Area of a Triangle
The plane closed figure, with three sides and three angles is called as a triangle.
Types of triangles:
Based on sides – a) Equilateral b) Isosceles c) Scalene
Based on angles – a) Acute angled triangle b) Right- angled triangle c) Obtuse angled triangle
Area of a triangle
In case of equilateral and isosceles triangles, if the length of the sides of triangles are given then,
we use Pythagoras theorem in order to find the height of a triangle.
Area of an equilateral triangle
Consider an equilateral ΔABC, with each side as a units. Let AO be perpendicular bisector of BC. In order to derive the formula for the area of equilateral triangle, we need to find height AO.
Using Pythagoras theorem,
Substitute AC=a,OC=a2 to find OA
We know the area of triangle is
∴Area of Equilateral triangle=√3a24
Area of an isosceles triangle
Consider an isosceles ΔABC with equal sides as a units and base as b unit.
The height of the triangle can be found by Pythagoras’ Theorem :
Area of triangle is A=12bh
Area of a triangle – By Heron’s formula
Area of a ΔABC, given sides a, b, c by Heron’s formula (Also known as Hero’s Formula) :
Find semi perimeter (s ) =a+b+c2
This formula is helpful to find area of a scalene triangle, given the lengths of all its sides.
Area of any polygon – By Heron’s formula
Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Heron’s formula.
Example :A park, in the shape of a quadrilateral ABCD, has ∠C=90∘, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m. How much area does it occupy?
⇒We draw the figure according to the information given.
The figure can be split into 2 triangles ΔBCD and ΔABD
From ΔBCD, we can find BD (Using Pythagoras’ Theorem)
Semi-perimeter for ΔBCD S1=12+5+132=15
Semi-perimeter ΔABD S2=9+8+132=15
Using Heron’s formula we find A1 and A2
Similarly we find A2 to be 35.49cm2.
The area of the quadrilateral ABCD=A1+A2=65.49 cm2