What is the Area of a Triangle?
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. $A = \frac{1}{2} \times b \times h$. To find the area of a triangle, we need to know the base (b) and height (h).
This formula is applicable to all types of triangles, whether scalene, isosceles or equilateral. The base and height of the triangle must be perpendicular to each other. The unit of area is measured in square units (m², cm²).
Standard formula for triangle area:
where b is the base and h is the height of the triangle
Example:
What is the area of a triangle with base b = 3 cm and height h = 4 cm?
Using the formula:
Area of a Triangle, $A = \frac{1}{2} \times b \times h$
$= \frac{1}{2} \times 3 \text{ (cm)} \times 4 \text{ (cm)}$
$= 1.5 \text{ (cm)} \times 4 \text{ (cm)}$
$= 6 \text{ cm}^2$
Apart from the above formula, we have Heron's formula to calculate the triangle's area when we know the length of its three sides. Also, trigonometric functions are used to find the area when we know two sides and the angle formed between them in a triangle.
Area Formulas for Different Triangle Types
Area of a Right-Angled Triangle
A right-angled triangle has one angle equal to 90°. The height of the triangle will be the length of the perpendicular side.
Area of a Right Triangle = $A = \frac{1}{2} \times \text{Base} \times \text{Height}$ (Perpendicular distance)
From the above figure, Area of triangle ACB = $\frac{1}{2} \times a \times b$
Area of an Equilateral Triangle
An equilateral triangle has all three sides equal. The perpendicular drawn from the vertex to the base divides the base into two equal parts.
Area of an Equilateral Triangle = $A = \frac{\sqrt{3}}{4} \times \text{side}^2$
Area of an Isosceles Triangle
An isosceles triangle has two equal sides and two equal angles opposite to these sides.
Area of an Isosceles Triangle = $A = \frac{1}{4}b\sqrt{4a^2 - b^2}$
Where a is the length of the two equal sides and b is the length of the third side (the base).
Area of a Triangle Using Heron's Formula
When we know the lengths of all three sides of a triangle, we can use Heron's formula:
where $s = \frac{a + b + c}{2}$ (semi-perimeter)
a, b, and c are the lengths of the three sides of the triangle