Area of a Triangle
Reference for triangle area formulas: A=½bh, Heron's formula from three sides, coordinate shoelace formula, and SAS with sine.
Step-by-step worked examples.
The Formula
A = ½ × b × h
This formula calculates the area of any triangle when you know the base and the perpendicular height.
The height must be measured at a right angle to the base.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the triangle |
| b | Length of the base |
| h | Perpendicular height (measured at 90° to the base) |
Example 1
Find the area of a triangle with base 10 cm and height 6 cm
A = ½ × b × h = ½ × 10 × 6
A = ½ × 60
A = 30 cm²
Example 2
A triangular sail has a base of 4.5 m and a height of 8 m. What is its area?
A = ½ × b × h = ½ × 4.5 × 8
A = ½ × 36
A = 18 m²
When to Use It
Use this formula when:
- You know (or can measure) the base and the perpendicular height
- Calculating the area of triangular land plots, roofs, or surfaces
- Breaking complex shapes into triangles to find their total area
- For triangles where you only know side lengths, use Heron's formula instead
Key Notes
- The height must be the perpendicular distance from the base to the opposite vertex — for an obtuse triangle the height falls outside the triangle, but the formula still works; using the slant side length instead is the most common error
- Any side can serve as the base — choose the one whose perpendicular height you know; all three base-height pairs give the same area, so b₁h₁ = b₂h₂ = b₃h₃ for any triangle
- When only side lengths are known (no height), use Heron's formula: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 — this is especially useful for land surveying and construction
- For triangles defined by coordinates (x₁,y₁), (x₂,y₂), (x₃,y₃), the shoelace formula gives the area directly: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| — no height measurement needed