Special Right Triangles Calculator

What Are Special Right Triangles?

Special right triangles are right triangles with fixed angle or side ratios that let you find all sides quickly without the full Pythagorean theorem every time. The two most common types are the 30°-60°-90° triangle and the 45°-45°-90° isosceles right triangle.

These triangles appear in trigonometry, geometry proofs, construction layout, and physics vector problems. You may also like our Similar Triangles Calculator and Pythagorean Calculator.

30°-60°-90° Side Ratios

If the shorter leg (opposite 30°) is $x$:

Longer leg (opposite 60°): $x\\sqrt{3}$
Hypotenuse: $2x$
Area: $\\frac{x^2\\sqrt{3}}{2}$

45°-45°-90° Side Ratios

If each equal leg is $x$:

Hypotenuse: $x\\sqrt{2}$
Area: $\\frac{x^2}{2}$

3-4-5 Pythagorean Triple

The sides scale as $3:4:5$. If the shortest side is $3k$, the other sides are $4k$ and $5k$. This triple is widely used in carpentry and surveying for checking right angles.

Frequently Asked Questions

How do I solve a 30-60-90 triangle from the hypotenuse?

If the hypotenuse is $c$, the shorter leg is $c/2$ and the longer leg is $c\\sqrt{3}/2$.

What are the angles in a 45-45-90 triangle?

The two acute angles are both 45° and the right angle is 90°. The two legs are equal in length.

Is 3-4-5 a Pythagorean triple?

Yes. Since $3^2 + 4^2 = 5^2$, any triangle with sides in ratio 3:4:5 is a right triangle.

Why are special right triangles useful in trigonometry?

They give exact values for sine, cosine, and tangent at 30°, 45°, and 60° without a calculator, which simplifies many identity proofs and unit-circle problems.

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