Area of a triangle

This is universal online calculator for the area of a triangle which uses different methods to find the area of a triangle, depending on your knowledge of the triangle.

While we do have a couple of calculators for finding the area of a triangle (see Area of triangle by coordinates and Calculator of an area of a triangle using Hero's formula), there are other methods, depending on what you know about the triangle. Here is the universal calculator where you can choose the formula to calculate a triangle area.

The following formulas are supported:

  • Half of base times height formula - if you know the base and the altitude of a triangle
  • Heron's formula - if you know all three sides of a triangle
  • Side-angle-side formula - if you know two sides and included angle
  • Coordinates formula - if you know the coordinates of the three vertices of a triangle
  • Equilateral triangle formula - formula for the equilateral triangle, which is simplified Heron's formula

You can find all formulas with descriptions below the calculator.

PLANETCALC, Area of a triangle

Area of a triangle

°
Digits after the decimal point: 2

Half of base times height formula

Heron's formula

Side angle side formula

Coordinates formula

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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PLANETCALC, Area of a triangle

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