Most right triangles force you to grind through the Pythagorean theorem or look up trig ratios. But two special triangles — the 45-45-90 and the 30-60-90 — have fixed angle combinations that lock their sides into predictable ratios. Memorize two patterns and you can write down exact side lengths instantly, radicals and all, with no calculator.
What Makes These Triangles “Special”?
- 45-45-90 triangle (isosceles right triangle): cut a square in half diagonally and you get this — two equal 45° angles and two equal legs.
- 30-60-90 triangle (scalene right triangle): slice an equilateral triangle down its line of symmetry and you get this — angles of 30°, 60°, and 90°.
If your foundation on ordinary right triangles is shaky, review our guide on pythagorean theorem step-by-step first, or see how flat shapes combine in our walkthrough on how to find area and perimeter of polygons. You can also verify triangle values and radical simplifications with the automated CalcSolver tool.
The 45-45-90 Triangle: Rules and Side Ratios
Because the two acute angles are equal, the sides opposite them are equal too — it’s isosceles:
- Leg 1: x
- Leg 2: x
- Hypotenuse: x√2
The rule: to find the hypotenuse from a leg, multiply the leg by √2. To find a leg from the hypotenuse, divide the hypotenuse by √2.
Worked Example 1: Finding the hypotenuse
A 45-45-90 triangle has a leg of 7 inches. Find the hypotenuse.
- Identify the given side: leg = x = 7
- Apply the rule: hypotenuse = x√2
- Compute: hypotenuse = 7√2 inches (≈ 7 × 1.414 ≈ 9.90 inches)
Worked Example 2: Finding a leg from the hypotenuse
A 45-45-90 triangle has a hypotenuse of 10 cm. Find each leg.
- Set up: x√2 = 10
- Divide by √2: x = 10 ÷ √2
- Rationalize (multiply top and bottom by √2): x = (10√2) ÷ 2 = 5√2 cm (≈ 7.07 cm)
Both legs measure 5√2 cm. Check: 5√2 × √2 = 5 × 2 = 10 ✓.
The 30-60-90 Triangle: Rules and Side Ratios
This triangle’s sides follow a strict 1 : √3 : 2 proportion:
- Shorter leg (opposite 30°): x
- Longer leg (opposite 60°): x√3
- Hypotenuse (opposite 90°): 2x
The key: always find the shorter leg (x) first. Then double it for the hypotenuse, or multiply by √3 for the longer leg.
The side across from the smallest angle is always the shortest side, and the side across from the 90° angle is always the longest. Match each side to its angle before touching the ratios.
Worked Example 3: Finding sides from the shorter leg
A 30-60-90 triangle has a shorter leg of 4 meters.
- Identify x: shorter leg = x = 4
- Longer leg: x√3 = 4√3 meters (≈ 6.93 m)
- Hypotenuse: 2x = 2 × 4 = 8 meters
Worked Example 4: Finding sides from the longer leg
A 30-60-90 triangle has a longer leg of 9 feet.
- Set up: x√3 = 9
- Solve for x: x = 9 ÷ √3
- Rationalize: x = (9√3) ÷ 3 = 3√3 feet (≈ 5.20 ft)
- Hypotenuse: 2x = 2 × 3√3 = 6√3 feet (≈ 10.39 ft)
Quick Reference
- 45-45-90: legs x and x → hypotenuse x√2
- 30-60-90: short leg x → long leg x√3 → hypotenuse 2x
3 Mistakes to Avoid
- Swapping √2 and √3: 45-45-90 uses √2 (two equal angles); 30-60-90 uses √3 (three different angles). That’s the memory hook.
- Making the hypotenuse x√3 in a 30-60-90: the hypotenuse is simply 2x — double the short leg. The √3 belongs only to the longer leg.
- Leaving a radical in the denominator: answers like 10/√2 must be rationalized to 5√2. Standard form never has a square root on the bottom.
These two patterns also shortcut trigonometry: sin 45° = √2/2, sin 30° = 1/2, cos 30° = √3/2 — all of them fall straight out of the side ratios above.