A 45-45-90 triangle is half a square. Its sides are in ratio 1 : 1 : √2.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The other one
A 30-60-90 triangle is half an equilateral triangle, with sides 1 : √3 : 2.
Shortest opposite smallest
The side opposite 30° is the shortest, and it is exactly half the hypotenuse.
Why memorise them
They appear constantly, and knowing them turns a calculator problem into mental arithmetic.
The 45-45-90 triangle
An isosceles right triangle has legs in ratio 1 : 1 and hypotenuse √2 times a leg. It is half a square cut along the diagonal, which is where the √2 comes from — it is the diagonal of a unit square.
The 30-60-90 triangle
Sides are in ratio 1 : √3 : 2. It is half an equilateral triangle cut down the middle, which is why the shortest side is exactly half the hypotenuse. Reconstructing it from the equilateral triangle means you can never quite forget it.
Exact values, no calculator
These two triangles give sin 30° = 1/2, cos 45° = √2/2 and tan 60° = √3 exactly. Exact values matter in calculus and physics, where a rounded decimal early in a derivation destroys the result.
Recognising them in a problem
Any equilateral triangle, square diagonal, or regular hexagon contains these triangles. Spotting one lets you write down lengths immediately instead of reaching for trigonometry.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.71
- cos — adjacent over hypotenuse0.71
- tan — opposite over adjacent1
- The marked angle45°
The hypotenuse is not given. It comes from 5² + 5² = 50, whose square root is 7.07.
Step 3: Watch an Example
One step at a time.
Watch Amara Use the 30-60-90
A 30-60-90 triangle has a hypotenuse of 10.
- Step 1
The side opposite 30° is half the hypotenuse.
Step 4: Your Turn
Practice makes it stick.
Half the Hypotenuse
Problem 1 of 2
A 30-60-90 with hypotenuse 12. What is the shortest side?
The Square
Problem 2 of 2
A 45-45-90 with legs 1. What is the hypotenuse squared?
Ratios Worth Knowing
1 of 8
30-60-90 with hypotenuse 20. Shortest side?
2 of 8
30-60-90 with shortest side 7. Hypotenuse?
3 of 8
45-45-90 with legs 6. What is the hypotenuse squared?
4 of 8
In a 45-45-90, are the two legs equal? 1 for yes, 0 for no.
5 of 8
Match each triangle to its side ratio.
Tap a card on the left to start.
6 of 8
Which angle faces the shortest side in a 30-60-90? Give the degrees.
7 of 8
A 45-45-90 is half of which shape? 1 for square, 2 for equilateral triangle.
8 of 8
30-60-90 with hypotenuse 30. Shortest side?
Step 5: Quick Check
Show what you know.
Question 1 of 1
30-60-90 with hypotenuse 18. Shortest side?
What You Learned
- A 45-45-90 triangle has sides in ratio 1 : 1 : √2.
- A 30-60-90 has sides 1 : √3 : 2, and the shortest side is half the hypotenuse.
- Both come from cutting a familiar shape in half.