The 30-60-90 triangle formula: the sides are in the ratio 1 : √3 : 2, so if the short leg is x, the long leg is x√3 and the hypotenuse is 2x. This 30-60-90 triangle calculator solves the whole triangle from any one measure. That can be a leg, the hypotenuse, the area, the perimeter, the altitude, the inradius or the circumradius. Every answer appears in exact radical form and as a decimal, next to a diagram drawn to scale. Below you’ll find the formulas, two proofs, a comparison with the 45-45-90 triangle, and practice problems with answers.
How to use the 30-60-90 triangle calculator
- Choose what you know, such as “Long leg” or “Area”.
- Type its value. Decimals (7.5), fractions (15/2) and radicals (4√3, 4 sqrt3 or √3/2) all work.
- Read all eight measures in the table. The exact column keeps √3 symbolic, so you can copy an answer in simplest radical form.
- Check the diagram. The triangle, its altitude to the hypotenuse and its incircle are drawn to scale.
30-60-90 triangle formula and side ratios
OpenStax Contemporary Mathematics puts the rule this way: there is a set ratio of one side to another in the 30°-60°-90° triangle, given as 1 : √3 : 2, or x : x√3 : 2x. The shortest side is opposite the 30° angle, the middle side is opposite 60°, and the hypotenuse is opposite 90°. To solve any 30-60-90 triangle, first turn the side you know into x, then scale.
| You know | Short leg a | Long leg b | Hypotenuse c |
|---|---|---|---|
| Short leg x | x | x√3 | 2x |
| Long leg y | y/√3 = y√3/3 | y | 2y/√3 = 2y√3/3 |
| Hypotenuse h | h/2 | h√3/2 | h |
Area, perimeter, altitude, inradius and circumradius
| Measure | Exact formula | Decimal multiplier | Why |
|---|---|---|---|
| Area | (√3/2)x² | 0.866x² | ½ × leg × leg |
| Perimeter | x(3 + √3) | 4.7321x | x + x√3 + 2x |
| Altitude to hypotenuse | (√3/2)x | 0.866x | leg × leg ÷ hypotenuse |
| Inradius r | x(√3 − 1)/2 | 0.366x | (a + b − c)/2 |
| Circumradius R | x | 1x | hypotenuse ÷ 2 |
The area is half the base times the height. In a right triangle the two legs are a base and its height, so area = ½ · x · x√3. The same area can be written as ½ × hypotenuse × altitude. Setting the two equal gives the altitude to the hypotenuse as (x · x√3) ÷ 2x = x√3/2. For the incircle, the Wichita State notes use equal tangent lengths to show that c = (a − r) + (b − r), so r = (a + b − c)/2 for any right triangle. Here that is (x + x√3 − 2x)/2 = x(√3 − 1)/2. The circumradius follows from Euclid III.31: the angle in a semicircle is right, so the hypotenuse of a right triangle is a diameter of its circumscribed circle. That makes R = c/2 = x, exactly the short leg.
Proof: why the sides are 1 : √3 : 2
Proof 1: half of an equilateral triangle
Start with an equilateral triangle with side 2x. All of its angles are 60°. Drop the altitude from the top vertex to the base. By symmetry it meets the base at a right angle, splits the base into two pieces of length x, and splits the top 60° angle into two 30° angles. Each half is a right triangle with angles 30°, 60° and 90°, a hypotenuse of 2x (an original side), and a short leg of x (half the base). The Pythagorean theorem gives the remaining leg: b² = (2x)² − x² = 3x², so b = x√3. That proves the ratio x : x√3 : 2x. It also explains why the short leg is exactly half the hypotenuse, the fact behind OpenStax’s ladder example.
Proof 2: from trigonometry
In a right triangle with hypotenuse c, the side opposite an angle θ is c·sin θ and the adjacent side is c·cos θ. OpenStax Algebra and Trigonometry gives the special-angle values sin 30° = cos 60° = 1/2 and sin 60° = cos 30° = √3/2. With c = 2x, the side opposite 30° is 2x · ½ = x and the side opposite 60° is 2x · √3/2 = x√3, the same ratio. The argument also runs backwards: if a right triangle’s short leg is exactly half its hypotenuse, then sin θ = ½ for the smallest angle, so the angles must be 30°, 60° and 90°.
Worked examples
Hypotenuse 10 (OpenStax Example 10.67)
Set 2x = 10, so x = 5. The sides are 5, 5√3 ≈ 8.6603 and 10. The area is 25√3/2 ≈ 21.6506, the perimeter is 15 + 5√3 ≈ 23.6603, and the inradius is (5√3 − 5)/2 ≈ 1.8301.
A 40-foot ladder at 30° (OpenStax Example 10.68)
A 40-foot ladder leans against a wall and makes a 30° angle with the ground. The wall height it reaches is opposite the 30° angle, so it is the short leg. With 2x = 40, the ladder reaches x = 20 feet up the wall. The foot of the ladder stands 20√3 ≈ 34.641 ft from the wall.
Long leg 11
x = 11/√3 = 11√3/3 ≈ 6.3509. The hypotenuse is 22√3/3 ≈ 12.7017 and the area is ½ · 11 · 11√3/3 = 121√3/6 ≈ 34.9297. Rationalizing, multiplying top and bottom by √3, is what turns 11/√3 into the simplest radical form 11√3/3.
From the area: A = 50
Solve (√3/2)x² = 50 for x² = 100/√3, so x = √(100/√3) ≈ 7.5984. The long leg is ≈ 13.1607 and the hypotenuse ≈ 15.1967. This involves a fourth root of 3, so the calculator shows decimals only for area inputs.
30-60-90 vs 45-45-90 triangles
These are the two special right triangles whose sides you can find from a single length without a calculator. OpenStax describes the 45°-45°-90° triangle as having two equal angles beside the right angle, and therefore two equal opposite sides. Its ratio is 1 : 1 : √2, or x : x : x√2. In OpenStax Example 10.69, a leg of 3 gives a hypotenuse of 3√2.
| Property | 30-60-90 | 45-45-90 |
|---|---|---|
| Comes from | Half an equilateral triangle | Half a square (cut on a diagonal) |
| Side ratio | 1 : √3 : 2 | 1 : 1 : √2 |
| Hypotenuse | 2x | x√2 |
| Area | (√3/2)x² | x²/2 |
| Perimeter | x(3 + √3) | x(2 + √2) |
| Altitude to hypotenuse | (√3/2)x | (√2/2)x |
| Inradius | x(√3 − 1)/2 | x(2 − √2)/2 |
| Circumradius | x | (√2/2)x |
| Isosceles? | No, all sides differ | Yes, the legs are equal |
A quick way to tell them apart: if the hypotenuse is exactly twice one leg, the triangle is a 30-60-90. If the legs are equal, it is a 45-45-90. The 45-45-90 entries come from the same general rules used above: area = ½ · leg · leg, altitude = legs ÷ hypotenuse, r = (a + b − c)/2 and R = c/2.
Practice problems
- The short leg is 7. Find the other sides. Long leg 7√3 ≈ 12.1244, hypotenuse 14.
- The hypotenuse is 18. Find the long leg. Short leg 9, so the long leg is 9√3 ≈ 15.5885.
- The long leg is 12. Find the short leg in simplest radical form. 12/√3 = 12√3/3 = 4√3 ≈ 6.9282. The hypotenuse is 8√3.
- The perimeter is 12. Find the short leg. x = 12/(3 + √3) = (18 − 6√3)/3 = 6 − 2√3 ≈ 2.5359.
- The inradius is 2. Find the hypotenuse. x = 2 · 2/(√3 − 1) = 2(√3 + 1) ≈ 5.4641, so c = 4 + 4√3 ≈ 10.9282.
- An equilateral triangle has side 10. Find its height and area. The height is the long leg of a 30-60-90 with hypotenuse 10: 5√3 ≈ 8.6603. The area is ½ · 10 · 5√3 = 25√3 ≈ 43.3013.
Enter any of these in the calculator to check your working. The exact column should match the radical answers above.
Common mistakes
- Putting √3 on the wrong leg. The √3 side is the longer leg, opposite 60°. If your “short” leg comes out longer than your “long” leg, you have them swapped.
- Multiplying when you should divide. From the long leg, divide by √3 to get x. From the hypotenuse, divide by 2.
- Leaving √3 in a denominator. Many courses want 4√3, not 12/√3. Multiply the top and bottom by √3.
- Rounding too early. Keep radicals until the last step. Rounding √3 to 1.7 makes a 1.8% error that compounds in areas.
- Assuming any triangle with a 30° angle is 30-60-90. The ratio only holds when there is also a right angle.
Special right triangles come up in construction layout too. The bolt circle calculator uses the same trigonometry to space holes around a circle. To turn a decimal answer back into a fraction for a tape measure, use the mixed fraction calculator.
Frequently asked questions
What is the 30-60-90 triangle formula?
The sides are in the ratio 1 : √3 : 2. If the short leg (opposite 30°) is x, the long leg (opposite 60°) is x√3 and the hypotenuse is 2x. OpenStax Contemporary Mathematics states it as x : x√3 : 2x.
How do I find the sides of a 30-60-90 triangle from the hypotenuse?
Halve the hypotenuse to get the short leg, then multiply the short leg by √3 for the long leg. A hypotenuse of 10 gives legs 5 and 5√3 ≈ 8.66, the example OpenStax works through.
How do I find the short leg from the long leg?
Divide the long leg by √3, which is the same as multiplying by √3/3. A long leg of 9 gives a short leg of 9/√3 = 3√3 ≈ 5.196 and a hypotenuse of 6√3 ≈ 10.392.
What is the area of a 30-60-90 triangle?
Area = ½ × short leg × long leg = ½ × x × x√3 = (√3/2)x². For a short leg of 5 the area is 25√3/2 ≈ 21.65 square units.
What is the perimeter of a 30-60-90 triangle?
Perimeter = x + x√3 + 2x = x(3 + √3) ≈ 4.732x. For x = 5 the perimeter is 15 + 5√3 ≈ 23.66.
Why is the 30-60-90 ratio 1 : √3 : 2?
Cut an equilateral triangle with side 2x in half along its altitude. Each half has angles 30°, 60° and 90°, a hypotenuse of 2x and a short leg of x. The Pythagorean theorem then gives the long leg √(4x² − x²) = x√3.
What is the difference between a 30-60-90 and a 45-45-90 triangle?
A 45-45-90 triangle is half of a square. Its legs are equal and the sides are in the ratio 1 : 1 : √2. A 30-60-90 triangle is half of an equilateral triangle, with ratio 1 : √3 : 2. Both are right triangles whose sides you can find from one length.
Which side is opposite the 60° angle?
The long leg, x√3. The short leg x is opposite 30°, and the hypotenuse 2x is opposite the right angle. The longest side is always opposite the largest angle.
What are the inradius and circumradius of a 30-60-90 triangle?
The circumradius equals half the hypotenuse, which is the short leg x, because the hypotenuse is a diameter of the circumscribed circle. The inradius is (a + b − c)/2 = x(√3 − 1)/2 ≈ 0.366x.
How do I write a 30-60-90 answer in simplest radical form?
Keep √3 as a symbol and rationalize denominators: 12/√3 = 12√3/3 = 4√3. The calculator accepts inputs like 4√3 or √3/2 and shows each answer in exact radical form next to its decimal.
Sources & method
- OpenStax Contemporary Mathematics §10.8 — Right triangle trigonometry: 30°-60°-90° and 45°-45°-90° triangles (ratios, Examples 10.67–10.69)
- OpenStax Algebra and Trigonometry 2e §7.2 — Right triangle trigonometry (special angles 30°, 45°, 60°; exact sine and cosine values)
- Wichita State University, Dept. of Mathematics — The inradius of a right triangle, r = (a + b − c)/2 and r = ab/(a + b + c)
- Clark University (D. E. Joyce) — Euclid’s Elements Book III Prop. 31: the angle in a semicircle is right
- Clark University (D. E. Joyce) — Area of a triangle: half the base times the height
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