# Area of triangle with vertices formula explained with example

Learn and know what is the formula we have to use for finding area of triangle with vertices i.e. when vertices are given. This concept comes in coordinate geometry in class 10. We know that there are formulas for area of triangle based on the type of the triangle. For example, there are formulas for area of equilateral triangle, area of scalene triangle, area of isosceles triangle and so on. These formulas we will learn in geometry. The formula for area of triangle with vertices comes in coordinate geometry. Now we will learn this formula.

## What are called as vertices of a triangle?

We know that triangle consists of 3 line segments. The meeting point of any two line segments, we call it as a vertex of the triangle. Totally for a triangle there exist 3 vertices.

## What is mean by of area of triangle in math?

The space occupied by the triangle is called as the area of triangle. We will measure the area in terms of square units.

## Formula used to find area of triangle with vertices:

Now we are going to learn how to determine the area of a triangle when vertices are given. Let PQR be a triangle with vertices P $\left( { x }_{ 1 },{ y }_{ 1 } \right)$, Q $\left( { x }_{ 2 },{ y }_{ 2 } \right)$ and R $\left( { x }_{ 3 },{ y }_{ 3 } \right)$.

The formula of area of triangle with vertices is given as

A = $\frac { 1 }{ 2 }$ | ${ x }_{ 1 }$ (${ y }_{ 2 }$${ y }_{ 3 }$) + ${ x }_{ 2 }$ (${ y }_{ 3 }$${ y }_{ 1 }$) + ${ x }_{ 3 }$ (${ y }_{ 1 }$${ y }_{ 2 }$) |

Examples:

Find the area of triangle with vertices (5, 8), (1, 7) and (10, 6)

Solution:

Let us take

(5,8) as $\left( { x }_{ 1 },{ y }_{ 1 } \right)$

(1 , 7) as $\left( { x }_{ 2 },{ y }_{ 2 } \right)$

(10, 6) as $\left( { x }_{ 3 },{ y }_{ 3 } \right)$

Substitute these values in the area of triangle formula.

A = $\frac { 1 }{ 2 }$ | ${ x }_{ 1 }$ (${ y }_{ 2 }$${ y }_{ 3 }$) + ${ x }_{ 2 }$ (${ y }_{ 3 }$${ y }_{ 1 }$) + ${ x }_{ 3 }$ (${ y }_{ 1 }$${ y }_{ 2 }$) |

= $\frac { 1 }{ 2 }$  | 5 (7 – 6) + 1 (6 – 8) + 10 (8 –7) |

= $\frac { 1 }{ 2 }$ | 5 (1) + 1 (-2) + 10 (1) |

= $\frac { 1 }{ 2 }$ | 5 -2 + 10 |

= $\frac { 1 }{ 2 }$  | 13 |

=  $\frac { 13 }{ 2 }$

= 6.5 square units. 