Right Triangle Formulas — Sides, Area, Trig Ratios

Right Triangle Formulas — Sides, Area, Trig Ratios

TL;DR
The right triangle formulas are the Pythagorean theorem a² + b² = c², area = 1/2 bh, perimeter = a + b + c, and the three trigonometric ratios sin, cos, tan. This article gives each formula with a short derivation, six worked examples, the 45-45-90 and 30-60-90 special-triangle ratios, and common mistakes.

The Formulas

For a right triangle with legs a and b, hypotenuse c, and the two acute angles ∠A and ∠B:

How the Pythagorean Theorem Is Derived

The Pythagorean theorem comes from counting area. By arranging four identical right triangles inside a large square, we can equate the area computed one way as (a + b)² and as the sum of the areas of the four triangles plus the inner square, leading to

a² + b² = c².

How Do You Find the Area of a Right Triangle?

In a right triangle, the area formula is simply
Area = 1/2 * base * height = 1/2 * b * a.

Examples of Right Triangle Formulas

Example 1

Find the hypotenuse of a right triangle with legs of 6 cm and 8 cm.
By the Pythagorean theorem:
c² = 6² + 8² = 36 + 64 = 100, therefore c = √100 = 10 cm.

Example 2

Find the other leg of a right triangle with hypotenuse 13 m and one leg 5 m.
Here, a² = c² - b² = 13² - 5² = 169 - 25 = 144, therefore a = √144 = 12 m.

Example 3

Find the area and perimeter of a right triangle with legs 9 in and 12 in.
Area = 1/2 * 9 * 12 = 54 in².
To find the perimeter, first find the hypotenuse: c = √(9² + 12²) = √225 = 15 in.
Perimeter = 9 + 12 + 15 = 36 in.

Example 4

Find sin A and the angle in a right triangle where ∠A opposes a side of 7 cm and the hypotenuse is 25 cm.
sin A = 7/25 = 0.28, thus ∠A = sin⁻¹(0.28) ≈ 16.3°.

Example 5

Find the hypotenuse and area of a 45-45-90 triangle with legs 5 units each.
Hypotenuse = 5√2 ≈ 7.07 units, Area = 1/2 * 5 * 5 = 12.5 units².

Example 6

A ramp rises 1.5 m over a horizontal run of 6 m. Find the ramp's length and angle with the ground.
c = √(1.5² + 6²) ≈ 6.18 m.
tan θ = 1.5/6 = 0.25, therefore θ = tan⁻¹(0.25) ≈ 14°.

Special Right Triangles — The Two Ratios Worth Memorising

Common Mistakes to Watch Out For

  1. Treating a leg like the hypotenuse. Always identify the longest side first.
  2. Forgetting the square root. Ensure to take the square root for your final answer.
  3. Mixing up opposite and adjacent sides. Anchor them to the correct angle.

Key Takeaways