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Area of a triangle

This is univeral 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 couple of calculators for finding the area of a triangle (see Area of triangle by coordinates and Calculator of area of a triangle using Hero's formula), there are other methods, depending on that you know about the triangle. So, here is the universal calculator where you can choose the formula to calculate the area of a triangle.

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 area of a triangle from the length of the base and the length or 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 length of the altitude.

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}.

Formula is named after Hero of Alexendria, 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 as SAS, this formula allows you to find the area of a triangle, if you know two sides and the angle at common vertex (included angle). The formula is

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

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

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

Coordinates formula

Using this formula, you can find the area of a triangle, if you know the cartesian coordinates of all three vertexes of a triangle. If the vertexes has 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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