A triangle, the simplest polygon in Geometry, has three sides, three vertices, and three edges. The area of a triangle is a measure of the space that it occupies in a two-dimensional plane. This article will delve into the concept of the area of a triangle and the different methods used to calculate it in coordinate geometry.
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The area of a triangle can be calculated using three different methods. Let's discuss each of them in detail.
If the base and altitude of the triangle are given, the area of the triangle can be calculated using the formula:
Area of the triangle, A = bh/2 square units
Here, 'b' and 'h' represent the base and altitude of the triangle, respectively.
When the lengths of all three sides of the triangle are known, the area can be calculated using Heron’s formula.
The formula to calculate the area of the triangle is:
In this formula, 'a', 'b', 'c' are the lengths of the sides of the triangle and 's' represents the semi perimeter.
The semi perimeter 's' is calculated using the formula:
If the vertices of a triangle are given, we first need to calculate the length of the three sides using the distance formula.
Let's take a triangle PQR, with coordinates P, Q, and R given as (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ), respectively.
From the figure, the area of triangle PQR can be calculated by drawing perpendicular lines QA, PB and RC from Q, P and R, respectively to the x – axis.
This forms three different trapeziums in the coordinate plane: PQAB, PBCR and QACR.
The area of ∆PQR is calculated as the sum of the areas of trapezium PQAB and trapezium PBCR, subtracted by the area of trapezium QACR.
Therefore, Area of ∆PQR = [Area of trapezium PQAB + Area of trapezium PBCR] - [Area of trapezium QACR]
What is the area of a triangle?
Area of the triangle is a measure of the space covered by the triangle in the two-dimensional plane.
How to find the area of a triangle when base and altitude are given?
When the base and altitude of the triangle are given, Area of the triangle, A = bh/2 square units.
How to calculate the area of a triangle using Heron’s formula?
When the length of three sides of the triangle are given, the area of a triangle can be found using the Heron’s formula, A = √s(s - a)(s - b)(s - c), where a, b, c are the side lengths of the triangle and s is the semi perimeter.
How to find the area of a triangle when the vertices are given in coordinate geometry?
If the vertices of a triangle are given, first we have to find the length of three sides of a triangle using the distance formula. Then, calculate the area of all the trapeziums formed in the coordinate plane.