# What is the area of a triangle?

Finds the area of a triangle from its base and height, three sides (Heron’s formula), two sides and the angle between them, two angles and a side, or three corner points.

- Page: https://www.acalculator.org/math/triangle-area-calculator
- JSON spec: https://www.acalculator.org/math/triangle-area-calculator.json
- Version: 18ba5e016abf

## Default answer

Example with the default inputs (What do you know? Three sides, Side a 5, Side b 6, Side c 7): The triangle has an area of 14.696938.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| method | What do you know? | Which measurements of the triangle you know. |
| base | Base | The length of the side the height is measured from. |
| height | Height | The distance from the base to the opposite corner, at a right angle. |
| a | Side a | The length of side a. |
| b | Side b | The length of side b. |
| c | Side c | The length of side c; with two angles, the side between them. |
| C | Angle C | The angle between sides a and b. |
| A | Angle A | One angle next to side c. |
| B | Angle B | The other angle next to side c. |
| x1 | x₁ | The x coordinate of the first corner. |
| y1 | y₁ | The y coordinate of the first corner. |
| x2 | x₂ | The x coordinate of the second corner. |
| y2 | y₂ | The y coordinate of the second corner. |
| x3 | x₃ | The x coordinate of the third corner. |
| y3 | y₃ | The y coordinate of the third corner. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| area | Area | The area inside the triangle, in square units. |
| perimeter | Perimeter | The sum of the three sides (not with base and height). |
| sideA | Side a | The length of side a. |
| sideB | Side b | The length of side b. |
| sideC | Side c | The length of side c. |

## Method

A = ½ × base × height; from three sides, Heron’s formula √(s(s − a)(s − b)(s − c)); from two sides and the angle between them, ½ab sin C.

## Assumptions

- Lengths are positive numbers in any one unit; the area is in that unit squared.
- Angles are more than 0° and less than 180°, in degrees or radians.
- Corner points are read exactly as typed (0.1 is 1/10), so their area is exact before rounding.

## Worked examples

1. method = bh, base = 10, height = 8 gives area = 40. Source: OpenStax, Prealgebra 2e, §9.4, area of a triangle (https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids).
2. method = sss, a = 5, b = 6, c = 7 gives area = 14.696938, perimeter = 18.
3. method = sas, a = 5, b = 6, C = 1.047198 gives area = 12.990381, sideC = 5.567764.
4. method = asa, A = 0.785398, B = 1.047198, c = 7 gives area = 15.532378.
5. method = points, x1 = 0, y1 = 0, x2 = 4, y2 = 0, x3 = 0, y3 = 3 gives area = 6, perimeter = 12, sideA = 5.
6. method = points, x1 = 0.1, y1 = 0.2, x2 = 0.4, y2 = 0.2, x3 = 0.1, y3 = 0.6 gives area = 0.06.

## FAQ

### What is the formula for the area of a triangle?

Area = ½ × base × height. The height is the distance from the base to the opposite corner, measured at a right angle to the base. A triangle with a base of 10 and a height of 8 has an area of ½ × 10 × 8 = 40.

### How do I find the area of a triangle with three sides?

Use Heron’s formula. Work out half the perimeter, s = (a + b + c) ÷ 2, then area = √(s(s − a)(s − b)(s − c)). For sides 5, 6, and 7: s = 9, and the area is √(9 × 4 × 3 × 2) = √216 = 14.70.

### How do I find the area from two sides and an angle?

When the angle is between the two sides, area = ½ab sin C. Sides 5 and 6 with 60° between them give ½ × 5 × 6 × sin 60° = 12.99.

### How do I find the area of a triangle from coordinates?

Use the shoelace formula: area = ½ |(x₂ − x₁)(y₃ − y₁) − (x₃ − x₁)(y₂ − y₁)|. The corners (0, 0), (4, 0), and (0, 3) give ½ × |4 × 3 − 0 × 0| = 6.

### Does it matter which side I call the base?

No. Any side can be the base, as long as the height is measured to that side. A long base goes with a short height and a short base with a tall height, and the product is the same.

### Why do my three sides give no area?

They do not make a triangle. Each side must be shorter than the other two together. Sides 3, 4, and 8 cannot meet, and 3, 4, and 7 lie flat with no area.

### What units is the area in?

The square of the unit of the lengths: sides in centimetres give an area in square centimetres (cm²), and sides in feet give square feet. Use one unit for every length.

## Sources

- OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh). https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- OpenStax, Algebra and Trigonometry 2e, §10.1 Non-right Triangles: Law of Sines (area = ½ab sin C) and §10.2 Non-right Triangles: Law of Cosines (Heron’s formula). https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-2-non-right-triangles-law-of-cosines
- W. Kahan, "Miscalculating Area and Angles of a Needle-like Triangle" (2014). https://people.eecs.berkeley.edu/~wkahan/Triangle.pdf
- Weisstein, Eric W. "Triangle Area", MathWorld (area from vertex coordinates). https://mathworld.wolfram.com/TriangleArea.html