How To Find Area of Triangle
Area of Triangles - Definition, Formulas, Types with Examples
Suppose your mom gives you a sheet of colored paper sheet for the decoration of the room and asks you to cut the same size of triangle from that sheet, Now, you will make sure that you have cut all the triangles of the same area for that you have to measure all the three sides of triangles and cut them in such a way that you get the same size of triangles.
Get JEE Prep Help Download NRI DASA/CIWG Ebook
Definition of Area of Triangle
The area or region occupied by the three sides of the triangle in a two-dimensional (2D) plane is called the Area of a Triangle. A triangle has three sides - Base, Hypotenuse, and Perpendicular or Height. A Triangle can be easily formed by joining the three dots in a plane or paper placed at different positions using a straight line.
The point at which the lines are intersecting each other is called Vertices while the space or arc created between them is called the Angle of a Triangle and the sum of all the interior angles is 180°.
SI Unit - The area of the Triangle is measured in meters square m² or centimeter square cm².
How To Find the Area of a Triangle
(i) Basic Formula
The area of the Triangle is defined as the area occupied by all three sides of the Triangle. It is equal to half of the Base x Height.
Area of Triangle = 1/2 x Base x Height
Note - To find the area of a triangle, we must know the base and height of that triangle.
An example for your better understanding of the concept -
Question - Find the Area of the Triangle with Base = 4cm, Height =7cm.
Answer - We have the formula
Area of the Triangle = 1/2x base x height
Therefore, = 1/2x4x7
= 14 cm²
(ii) Using Heron's Formula
This is another method or formula for finding out the area of a triangle. For using the method or formula, we first need to have measurements or lengths of all the sides of the triangle. This is a two-step method i.e. first to find out the semi-perimeter (s) of the triangle and then, put the values into an equation or formula. Below, we have Heron's Formula.
where s represents the semi-perimeter of the triangle which is found using the formula, s = (a + b + c)/2 where a, b, and c are the sides of the triangle.
Types of Triangle
There are mainly three types of the triangle for which we can find out the area -
(i) Equilateral Triangle
(ii) Right Angle Triangle
(iii) Isosceles Triangle
1. Equilateral Triangle - An equilateral triangle is a triangle whose all the sides are of equal sizes also when the perpendicular is drawn from any vertex of the triangle, it must be divided into two equal triangles.
The Mathematical Formula of an Equilateral Triangle to find out the area is
Area = (√3)/4 × side²
For Example:-
Question- Find the area of the equilateral triangle of side = 2 cm.
Answer- We know that the formula for the area of the equilateral triangle
Area = (√3)/4 × side²
Now, putting the values into the equation, we have
= (√3)/4 × 2²
= √3 or 1.74 cm²
2. Right AngleTriangle - A right triangle whose one angle is equal to 90°or one side (height) is perpendicular to the base of the triangle.
The Mathematical Formula of an Equilateral Triangle to find out the area is
Area = 1/2 x Base x Height (Perpendicular Distance)
For Example:-
Question - Find the area of a right-angle triangle with Base = 7 cm and Height = 6 cm.
Answer - We know
Area of the Triangle = 1/2x base x height
Therefore, = 1/2x 7x 6
= 21 cm²
3. Isosceles Triangle - An isosceles triangle is a triangle whose any of two sides are equal in size and whose angles are equal in opposite directions.
The Mathematical Formula of the Isosceles Angle Triangle is -
A = ½[√(a2 − b2 ⁄4) × b]
For Example:-
Question - Find the area of the isosceles angle Triangle of a = 7 cm and b = 6 cm.
Answer - We have the formula for finding the area of an isosceles triangle,
A = ½[√(a2 − b2 ⁄4) × b]
Putting the values into an equation, we have
A = 1/2[√7^2 - 6^2/4) x 6]
= 1/2[ √49-36/4) x 6 ]
= 1/2[ √13/4 x 6 ]
= 1/2 [ 3√13]
= 3√13/2 cm²
Important Questions On Area of Triangles
Question 1. Which of the triangles have the same side lengths?
Answer: Equilateral Triangle
Question 2. The area of an equilateral triangle with side length a is equal to:
Answer: (√3/4) a2
Question 3. If the perimeter of a triangle is 100 cm and the length of two sides are 30 cm and 40 cm, the length of the third side will be:
Answer: 30 cm
Question 4. The height of an equilateral triangle of side 5 cm is:
Answer: 4.33 cm
Question 5. The ratio of the areas of two similar triangles is equal to
Answer: Square of the ratio of their corresponding sides.
Question 6. Which of the following is not a similarity criterion for two triangles?
Answer: ASA
Question 7. The sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio
Answer: 16:81