squares used in technical drawing are often 45-45-90 and 30-60-90 triangles. Hexagonal tiles, bolts, and honeycomb cells are made of equilateral triangles, so 30-60-90 triangles give their dimensions.
Worked example
A regular hexagon with side 10 cm splits into six equilateral triangles. Each has area \(\frac{\sqrt{3}}{4}(100) = 25\sqrt{3}\), so the hexagon's area is \(150\sqrt{3} \approx 259.8\) square centimetres. Its distance across the flats is twice a triangle's height: \(2 \times 5\sqrt{3} = 10\sqrt{3} \approx 17.3\) cm, the size of spanner needed for a hexagonal nut of that size.
Key takeaways
A 45-45-90 triangle has equal legs and a hypotenuse root 2 times a leg; divide the hypotenuse by root 2 to get a leg. A 30-60-90 triangle, half an equilateral triangle, has sides in the ratio 1 : root 3 : 2, with the short side opposite 30 degrees; find the short side first. An equilateral triangle of side s has height root 3 over 2 times s and area root 3 over 4 times s squared. These ratios give exact trigonometric values and dimensions of hexagons and set squares.
Flashcard
Isosceles right triangle
Check your understanding
In a 30-60-90 triangle, the shortest side is 4. What is the hypotenuse?