The 30-60-90 Triangle Calculator finds all sides, area, perimeter, and altitude of a special right triangle from one known side. Based on the exact 1:√3:2 ratio, it delivers instant results for geometry, trigonometry, and engineering problems.
5
units
8.6603
units
10
units
21.6506
sq units
23.6603
units
4.3301
units
5
units
8.6603
units
10
units
21.6506
sq units
23.6603
units
4.3301
units
The calculator for the 30-60-90 triangle computes every property of this special right triangle — all three sides, area, perimeter, and altitude to the hypotenuse — from a single input. Because the side ratios are fixed at exactly 1 : √3 : 2, one measurement is always enough to determine the entire triangle.
The 30-60-90 triangle is one of the two special right triangles in Euclidean geometry. Its side ratios are derived by bisecting an equilateral triangle along its altitude. If the equilateral triangle has side length 2a, the bisection produces two 30-60-90 triangles where:
short leg = a | long leg = a√3 | hypotenuse = 2a
This geometric origin explains why √3 appears so naturally in the triangle and why it is inseparable from equilateral triangle geometry. The equilateral triangle calculator and this tool share the same underlying ratio.
The 30-60-90 triangle is the source of the most commonly memorized trigonometric values in mathematics. Reading directly from the side ratios:
sin 30° = 1/2 cos 30° = √3/2 tan 30° = 1/√3
sin 60° = √3/2 cos 60° = 1/2 tan 60° = √3
These exact values — not decimal approximations — are used throughout physics, engineering, and computer graphics whenever 30° or 60° angles appear. The triangle calculators category includes tools for all standard triangle types that build on these fundamentals.
The calculator takes the shortest side (opposite the 30° angle) as its primary input. If you know a different side, convert it first using the fixed ratios:
short side = long side / √3 short side = hypotenuse / 2
For example, if the hypotenuse is 10, the short side is 5. Enter 5, and the online calculator instantly returns the long side (5√3 ≈ 8.660), area, and perimeter. The right triangle calculator handles the general case when angles are not fixed at 30-60-90.
With the short leg a as the base input, the area and perimeter follow directly:
Area = (√3 / 2) × a2
Perimeter = a × (3 + √3)
The altitude from the right angle to the hypotenuse is h = a√3 / 2, which equals the long leg divided by 2. These compact formulas make the 30-60-90 triangle especially efficient for rapid hand calculations in construction and engineering.
The 30-60-90 triangle appears throughout applied mathematics and physical design. Roof pitch calculations frequently involve 30° or 60° inclines — a 30° ramp rises exactly 1 unit for every 2 units of ramp length. Hexagonal bolt heads, honeycomb structures, and graphene lattices all encode 60° angles, making 30-60-90 geometry essential in mechanical engineering and materials science. The 45-45-90 triangle calculator covers the complementary special right triangle used when equal legs are required.
A 30-60-90 triangle has angles of 30°, 60°, and 90°, with sides in the exact ratio $$1 : \sqrt{3} : 2$$. Let $$x$$ be the shortest side (opposite the 30° angle).
Side Lengths:
$$\text{Short side} = x \quad (\text{opposite } 30°)$$
$$\text{Medium side} = x\sqrt{3} \quad (\text{opposite } 60°)$$
$$\text{Hypotenuse} = 2x \quad (\text{opposite } 90°)$$
Area: The two legs are the short and medium sides:
$$A = \frac{1}{2} \cdot x \cdot x\sqrt{3} = \frac{\sqrt{3}}{2} x^2$$
Perimeter:
$$P = x + x\sqrt{3} + 2x = x(3 + \sqrt{3})$$
Altitude to Hypotenuse:
$$h = \frac{x \cdot x\sqrt{3}}{2x} = \frac{x\sqrt{3}}{2}$$
The derivation of the side ratios comes from bisecting an equilateral triangle of side $$2x$$. The altitude of the equilateral triangle, $$\frac{\sqrt{3}}{2} \cdot 2x = x\sqrt{3}$$, becomes the medium side of the 30-60-90 triangle.
The Short Side is opposite the 30° angle and is always the smallest side. It equals exactly half the hypotenuse.
The Medium Side is opposite the 60° angle and equals the short side multiplied by $$\sqrt{3} \approx 1.732$$. It is the longer leg of the right triangle.
The Hypotenuse is opposite the 90° angle and equals exactly twice the short side. It is always the longest side.
The Area equals $$\frac{\sqrt{3}}{2} x^2$$, growing quadratically with the short side. This is also half the area of the equilateral triangle from which the 30-60-90 triangle is derived.
The Perimeter is the sum of all three sides: $$x(3 + \sqrt{3})$$.
The Altitude to Hypotenuse is the perpendicular distance from the right-angle vertex to the hypotenuse, useful for area verification and geometric constructions.
Inputs
Results
Medium side = 5√3 = 8.6603. Hypotenuse = 2*5 = 10. Area = 0.5 * 5 * 8.6603 = 21.6506. Perimeter = 5 + 8.6603 + 10 = 23.6603. Altitude to hypotenuse = 5 * 8.6603 / 10 = 4.3301. Verification: sin(30°) = 5/10 = 0.5. ✓
Inputs
Results
This is the unit 30-60-90 triangle: sides 1, √3, 2. Area = √3/2 = 0.866. These are the exact values used to define sin(30°)=1/2, cos(30°)=√3/2, tan(30°)=1/√3. The altitude to hypotenuse equals √3/2, the same as the area since the hypotenuse is 2.
Divide the hypotenuse by 2 to get the short side, then enter that value. For example, if the hypotenuse is 14, the short side is 14/2 = 7. Enter 7 as the shortest side.
Divide the medium side by √3 (approximately 1.7321) to get the short side. For example, if the medium side is 12, the short side is 12/1.7321 ≈ 6.928. Enter that value.
It comes from bisecting an equilateral triangle along its altitude. The equilateral triangle (all sides equal, all angles 60°) splits into two congruent 30-60-90 triangles. The half-base is 1, the altitude is √3, and the original side is 2 (relative to the half-base).
sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3. For 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3. These exact values are foundational in trigonometry.
A 30° angle creates a gentle slope (rise-to-run ratio of 1:√3 ≈ 1:1.73). This is used for wheelchair ramps, staircase design, and roof pitches. Hexagonal bolt patterns and 60° chamfers also rely on 30-60-90 geometry.
A 30-60-90 triangle has sides in ratio 1:√3:2 with three different angles and three different side lengths. A 45-45-90 triangle has sides in ratio 1:1:√2 with two equal legs and two equal angles. The 30-60-90 is scalene; the 45-45-90 is isosceles.
The altitude from the right angle (90°) to the hypotenuse equals the long leg divided by 2, or equivalently a√3/2 where a is the short side. For a triangle with short side 4, the hypotenuse is 8 and the altitude to the hypotenuse is 4√3/2 ≈ 3.464. This altitude also equals the geometric mean relationship between the two segments it creates on the hypotenuse.
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