The triangle is the building block of area — every polygon can be cut into triangles. Two formulas cover every case: base-and-height when you can measure the height, and Heron’s formula when you only know the three sides. This guide makes both easy.
What’s in this guide
The triangle area formula
Base and height explained
The base is any one side you choose. The height is the perpendicular distance from that base to the opposite corner — the dashed line in the diagram. It is not the length of a slanted side, which is a common trap.
Height always meets the base at a right angle. On a slanted triangle, drop a straight line from the peak to the base line.
Three sides: Heron's formula
Worked examples
Base & height: base 18 ft, height 12 ft.
| Step | Value |
|---|---|
| Multiply | 18 × 12 = 216 |
| Halve | 216 ÷ 2 = 108 sq ft |
Heron (sides 13, 14, 15 ft):
| Step | Value |
|---|---|
| Semi-perimeter s | (13+14+15)/2 = 21 |
| Product | 21 × 8 × 7 × 6 = 7,056 |
| Area | √7,056 = 84 sq ft |
Where triangle area shows up
| Application | Triangle role |
|---|---|
| Gable walls | The peak above the eave line |
| Roof planes | Hip and gable roof surfaces |
| Land plots | Triangulating irregular parcels |
| Garden beds | Corner and wedge beds |
| Shade sails / awnings | Triangular fabric spans |
Common mistakes
Height is the perpendicular distance, not the sloped edge. The sloped side is always longer.
base × height is the area of a rectangle around the triangle. The triangle is half of it.
If any side is longer than the other two combined, the sides can't form a triangle and Heron's formula returns an error.
Key takeaways
- Triangle area = ½ × base × height.
- Height is perpendicular to the base, not a slanted side.
- Know only the sides? Use Heron's formula.
- Every polygon can be split into triangles.
Related calculators & guides
Frequently asked questions
- How do I calculate the area of a triangle?
- Use ½ × base × height. Pick any side as the base, measure the perpendicular height to the opposite vertex, multiply, and halve. A 18 ft base with 12 ft height = ½ × 18 × 12 = 108 sq ft.
- How do I find a triangle's area from three sides?
- Use Heron's formula. Compute the semi-perimeter s = (a+b+c)/2, then area = √[s(s−a)(s−b)(s−c)]. This needs no height — just the three side lengths.
- What is the height of a triangle?
- The perpendicular distance from the base to the opposite vertex — not the length of a slanted side. It forms a right angle with the base.
- Does the triangle type change the formula?
- No. ½ × base × height works for every triangle. Heron's formula also works for any triangle given three sides. Right triangles are just convenient because the two legs are base and height.