Two triangles whose sides you can write down without measuring anything.
Find the hypotenuse
A 45°-45°-90° triangle has legs of 7 cm. How long is the hypotenuse?
In a 45-45-90 triangle the hypotenuse is always the leg times √2, so 7√2 ≈ 9.9 cm. Check with Pythagoras: 49 + 49 = 98, and √98 = 7√2 ✓ C doubles the leg, which would make the hypotenuse longer than the two legs combined — impossible in any triangle. D uses √3, which belongs to the 30-60-90.
| Triangle | Sides in the order of the angles |
|---|---|
| 45-45-90 | x : x : x√2 |
| 30-60-90 | x : x√3 : 2x |
List the sides in the same order as the angles they face. In a 30-60-90, the shortest side faces the 30°, the middle faces the 60°, and the hypotenuse faces the right angle.
Two equal legs and a right angle: an isosceles right triangle. Its angles are always 45°, 45° and 90°.
cut a square along its diagonal
two equal legs
x² + x² = h²
Pythagoras
h² = 2x², h = x√2
there is the √2
Not a formula to memorise so much as one you can rebuild in fifteen seconds if it slips.
cut an equilateral triangle in half
side 2x
the base halves to x
the short side
x² + h² = (2x)² = 4x²
Pythagoras
h² = 3x², h = x√3
there is the √3
Not twice the middle one. The short side faces the 30° angle, and everything else is measured from it — getting that wrong scrambles the whole triangle.
30-60-90, hypotenuse 20
find the other two
2x = 20, so x = 10
the short side
middle = 10√3 ≈ 17.3
x√3
45-45-90, hypotenuse 12
find a leg
x√2 = 12, x = 12√2
rationalise
= 6√2 ≈ 8.49
shorter than 12 ✓
Always find x first — the short side in a 30-60-90, a leg in a 45-45-90. Every other side is written in terms of it.
These are also where the exact trigonometric values come from: sin 30° = 12, sin 45° = 1√2, and tan 60° = √3 all read straight off these two triangles.
The middle side of a 30-60-90 is x√3. Divide it by √3 to get x, then double x for the hypotenuse.
In a 30-60-90 triangle the longer leg is 11√3. Find the other two sides.
x√3 = 11√3
the middle side is x√3
x = 11
the short side
hypotenuse = 2x = 22
check: 121 + 363 = 484 = 22² ✓
When the longer leg has no √3 in it: the longer leg is 6.
x√3 = 6, x = 6√3
divide by √3
x = 6√33 = 2√3
rationalise
hypotenuse = 4√3 ≈ 6.93
longer than 6 ✓
Doubling the longer leg gives a "hypotenuse" of 22√3 ≈ 38.1 for the first triangle — far too long, since 22√3 is twice 11√3. Only the SHORT side doubles to the hypotenuse; reach it first.
The same two ratios, inside other shapes and with awkward starting numbers.
An equilateral triangle has sides of 10 cm. Find its height and area.
short side 5, height 5√3
half of it is a 30-60-90
area = (12)(10)(5√3) = 25√3
≈ 43.3 cm²
The height is the middle side, not the hypotenuse; the full side is the hypotenuse.
A square has a diagonal of 8 cm. Find its side and area.
x√2 = 8, x = 8√2 = 4√2
the diagonal is the hypotenuse
area = (4√2)² = 32 cm²
16 × 2
Going from the diagonal back to the side means dividing by √2, not multiplying.
In a 30-60-90 triangle the short side is √3. Find the other two sides.
longer leg = √3 × √3 = 3
√3 times √3 is a whole number
hypotenuse = 2√3 ≈ 3.46
check: 3 + 9 = 12 = (2√3)² ✓
The short side need not be a whole number; the pattern x : x√3 : 2x holds anyway.
Expect one side of a 45-45-90 or 30-60-90 triangle given and another asked — often the diagonal of a square or the height of an equilateral triangle — with answers left in √2 or √3 form. The fastest route is always to find x first (a leg, or the short side), then write every other side from it. The 10-second check: the hypotenuse must be the longest side, about 1.4 times a leg or exactly twice the short side.
Find the sides
In a 30°-60°-90° triangle the side opposite the 30° angle is 6 cm. How long is the hypotenuse?
The side opposite 30° is the short side, x = 6, and the hypotenuse is 2x = 12 cm. B is the middle side, opposite the 60° angle. C uses √2, which belongs to the 45-45-90. Check: 6² + (6√3)² = 36 + 108 = 144 = 12² ✓
Hint: the side facing the smallest angle is the shortest, and the hypotenuse is exactly twice it.
