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The area of a triangle on three sides (Herons formula)

Knowing the three sides of a triangle, you can calculate its area by Heron's formula.

The online calculator below allows you to find:
- The area of the triangle on the three sides of the triangle;
- The side of the triangle through the other sides and the area of the triangle.

When calculating the side of a triangle, keep in mind that the equation of the 4th degree is solved, and there can be more than one solution of the equation. The calculator uses only one valid solution.

The area of a triangle is one of the main parameters that determine its shape. One of the most famous ways to calculate the area of a triangle is to use Heron's formula. Heron's formula allows you to calculate the area of a triangle if the lengths of its three sides are known.

Heron's formula looks like this: S = √(p(p-a)(p-b)(p-c)), where S is the area of the triangle, a, b and c are the lengths of the three sides, p is the half-perimeter of the triangle, equal to (a+ b+c)/2.

Before applying Heron's formula, it is necessary to calculate the semiperimeter of a triangle. This can be done by adding the lengths of all three sides and dividing by 2. After that, we can substitute the lengths of the sides and the semi-perimeter into Heron's formula and calculate the area of the triangle.

To illustrate this formula, let's look at an example. Suppose we have a triangle with sides of length 3, 4, and 5. First, we calculate the semi-perimeter of the triangle: p = (3+4+5)/2 = 6. Then we substitute the values of the sides and semi-perimeter into Heron's formula: S = √ (6(6-3)(6-4)(6-5)) = √(6*3*2*1) = √36 = 6. So the area of a triangle is 6 square units.

Heron's formula is one of the main ways to calculate the area of a triangle and is widely used in geometry, mechanics, physics and other scientific fields. It is based on the properties of a triangle and allows you to calculate the area of a triangle if you know the lengths of its sides.

It is important to note that Heron's formula can only be used if the lengths of all three sides of a triangle are known. If only two sides and the angle between them are known, or one side and the height lowered to this side, then other formulas should be used to calculate the area of the triangle.

In conclusion, Heron's formula allows you to calculate the area of a triangle if the lengths of its three sides are known. It is one of the main ways to calculate the area of a triangle and is widely used in scientific fields.
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