Triangle Formulas
Essential triangle formulas: area, perimeter, Heron's formula, and trigonometric relations with worked examples.
Basic Triangle Formulas
Area = ½ × base × height
Perimeter = a + b + c
Perimeter = a + b + c
Heron's Formula
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s − a)(s − b)(s − c)]
Area = √[s(s − a)(s − b)(s − c)]
Heron's formula lets you find the area of any triangle when you know all three side lengths, without needing the height.
Trigonometric Area Formula
Area = ½ × a × b × sin(C)
Where a and b are two sides, and C is the angle between them.
Variables
| Symbol | Meaning |
|---|---|
| a, b, c | Lengths of the three sides |
| h | Height (perpendicular distance from base to opposite vertex) |
| s | Semi-perimeter = (a + b + c) / 2 |
| A, B, C | Angles opposite to sides a, b, c respectively |
Law of Sines and Cosines
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines: c² = a² + b² − 2ab·cos(C)
Law of Cosines: c² = a² + b² − 2ab·cos(C)
Example 1 — Basic Area
Find the area of a triangle with base 12 cm and height 8 cm.
Area = ½ × 12 × 8
Area = 48 cm²
Example 2 — Heron's Formula
Find the area of a triangle with sides 5, 7, and 10.
s = (5 + 7 + 10) / 2 = 11
Area = √[11(11−5)(11−7)(11−10)]
= √[11 × 6 × 4 × 1] = √264
Area = 16.25 square units
Example 3 — Trigonometric Area
Two sides of a triangle are 9 and 14, with an included angle of 40°. Find the area.
Area = ½ × 9 × 14 × sin(40°)
= 63 × 0.6428
Area = 40.50 square units
When to Use Each Formula
- ½ × base × height: When you know the base and height
- Heron's formula: When you know all three sides but not the height
- Trigonometric: When you know two sides and the included angle
- Law of Sines: When you know two angles and a side (AAS or ASA)
- Law of Cosines: When you know three sides (SSS) or two sides and included angle (SAS)