5 Formulas to Calculate the Area of a Triangle

A triangle is a special case of polygon with three sides. It can be formed by three non-parallel lines intersecting each other. The area of a triangle refers to the area enclosed by its three sides. Depending on given conditions, you may choose different formula to calculate the area of a triangle.

1. Area of a triangle by base and height


The formula for the area of a triangle by its base and height is


A = \dfrac{1}{2}bh


where b stands for the base and h stands for the altitude (height) drawn to that base.


How to derive the formula? Well, consider a diagonal divides a parallelogram into two congruent triangles. The area of the triangle will be one half of the area of the parallelogram.


Based on the formula of the area of a parallelogram, that is,


A_p = bh


The area of a parallelogram A_p is product of the base b and the height h of the parallelogram, we get the formula of the area for the triangle.

2. Trigonometry Formula for the Area of a Triangle


Given two sides and the Included angle of the two sides, the area of a triangle can be expressed as,


A = \dfrac{1}{2}ab\sin C


That is, the area of a triangle is the half of the product of two sides and the sines of the included angle.

If angle C is equal to 90 \degree , we get the formula for the area of right triangle,


A = \dfrac{1}{2}ab

3. Area of a triangle by the length of three sides - Heron’s formula

If three sides of a triangle are known value, the area of the triangle can be calculated with Heron’s formula.



A = \sqrt{s(s-a)(s-b)(s-c)} ,


in which s =\dfrac{a+b+c}{2} ,


Heron's formula can also be written as


A = \dfrac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}

4. Area of a triangle with an inscribed circle

Given three sides of a triangle and radius of its inscribed circle, the triangle can be divided into three sub-triangles with the same height. So the area of the triangle is the sum of the three triangles.


A = \dfrac{1}{2}(a+b+c)r
(1)


By Heron’s formula , we get,


r = \dfrac{A}{s}


= \dfrac{ \sqrt{s(s-a)(s-b)(s-c)}}{s}


=\sqrt{\dfrac{(s-a)(s-b)(s-c)}{s} }
(2)

5. Area of a triangle with circumcircle circle

Given three sides and the radius of the circumcircle circle of a triangle, the area of a triangle can be expressed in simple form .


According g to The Law of Sines,


\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R


in which R is the radius of the circumcircle circle of the triangle.


Then


\sin C = \dfrac{c}{2R}


The equation related to a side and the sines of its opposite angle of a triangle can also be derived geometrically.


Substitute to the trigonometry area formula, we get


A = \dfrac{abc}{4R}

Steven Zheng Steven Zheng posted 1 month ago

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