Free 30 60 90 Triangle Calculator

Understanding the 30‑60‑90 Triangle

The 30‑60‑90 triangle is a special right triangle whose three interior angles are 30°, 60°, and 90°. Because these angles are fixed, the side lengths always obey a constant ratio, making it easy to solve for any unknown dimension when only one side is given. A dedicated 30‑60‑90 triangle calculator can instantly compute the missing side, area, or perimeter using this ratio. The tool is especially useful for students, architects, and anyone working with right triangles.

Core Side Ratios

Let the shorter leg (opposite the 30° angle) be denoted as $a$. Then:

• The longer leg (opposite 60°) equals $a\sqrt{3}$.
• The hypotenuse (opposite 90°) equals $2a$.

Consequently, the side ratio is $1 : \sqrt{3} : 2$, while the angle ratio is $1 : 2 : 3$ for $30° : 60° : 90°$. This triangle is unique among right triangles because its angles form an arithmetic progression.

Key Formulas

From the side definitions we obtain the area and perimeter:

$$\text{Area} = \frac{a^{2}\sqrt{3}}{2}$$ $$\text{Perimeter} = a + a\sqrt{3} + 2a = a(3 + \sqrt{3})$$

Solving When Different Sides Are Known

Depending on which side you start with, the 30‑60‑90 triangle formulas adapt:

• If the shorter leg $a$ is known:
  Longer leg $b = a\sqrt{3}$, hypotenuse $c = 2a$.

• If the longer leg $b$ is known:
  Shorter leg $a = \dfrac{b\sqrt{3}}{3}$, hypotenuse $c = \dfrac{2b\sqrt{3}}{3}$.

• If the hypotenuse $c$ is known:
  Shorter leg $a = \dfrac{c}{2}$, longer leg $b = \dfrac{c\sqrt{3}}{2}$.

These three cases cover every input scenario and form the basis of any 30‑60‑90 triangle solver.

How the Side Ratios Are Derived

Two approaches confirm the same relationships:

1. Equilateral triangle half — Splitting an equilateral triangle of side $c$ along its altitude gives two 30‑60‑90 triangles. The altitude of an equilateral triangle is $h = \dfrac{c\sqrt{3}}{2}$, which equals the longer leg $b$. The half‑base equals the shorter leg $a = c/2$. Therefore $b = \dfrac{c\sqrt{3}}{2} = a\sqrt{3}$.

2. Trigonometric proof — In the 30‑60‑90 triangle:
   $\sin 30^{\circ} = \dfrac{a}{c} = \dfrac{1}{2} \Rightarrow c = 2a$
   $\sin 60^{\circ} = \dfrac{b}{c} = \dfrac{\sqrt{3}}{2} \Rightarrow b = c\dfrac{\sqrt{3}}{2} = a\sqrt{3}$

Both methods produce the same consistent side ratio.

Worked Example: Longer Leg Given

Assume the longer leg is 11 inches. Applying the formulas:

• Shorter leg: $a = \dfrac{11\sqrt{3}}{3} \approx 6.35\ \text{in}$
• Hypotenuse: $c = 2a \approx 12.7\ \text{in}$
• Area: $\dfrac{a^{2}\sqrt{3}}{2} \approx 34.9\ \text{in}^{2}$
• Perimeter: $a(3 + \sqrt{3}) \approx 30.05\ \text{in}$

Why Use This Tool?

The 30‑60‑90 triangle calculator removes the need to memorize or manually apply each formula. Enter any one known side — whether short leg, long leg, or hypotenuse — and the calculator returns the other sides, the area, and the perimeter instantly. This reduces errors and speeds up problem solving for any 30‑60‑90 triangle.