Triangle Angles and Side Calculation Online
γ = 90° | Altitude (h) is perpendicular to side c
1. Basic Anatomy
A right triangle is defined by having exactly one 90° angle. This simple shape is the basis for measuring distances and angles in the physical world.
- Hypotenuse (c): The longest side, always opposite the right angle.
- Legs (a, b): The two sides that meet to form the right angle.
- Altitude (h): The height measured from the right angle vertex perpendicular to the hypotenuse.
2. Key Formulas
For any right triangle with legs a, b and hypotenuse c:
$$ Perimeter = a + b + c$$ $$ Area = ½ × a × b $$ $$ Area = ½ × c × h $$Pythagorean Triples: Sets of three integers that perfectly fit the triangle (e.g., 3-4-5 or 5-12-13).
3. Special Right Triangles
These triangles have fixed ratios, allowing you to find missing sides instantly.
30°-60°-90° Triangle
$$ The sides follow the ratio 1 : √3 : 2. $$ $$ Short leg (opposite 30°): x $$ $$ Long leg (opposite 60°): x\sqrt{3} $$ $$ Hypotenuse (opposite 90°): 2x $$45°-45°-90° Triangle
Also known as an Isosceles Right Triangle. The sides follow the ratio 1 : 1 : √2.
- $$Legs: x$$
- $$Hypotenuse: x\sqrt{2}$$
Quick Comparison
| Type | Angles | Side Ratio |
|---|---|---|
| Standard Right | 90°, α, β | $$a^2 + b^2 = c^2$$ |
| Special | 30°, 60°, 90° | $$1 : \sqrt{3} : 2$$ |
| Isosceles | 45°, 45°, 90° | $$1 : 1 : \sqrt{2}$$ |