# Triangle Area Calculator

This calculator is designed to find the area of a triangle using different formulas, depending on the available information about the triangle.

### This page exists due to the efforts of the following people:

#### Timur

Created: 2019-05-06 07:31:37, Last updated: 2023-04-06 02:45:34

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If you know the base and the altitude of the triangle, you can use the Half of base times height formula. This formula takes half of the base length and multiplies it by the altitude length to find the area.

If you know all three sides of the triangle, you can use Heron's formula. This formula uses the three side lengths to calculate the semiperimeter, which is then used to find the area of the triangle. Heron's formula is useful when you do not know the height of the triangle or when the triangle is not a right triangle.

If you know two sides and the included angle of the triangle, you can use the Side-angle-side formula to find the area. This formula uses the two side lengths and the included angle to calculate the area of the triangle.

If you know the coordinates of the three vertices of the triangle, you can use the Coordinates formula. This formula uses the coordinates of the vertices to find the length of the sides of the triangle and then uses Heron's formula to find the area.

Finally, if you are dealing with an equilateral triangle, there is a simplified formula that can be used. This formula is derived from Heron's formula and uses only the length of one side to calculate the area.

You can find all formulas below the calculator.

#### Area of a triangle

°
Digits after the decimal point: 2

#### Equilateral triangle formula

Area of the triangle

### Half of base times height formula

You can find the triangle area from the length of the base and the length of the corresponding altitude. Any side can be the base, but the altitude must correspond to the base. The formula is

$A=\frac{ah}{2}$

where A is the area, a is the length of the base, h is the altitude's length.

### Heron's formula

You can find the area of a triangle if you know the lengths of all sides. The formula is

$A=\sqrt{p(p-a)(p-b)(p-c)}$

where A is the area, a, b, c are the lengths of the sides, p is the perimeter divided by 2 (semi-perimeter) $\frac{a+b+c}{2}$.

The formula is named after Hero of Alexandria, a Greek Engineer and Mathematician, in 10 - 70 AD. A proof can be found in his book Metrica written around 60 AD.

### Side-angle-side formula

Also referred to as SAS, this formula allows you to find a triangle area if you know two sides and the angle at a common vertex (included angle). The formula is

$A=\frac{ab \sin \alpha}{2}$

where A is the area, a and b are the sides' lengths, alpha is the angle at the common vertex.

Actually, this form directly follows from half of the base times height formula because the triangle's height would be $a \sin \alpha$.

### Coordinates formula

Using this formula, you can find a triangle area if you know the cartesian coordinates of all three vertexes of a triangle. If the vertexes have coordinates like (x1, y1), (x2, y2) and (x3, y3) then the formula is

$A=|\frac{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}{2}|$

where A is the area, and x and y are coordinates of triangle vertexes.

Well, you can also compute the cartesian distance between the vertexes and use Heron's formula.

### Equilateral triangle formula

This is just simplified Heron's formula because all sides are equal. The formula is

$A=\frac{\sqrt{3}}{4}a^2$

where A is the area, a is the length of the side.

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