How to Solve a Right Triangle: A Complete Guide

The Anatomy of a Right Triangle

A right triangle (or right-angled triangle) is one of the most foundational shapes in geometry. Solving a right triangle means finding all of its missing side lengths and angles based on the information you already know.

To solve these geometry problems quickly, you can always use our Right Angle Calculator Triangle tool. However, understanding the underlying math is essential for students and professionals alike. Every right triangle consists of three sides and three angles:

• Leg a: One of the perpendicular sides forming the right angle.
• Leg b: The other perpendicular side (often called the base).
• Hypotenuse (c): The longest side, directly opposite the 90° angle.
• Right Angle: The 90° angle formed by the intersection of Leg a and Leg b.
• Acute Angles (A & B): The remaining two angles, which will always add up to exactly 90°.

Finding Missing Sides

If you know two sides of a right triangle, you can find the third side using the Pythagorean theorem:

a² + b² = c²

Example: If Leg a is 3 and Leg b is 4, you can find the hypotenuse:

• 3² + 4² = c²
• 9 + 16 = c²
• 25 = c²
• c = 5

Finding Missing Angles

If you know at least two sides, you can find the missing acute angles using inverse trigonometric functions (SOH CAH TOA).

• Sine (sin): Opposite / Hypotenuse
• Cosine (cos): Adjacent / Hypotenuse
• Tangent (tan): Opposite / Adjacent

If you only know one side and one acute angle, you can use regular Sine, Cosine, or Tangent to find the remaining sides.

Calculating Area and Perimeter

• Area: The space inside the triangle. Formula: Area = (a × b) / 2
• Perimeter: The total distance around the outside. Formula: Perimeter = a + b + c