Solve a special right triangle
Solved from two angles and a side that isn’t between them (AAS).
| Part | Value | Exact |
|---|---|---|
| Side a given | 5 | |
| Side b | 8.6603 | 5√3 |
| Side c | 10 | |
| Angle A given | 30° | |
| Angle B | 60° | |
| Angle C given | 90° | |
| Area | 21.6506 | 25√3/2 |
| Perimeter | 23.6603 | 15 + 5√3 |
| Semiperimeter s | 11.8301 | (15 + 5√3)/2 |
| Height to a (h_a) | 8.6603 | 5√3 |
| Height to b (h_b) | 5 | |
| Height to c (h_c) | 4.3301 | 5√3/2 |
| Inradius r | 1.8301 | (5√3 − 5)/2 |
| Circumradius R | 5 |
Working
- Angle sum. B = 180° − C − A = 180° − 90° − 30° = 60°
- Law of sines. b = a sin B / sin A = 5 × sin 60° / sin 30° = 8.6603 (exactly 5√3)
- Law of sines. c = a sin C / sin A = 5 × sin 90° / sin 30° = 10
- Perimeter. a + b + c = 5 + 8.6603 + 10 = 23.6603 (exactly 15 + 5√3)
- Heron's formula. s = (a + b + c)/2 = 11.8301; area = √(s(s − a)(s − b)(s − c)) = √(11.8301 × 6.8301 × 3.1699 × 1.8301) = 21.6506 (exactly 25√3/2)
- Area from two sides. ½ab sin C = ½ × 5 × 8.6603 × sin 90° = 21.6506, the same area
- Heights. h = 2 × area / base: h_a = 8.6603, h_b = 5, h_c = 4.3301
- Inradius and circumradius. r = area / s = 1.8301; R = a / (2 sin A) = 5
The 30-60-90 triangle
Its angles are 30°, 60° and 90°, and its sides are always in the ratio 1 : √3 : 2. The short leg faces 30°, the long leg faces 60°, and the hypotenuse is twice the short leg.
Know any one side and you know them all. Pick the triangle and the side above, type its length, and the other two come out exact, such as 5√3 rather than 8.6603.
Roots work as input too. A long leg of 4√3 gives a short leg of 4 and a hypotenuse of 8.
30-60-90 rules
| You know | Short leg | Long leg | Hypotenuse |
|---|---|---|---|
| Short leg x | x | x√3 | 2x |
| Long leg y | y/√3 = y√3/3 | y | 2y√3/3 |
| Hypotenuse h | h/2 | h√3/2 | h |
Where 1 : √3 : 2 comes from
Start with an equilateral triangle of side 2. Every angle is 60°. Cut it in half with a line from the top straight down to the middle of the base.
Each half becomes a right triangle with a hypotenuse of 2, a base of 1 and angles of 30°, 60° and 90°. Pythagoras gives the height, √(2² − 1²) = √3. That's the whole ratio.
Exact trig values come from it too. sin 30° = 1/2, cos 30° = √3/2 and tan 60° = √3, all read off the same three sides.
Worked examples
A hypotenuse of 10. Halve it for the short leg, 5. Multiply that by √3 for the long leg, 5√3, about 8.6603.
A long leg of 6. Divide by √3 for the short leg, 6/√3 = 2√3, about 3.4641. Double that for the hypotenuse, 4√3, about 6.9282.
A short leg of 7. Multiplying by √3 gives a long leg of 7√3, about 12.1244, and doubling gives a hypotenuse of 14. The area is half the product of the legs, ½ × 7 × 7√3 = 49√3/2, about 42.4352.
The 45-45-90 triangle
Our second special right triangle has angles of 45°, 45° and 90°. Its two legs are equal, so it's isosceles, and the sides run 1 : 1 : √2.
Half a square, in other words. Cut a square of side 1 along a diagonal and each half has legs of 1 and a hypotenuse of √(1² + 1²) = √2.
To solve one, switch Triangle to 45-45-90 in the calculator above and pick a leg or the hypotenuse.
45-45-90 rules
| You know | Each leg | Hypotenuse |
|---|---|---|
| A leg x | x | x√2 |
| Hypotenuse h | h/√2 = h√2/2 | h |
45-45-90 examples
Legs of 7 give a hypotenuse of 7√2, about 9.8995.
A hypotenuse of 10 gives legs of 10/√2 = 5√2 each, about 7.0711. The area is ½ × (5√2)² = 25.
A square tile 30 cm on a side has a diagonal of 30√2 cm, about 42.43 cm, the hypotenuse of the two triangles the diagonal makes.
Exact trig values from the two triangles
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Where they turn up
The unit circle uses them for the points at 30°, 45° and 60°, and every other angle on it that's a multiple of 30° or 45° is a reflection of one of them.
For a right triangle with other angles, use the right triangle calculator.
Questions
What is the ratio of a 30-60-90 triangle?
1 : √3 : 2. The short leg, opposite 30°, is 1 part, the long leg is √3 parts and the hypotenuse is 2.
What is the ratio of a 45-45-90 triangle?
1 : 1 : √2. The legs are equal and the hypotenuse is a leg times √2.
How do I find the short leg from the long leg?
Divide by √3. A long leg of 6 gives 6/√3 = 2√3.
Why is the hypotenuse twice the short leg?
The 30-60-90 triangle is half an equilateral triangle. The hypotenuse is a full side of it and the short leg is half a side.
Which is the long leg?
Whichever faces the 60° angle. Bigger angles always face longer sides.
What are the special right triangles?
Usually the 30-60-90 and the 45-45-90, whose angles make the sides come out in simple ratios. Some lists add side-based ones such as 3, 4, 5.
Sources
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