What is an Isosceles Triangle?
A triangle is said to be isosceles if it has at least two congruent sides. The two congruent sides of an isosceles triangle are the legs and the third side is known as the base. The angle between the legs is called vertex angle and the two angles adjacent to the base are called base angles.
Important Formulas Related to an Isosceles Triangle:
The formula to find the area of an isosceles triangle is Area \( (A)~=~\frac{1}{2}~\times~b~\times~h \)
Where, ‘b’ is the base and ‘h’ is the height(altitude) of the triangle.
The perimeter of the isosceles triangle is calculated by using the formula:
Perimeter \( (P)~=~2a~+~b \)
Here, ‘a’ refers to the length of the legs and ‘b’ is the length of the third side, that is the base.
What is an Equilateral Triangle?
An equilateral triangle is a triangle in which all three sides are congruent and all the angles are equal in measure. It is also known as an equiangular triangle since the measure of each angle is 60°.
Important Formulas Related to an Equilateral Triangle
The formulas to get the area and perimeter of an equilateral triangle are:
Area of an equilateral triangle \( ~=~\frac{\sqrt{3}}{4}~a^2 \)
Perimeter of an equilateral triangle \( ~=~3a \)
Here, ‘a’ refers to the side length of an equilateral triangle.
Difference Between Isosceles and Equilateral Triangles
Solved Examples
Example 1:
The length of each side of a traffic sign board is 30 centimeters, what will be its area?
Solution:
As mentioned, each side of the traffic sign board is 30 cm, that is a = 30 cm, since it is an equilateral triangle.
Use the formula of the area of an equilateral triangle to find the area of the sign board.
\( A~=~\frac{\sqrt{3}}{4}~a^2 \) [Write the formula]
\( ~~=~\frac{\sqrt{3}}{4}~\times~(30)^2 \) [Substitute 30 for a]
\( ~~=~\frac{\sqrt{3}}{4}~\times~900 \) [Find the square of 30]
\( ~~=~\sqrt{3}~\times~225 \) [Divide]
\( ~~=~1.732~\times~225\) [Substitute 1.732 for \( \sqrt{3} \)]
\( ~~=~389.7 \) [Multiply]
Therefore, the area of the traffic sign board is 389.7 square centimeters.
Example 2:
If the base length of a cloth hanger is 33 centimeters and the two equal sides have a length of 27 centimeters. Find the perimeter of the hanger.
Solution:
The base length of a cloth hanger is 33 cm and the two equal sides are 27 cm.
We can say that the cloth hanger is an isosceles triangles as it has two equal sides.
b = 33 cm
a = 27 cm
Use the formula for the perimeter of the isosceles triangle to find the perimeter of the hanger.
\( P~=~2a~+~b \) [Write the formula for perimeter]
\( ~~=~2(27)~+~33 \) [Substitute the values]
\( ~~=~54~+~33 \) [Multiply]
\( ~~=~87 \) [Add]
Thus, the perimeter of the cloth hanger is 87 centimeters.
Example 3:
What will be the value of the vertex angle if the sum of the two base angles is 130°?
Solution:
Given that the sum of two base angles is 130°.
The sum of all the interior angles of the triangle is 180°, we need to subtract the base angles from the total sum to get the vertex angle.
That is,
Vertex angle = sum of all the angles – base angles
= 180°– 130°
= 50°
Thus, the vertex angle of the triangle is 50°.
On the basis of side length, there are three types of triangles:
- Isosceles triangle
- Equilateral triangle
- Scalene triangle
The name of a regular triangle is an equilateral triangle.
The isosceles triangle is divided into three different types namely, isosceles acute triangle, isosceles right triangle, and isosceles obtuse triangle.
Each vertex angle of an equilateral triangle is 60 degrees.