Multi-formula geometry tool
Triangle Area Calculator
Calculate the area of any triangle, choose from three methods: base & height, two sides & angle, or all three sides. Step-by-step working shown for every calculation.
- 3 calculation methods
- Step-by-step working
- All triangle types
- 100% free
Core calculator
Triangle Area Calculator
Formulas
Triangle Area Formulas
Area = (b × h) / 2
Use when: You know the base and the perpendicular height.
The height must be perpendicular to the base, not the slanted side length.
Area = (1/2) × a × b × sin(C)
Use when: You know two sides and the angle between them.
This works for any triangle type and does not require finding the height first.
s = (a + b + c) / 2
Area = √(s(s−a)(s−b)(s−c))
Use when: You know all three side lengths but not the height or angles.
s is the semi-perimeter, and the side lengths must satisfy the triangle inequality.
All three methods give the same area for the same triangle. They are simply different routes to the same answer depending on what information you have.
Method choice
Which Method Should You Use?
Do you know the base and height?
Yes: Use Method 1 (Base & Height), the simplest option.
Do you know two sides and the angle between them?
Yes: Use Method 2 (SAS), no height needed.
Do you only know the three side lengths?
Yes: Use Method 3 (Heron's Formula), which works from sides alone.
For a right triangle, the two legs are already perpendicular, so they act as base
and height directly: Area = (leg₁ × leg₂) / 2. If you need to find those legs first,
use the Right Triangle Calculator.
Examples
Worked Examples
One example for each method, showing the full calculation from start to finish.
Example 1: A triangle with base 10 cm and height 6 cm
| Identify values | b = 10 cm, h = 6 cm |
|---|---|
| Apply formula | Area = (b × h) / 2 |
| Calculate | (10 × 6) / 2 = 60 / 2 |
| Result | Area = 30 cm² |
Example 2: Two sides of 7 m and 9 m with an included angle of 60°
| Identify values | a = 7 m, b = 9 m, C = 60° |
|---|---|
| sin(60°) | √3/2 ≈ 0.8660 |
| Apply formula | Area = (1/2) × 7 × 9 × sin(60°) |
| Calculate | (1/2) × 7 × 9 × 0.8660 = (1/2) × 54.558 |
| Result | Area ≈ 27.279 m² |
Example 3: A triangle with sides 5, 6, and 7
| Semi-perimeter | s = (5 + 6 + 7) / 2 = 18 / 2 = 9 |
|---|---|
| s − a | 9 − 5 = 4 |
| s − b | 9 − 6 = 3 |
| s − c | 9 − 7 = 2 |
| Product | 9 × 4 × 3 × 2 = 216 |
| Square root | Area = √216 = 6√6 ≈ 14.697 |
Special cases
Area Formulas for Special Triangles
Right Triangle
Formula: Area = (a × b) / 2
The two legs are perpendicular, so they serve directly as base and height.
Example: Legs 3 and 4 → Area = 6
Right Triangle Calculator →45-45-90 Triangle
Formula: Area = a² / 2
Both legs are equal, so the area simplifies immediately. The hypotenuse form is c² / 4.
Example: Leg = 6 → Area = 18
45-45-90 Calculator →30-60-90 Triangle
Formula: Area = a²√3 / 2
Here a is the short leg, and the long leg is a√3.
Example: Short leg = 4 → Area = 8√3 ≈ 13.856
30-60-90 Calculator →FAQ
Frequently Asked Questions
The most common formula is Area = (base × height) / 2, where the height is
perpendicular to the base. If you know two sides and the included angle, use
Area = (1/2) × a × b × sin(C). If you know all three sides, use Heron's
formula.
For a right triangle, the two legs are perpendicular to each other, so they serve as the base
and height. Area = (leg₁ × leg₂) / 2. If you need to find the legs first, use
the Pythagorean theorem or the Right Triangle Calculator.
Use either Method 2 (SAS) or Method 3 (Heron's formula). Both avoid needing the perpendicular height explicitly.
No. The base can be any side of the triangle. The height is always the perpendicular distance from the chosen base to the opposite vertex, regardless of the triangle's orientation.
The semi-perimeter s is half the perimeter:
s = (a + b + c) / 2. It is the intermediate value used in Heron's formula.
Not with these three methods alone. You need at least two pieces of length information, or enough angle information to derive a second side first.
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