# 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:

- **Pythagorean theorem:** a² + b² = c²  
- **Area:** 1/2 * b * a  
- **Perimeter:** a + b + c  
- **Trig ratios:**  
  - sin A = opposite/hypotenuse  
  - cos A = adjacent/hypotenuse  
  - tan A = opposite/adjacent

### 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

- **45-45-90 (isosceles):** ratio 1:1:√2.
- **30-60-90:** ratio 1:√3:2.

## 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
- The right triangle formulas start from the Pythagorean theorem, area = 1/2 × base × height, and include sin, cos, tan ratios.  
- The 45-45-90 and 30-60-90 triangles are key ratios worth memorising.  
- The most common mistake is treating a leg as the hypotenuse.