The two special right triangles are the 45-45-90 triangle, with sides in the ratio ( 1 : 1 : sqrt2 ) (hypotenuse = leg × ( sqrt2 )), and the 30-60-90 triangle, with sides in the ratio ( 1 : sqrt3 : 2 ) (hypotenuse = 2 × short leg, long leg = short leg × ( sqrt3 )). As a result, they give the exact trig values of ( 30^circ ), ( 45^circ ) and ( 60^circ ).
Special right triangles are right triangles whose angles make the side lengths follow a fixed, simple ratio. Know the ratio and you can find every side from just one, without trig tables or a calculator. There are two:
| Triangle | Angles | Side ratio |
|---|---|---|
| Isosceles right | ( 45^circ )-( 45^circ )-( 90^circ ) | ( 1 : 1 : sqrt2 ) |
| Half-equilateral | ( 30^circ )-( 60^circ )-( 90^circ ) | ( 1 : sqrt3 : 2 ) |
They also give the exact values ( sin 30^circ = frac12 ), ( cos 45^circ = frac{sqrt2}{2} ) and the rest of the trigonometric functions table, and they’re the reason the unit circle has the coordinates it does.
The 45-45-90 triangle
The two legs are equal, and the hypotenuse is a leg times ( sqrt2 ):
$$text{leg} : text{leg} : text{hypotenuse} = x : x : xsqrt2$$
Where it comes from. Cut a square of side x along its diagonal. Each half is a right triangle with two ( 45^circ ) angles and legs x. The Pythagorean theorem gives the hypotenuse: ( sqrt{x^2 + x^2} = sqrt{2x^2} = xsqrt2 ).
Example 1: leg to hypotenuse
The legs are 5. The hypotenuse is ( 5sqrt2 approx 7.071 ).
Example 2: hypotenuse to leg
The hypotenuse is 10. Consequently, each leg is ( frac{10}{sqrt2} = frac{10sqrt2}{2} = 5sqrt2 approx 7.071 ).
Example 3: the diagonal of a square
A square tile has side 30 cm. Its diagonal is ( 30sqrt2 approx 42.43 ) cm.
The 30-60-90 triangle
The short leg (opposite ( 30^circ )) is x, the long leg (opposite ( 60^circ )) is ( xsqrt3 ), and the hypotenuse is ( 2x ):
$$text{short leg} : text{long leg} : text{hypotenuse} = x : xsqrt3 : 2x$$
Where it comes from. Take an equilateral triangle with side ( 2x ) (all angles ( 60^circ )) and draw a height from the top. It splits the base into two halves of length x and also splits the top angle into two ( 30^circ ) angles. Each half is a 30-60-90 triangle with hypotenuse ( 2x ) and short leg x. The height is ( sqrt{(2x)^2 – x^2} = sqrt{3x^2} = xsqrt3 ).
Memory check. The smallest side is opposite the smallest angle. Also, the hypotenuse is always exactly twice the short leg.
Example 4: short leg given
The short leg is 4. The long leg is ( 4sqrt3 approx 6.928 ) and the hypotenuse is 8.
Example 5: hypotenuse given
The hypotenuse is 14. Its short leg is half of it, 7, and the long leg is ( 7sqrt3 approx 12.12 ).
Example 6: long leg given
The long leg is 9. Then ( xsqrt3 = 9 ), so the short leg is ( x = frac{9}{sqrt3} = 3sqrt3 approx 5.196 ). As a result, the hypotenuse is ( 6sqrt3 approx 10.39 ).
Example 7: height of an equilateral triangle
An equilateral triangle has side 12. Its height is the long leg of a 30-60-90 triangle with short leg 6, so ( h = 6sqrt3 approx 10.39 ), and its area is ( frac12 cdot 12 cdot 6sqrt3 = 36sqrt3 approx 62.35 ). In general, an equilateral triangle with side s has area ( frac{sqrt3}{4}s^2 ).
Exact trig values from special triangles
Using SOH-CAH-TOA on the two triangles (with ( x = 1 )):
| Angle | ( sin ) | ( cos ) | ( tan ) |
|---|---|---|---|
| ( 30^circ ) | ( frac12 ) | ( frac{sqrt3}{2} ) | ( frac{1}{sqrt3} = frac{sqrt3}{3} ) |
| ( 45^circ ) | ( frac{1}{sqrt2} = frac{sqrt2}{2} ) | ( frac{sqrt2}{2} ) | 1 |
| ( 60^circ ) | ( frac{sqrt3}{2} ) | ( frac12 ) | ( sqrt3 ) |
For example, in the 30-60-90 triangle with sides 1, ( sqrt3 ), 2, the side opposite ( 30^circ ) is 1 and the hypotenuse is 2, so ( sin 30^circ = frac12 ).
Check with the triangle calculator
Enter two legs of 5 and a ( 90^circ ) angle between them; the calculator confirms the hypotenuse ( 5sqrt2 approx 7.071 ) and the two ( 45^circ ) angles. For example, try ( a = 4 ), ( b = 4sqrt3 ) (type 4sqrt(3)) for a 30-60-90 triangle.
The full-page triangle calculator solves any triangle, and the trig calculator gives exact values for every special angle.
Other common right triangles
These aren’t “special” in the same angle sense, but their sides are whole numbers (Pythagorean triples), so they show up constantly:
| Triple | Check |
|---|---|
| 3, 4, 5 | ( 9 + 16 = 25 ) |
| 5, 12, 13 | ( 25 + 144 = 169 ) |
| 8, 15, 17 | ( 64 + 225 = 289 ) |
| 7, 24, 25 | ( 49 + 576 = 625 ) |
Multiples work too: 6-8-10 and 9-12-15 are scaled 3-4-5 triangles. The distance formula often produces them.
Where you’ll use special right triangles
- Trigonometry and the unit circle. Every exact trig value at multiples of ( 30^circ ) and ( 45^circ ) comes from these two triangles.
- Geometry. Diagonals of squares, heights of equilateral triangles and hexagons, and areas of regular polygons.
- Calculus. Evaluating limits and integrals at special angles, such as ( int_0^{pi/4}sec^2 x,dx = tanfrac{pi}{4} = 1 ) (see the integral of sec²x), and setting up related rates problems.
- Physics. Splitting a force or velocity at ( 30^circ ), ( 45^circ ) or ( 60^circ ) into components, as in projectile motion.
For triangles without a right angle, use the law of sines and cosines.
Common mistakes
- Swapping the legs in a 30-60-90 triangle. The ( sqrt3 ) side is opposite ( 60^circ ), the longer leg, not the shorter one.
- Multiplying instead of dividing. From the hypotenuse of a 45-45-90 triangle, divide by ( sqrt2 ) to get a leg.
- Hypotenuse ( sqrt3 ) times the short leg. It’s 2 times the short leg.
- Not rationalizing. Many courses want ( frac{sqrt2}{2} ) rather than ( frac{1}{sqrt2} ); they’re equal.
Practice problems
- A 45-45-90 triangle has legs of 8. Find the hypotenuse.
- A 45-45-90 triangle has hypotenuse ( 6sqrt2 ). Find a leg.
- A 30-60-90 triangle has short leg 10. Find the other sides.
- A 30-60-90 triangle has hypotenuse 20. Find both legs.
- A 30-60-90 triangle has long leg ( 5sqrt3 ). Find the hypotenuse.
- Find the height of an equilateral triangle with side 8.
Answers: (1) ( 8sqrt2 ); (2) 6; (3) long leg ( 10sqrt3 ), hypotenuse 20; (4) 10 and ( 10sqrt3 ); (5) 10; (6) ( 4sqrt3 approx 6.93 ).
FAQ
What are the special right triangles?
The 45-45-90 triangle, with sides in the ratio ( 1 : 1 : sqrt2 ), and the 30-60-90 triangle, with sides in the ratio ( 1 : sqrt3 : 2 ).
What is the rule for a 30-60-90 triangle?
The hypotenuse is twice the short leg, and the long leg is the short leg times ( sqrt3 ).
What is the rule for a 45-45-90 triangle?
The legs are equal, and the hypotenuse is a leg times ( sqrt2 ).
Why are they called special?
Because their angles force simple, exact side ratios, so you can find every side and trig value without a calculator.
Is 3-4-5 a special right triangle?
It’s a Pythagorean triple with whole-number sides, but its angles (about ( 36.87^circ ) and ( 53.13^circ )) aren’t special, so it isn’t usually counted as one.
Further reading
- Special right triangle (Wikipedia) — angle-based and side-based special triangles with proofs.
- Cheat Sheets and Tables (Paul’s Online Math Notes) — a printable trig cheat sheet with the special-angle values and identities.
Calculators for this topic
Keep learning
The Unit Circle: Angles, Coordinates and Trig Values
The unit circle explained: why the coordinates are cos and sin, every special angle in degrees and radians, tangent on the unit circle, and memory tricks.
Trigonometric Functions: Sin, Cos, Tan, Table and Graphs
The six trig functions with SOH-CAH-TOA and the unit circle, a table of exact values (sin 30, cos 180, sec 45), graphs, period and amplitude.