Special Right Triangles: 30-60-90 and 45-45-90 Without a Calculator

The two right-triangle shapes the Texas CBE loves to test. Memorize one ratio for each and skip the Pythagorean arithmetic on roughly 1 in 6 Geometry questions.

7 분 TEKS 7B,9B 기하학

Why “special” means “skip the calculator”

The Pythagorean theorem always works on a right triangle — but there are two specific shapes that show up so often the CBE designs questions around them. For these, you don't need a² + b² = c². You just need a single multiplication.

Big idea

The 30-60-90 and 45-45-90 triangles have fixed side ratios. Memorize the ratio once, and every problem of either type becomes a 5-second answer.

Meet the two shapes

Two triangles, two ratios. Memorize once — use forever.

The 45-45-90 rule

This is half a square cut along the diagonal. The two legs are equal, and the hypotenuse is just leg × √2.

Both legs: x Hypotenuse: x√2 If you know the leg, multiply by √2 to get the hypotenuse. If you know the hypotenuse, divide by √2 to get a leg.

Worked example

Leg = 8 Hypotenuse = 8 √2 ≈ 11.3 Want it without a calculator? Just leave it as 8√2 — the CBE often lists answers in radical form.

Where you'll see it

Squares (the diagonal of any square is side × √2), isosceles right triangles, 45° ramps, and any “diagonal of a square” problem. If you see “square” and “diagonal” in the same sentence, this rule applies.

Practice

Try a 45-45-90 problem

In a 45-45-90 triangle, if one leg is 8, what is the length of the hypotenuse?

Open the question →

The 30-60-90 rule

This is half an equilateral triangle cut down the middle. The three sides have a strict ratio: 1 : √3 : 2. Where you stand on that ratio depends on which side you call x.

Short leg (opposite 30°): x Long leg (opposite 60°): x√3 Hypotenuse (opposite 90°): 2x The shortest side is always opposite the smallest angle. The longest side is always opposite the right angle.

Anchor on the short leg

The fastest way to solve any 30-60-90 problem: find the short leg first (call it x), then the other two sides are x√3 and 2x. Whatever you're given, work back to x.

Worked example — given the short leg

Short leg = 7 ⇒ x = 7 Long leg = 7√3 ≈ 12.1 Hypotenuse = 2 · 7 = 14

Practice

Try the “given the short leg” case

In a 30-60-90 triangle, the shorter leg is 7. What is the hypotenuse?

Open the question →

Worked example — given the hypotenuse

Same triangle, harder direction. Hypotenuse = 16, find the legs.

Hypotenuse = 2x = 16 ⇒ x = 8 Short leg (a) = x = 8 Long leg (b) = x√3 = 8√3 ≈ 13.86 Always solve for x first — everything else falls out.

Practice

Solve from the hypotenuse

In the 30-60-90 triangle, the hypotenuse is 16. Find the lengths of the two legs.

Open the question →

The most common mix-up

Don't swap √2 and √3

Both rules involve a square root, and students mix them up under time pressure. Anchor the difference:

45-45-90 → uses √2 → comes from a square cut diagonally (2 sides of a square)
30-60-90 → uses √3 → comes from an equilateral triangle cut in half (3 angles of a triangle)

Quick reference table

The whole topic on one page. Tape this to your study desk.

3-second recap

45-45-90 → legs equal (x), hypotenuse x√2. Think “square's diagonal.”
30-60-90 → ratio x : x√3 : 2x. Anchor on the short leg.
If the answer choices use radicals (√2, √3), you almost certainly have a special right triangle problem.
Both shapes still satisfy a² + b² = c² — the ratios are just shortcuts so you skip the squaring.

Up next: Triangle Congruence & Similarity — the rules that decide whether two triangles are the same shape, the same size, or proportional.