The 30-60-90 Triangle
Two triangles have side ratios exact enough to write down, which makes them the only ones you can solve in your head. Both come from cutting a shape you already know in half.
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Every right triangle has ratios between its sides. For almost all of them those ratios are ugly decimals that need a calculator. For exactly two, they are short exact expressions you can write down - which is why those two turn up in question after question.
Switch between them below, and change the size.
Move the slider. Every side changes and the three angles do not - which is why one remembered ratio settles a triangle of any size.
Where the 30-60-90 comes from
Start with an equilateral triangle: three equal sides, three 60° angles. Cut it straight down the middle, from the top vertex to the midpoint of the base.
The cut is perpendicular to the base, so you have a right angle. One of the 60° angles at the base survives untouched. The 60° at the top is split in half, so it becomes 30°. That is your 30-60-90 triangle, and it came from a shape whose sides you already knew.
Now read the sides off. If the equilateral triangle had sides of 2, the hypotenuse of the half is still 2 and the base has been halved to 1. The height comes from Pythagoras:
h^2 = 2^2 - 1^2 = 3, so h = sqrt(3).
Sides of 1, sqrt(3) and 2 — that is the ratio, and it is exact because it came from an exact construction rather than a measurement.
Which side goes where
The rule that prevents most mistakes: the biggest side is opposite the biggest angle.
| angle | side | length |
|---|---|---|
| 30° | short leg | x |
| 60° | long leg | x√3 |
| 90° | hypotenuse | 2x |
So the hypotenuse is twice the short side, never twice the long one. And since sqrt(3) is about 1.73, the long leg sits between the two - which is a quick sanity check on any answer you write.
The 45-45-90 is a square cut in half
Same trick, different shape. Take a square with sides of 1 and cut it along its diagonal. Both remaining angles are 45°, the two legs are equal, and the diagonal is the hypotenuse:
d^2 = 1^2 + 1^2 = 2, so d = sqrt(2).
Ratio 1 : 1 : sqrt(2). This one shows up wherever squares do - diagonals of screens, tiles, chessboards.
Finding a missing side
Always work back to the short side, then scale up.
Given the hypotenuse. A 30-60-90 triangle with hypotenuse 14: halve it, so the short side is 7, and the long leg is 7sqrt(3).
Given the long leg. A 30-60-90 triangle with long leg 9: divide by sqrt(3) to get the short side, 3sqrt(3), so the hypotenuse is 6sqrt(3).
Given a leg of a 45-45-90. The other leg matches it, and the hypotenuse is that leg times sqrt(2).
Move the slider in the figure while you read those - the arithmetic above is just the readout at three different sizes.
Try one
A 30-60-90 triangle has a short side of 6. What is its long leg?
Leave it as 6sqrt(3) rather than 10.39. The exact form is what the next step of a problem can actually use, and it is what an examiner is looking for.
The exact trig values fall straight out
Because these two triangles have exact sides, they have exact sine and cosine values - which is where the numbers in every unit-circle table come from:
| angle | sin | cos |
|---|---|---|
| 30° | 1/2 | √3/2 |
| 45° | √2/2 | √2/2 |
| 60° | √3/2 | 1/2 |
Three distinct values across the whole first quadrant, and 30° and 60° are the same pair swapped - which is exactly what you would expect from one triangle read from its two different acute angles.
Common questions
- What are the sides of a 30-60-90 triangle?
- In the ratio 1 : √3 : 2. The shortest side is opposite the 30 degree angle, the longest is the hypotenuse and is exactly twice the shortest, and the remaining side is the shortest times the square root of 3. Every 30-60-90 triangle in existence has those proportions, whatever its size.
- Where does the 30-60-90 ratio come from?
- Cut an equilateral triangle in half down its height. The two 60 degree angles at the base survive, the 60 at the top is halved to 30, and the cut makes a right angle. The base is halved too, so the short side is exactly half the original side - which is why the hypotenuse is twice the short side. The height then follows from Pythagoras.
- What are the sides of a 45-45-90 triangle?
- In the ratio 1 : 1 : √2. Its two legs are equal because the two 45 degree angles are equal, and the hypotenuse is a leg times the square root of 2 - which is Pythagoras applied to a square cut along its diagonal, since that is exactly what the triangle is.
- How do I find a missing side in a special triangle?
- Find the short side first, then scale. If you are given the hypotenuse of a 30-60-90 triangle, halve it to get the short side; if you are given the long leg, divide by the square root of 3. Once the short side is known, the other two are it times √3 and it times 2.
- Why should I memorise these when a calculator exists?
- Because the answers are exact and a calculator's are not. A calculator gives 8.660254 where the truth is 5√3, and an exact value is what later work needs - simplifying, cancelling, or leaving an answer in surd form. Exam questions also use these two triangles constantly precisely because they can be answered without a calculator.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
