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In the mathematical community, it is generally believed that the triangle area formula has different formulas according to different known conditions, such as base times height formula, Helen’s formula, sine and cosine formula, angle bisquer formula and two angles and other basic formulas, and various indirect methods for calculating the triangle area can also be derived from basic formulas according to different application scenarios and known conditions. This paper focuses on the formulas commonly used in industry to calculate the area of a triangle.

1, the classic base times height formula

Base times height formula: applies to triangles where the height of an angle and the length of the adjacent sides are known, the formula is

Where a is the length of the base of the triangle, h is the height of the triangle (the edge of the base of a) and S is the area of the triangle.

For example, if we know that the length of the base of a triangle is 5 and the height of the base edge is 3, we can use the height formula to calculate the area of the triangle as follows:

The area of the triangle is therefore 7.5.

2. Helen’s formula (trilateral method)

Helen’s formula: for triangles with known three sides, the formula is

Where s is half the circumference of the triangle, a, b and c are the three sides of the triangle and S is the area.

For example, if you have a triangle with the three sides 3, 4 and 5, you can use Helen’s formula to calculate its area. First work out the value of s: s = (3 + 4 + 5) / 2 = 6. Then use Helen’s formula:

So the area of this triangle is the whole number 6.

3, use the sine and cosine formulae (angle between two sides)

The sine and cosine formulae are applied to triangles with known sides and angles as follows:

Where a and b are the lengths of the two sides, C is the degree of the angle between the two sides, and sin C is the sine of C.

It can also be written as:

The lengths of the three sides of the triangle can be obtained by using the cosine theorem formula, and then the half circumference can be calculated according to the Helen formula, and finally the area of the triangle can be obtained by substituting the triangle area formula.

Finally, there are a lot of derivations based on the base times the height, sine and cosine, Helen’s formula, other indirect derivation of the triangle area formula according to specific input conditions, are introduced in the relevant articles, interested can search for their own view.

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