Formula for the Perimeter of an Isosceles Triangle
The perimeter of any two dimensional shape is the total length of its boundary. Here in the image we can observe a triangle with two congruent sides. We can add the length of all three sides of the triangle to get its perimeter.
The formula used to find the perimeter of an isosceles triangle is,
Where, P is the perimeter of isosceles triangle
\( a \) is the measure of congruent edges (or sides)
And \( b \) is the measure of the third side.
As discussed above the perimeter is a measure of length itself, so the unit of measurement of perimeter is the same as that of length, which are, millimeters, centimeters, meters, kilometers, feet, inches, yards or miles.
Solved Examples
Example 1: A warning sticker is in the shape of an isosceles triangle. The length of the congruent edges is 1.4 feet each, and the length of the third edge is 1.2 feet. Find the perimeter of the sticker.
Solution:
\( P~=~2a~+~b \) Write the formula for the perimeter
\( P~=~2~\times~1.4~+~1.2 \) Substitute 1.4 for \(a \) and 1.2 for \( b \)
\( P~=2.8~+~1.2 \) Multiply
\( P~=~4 \) Add
So, the perimeter of the triangular warning sticker is 4 feet.
Example 2: The perimeter of an isosceles triangle is 23 yards. Find the measure of the congruent edges if the third edge is 7 yards.
Solution:
\( P~=~2a~+~b \) Write the formula for perimeter
\( 23~=~2a~+~7 \) Substitute 23 for P and 7 for \( b \)
\( 23~-~7~=~2a~+~7~-~7 \) Subtract 7 from each side
\( 16~=~2a \) Simplify
\( \frac{16}{2} ~=~\frac{2a}{2}\) Divide each side by 2
\( 8~=~a\) Simplify
So, the measure of congruent edges is 8 yards each.
Example 3: Find the perimeter of an isosceles right angle triangle whose hypotenuse is 6 inches.
Solution:
We have been given that the triangle is right angled and isosceles with the hypotenuse measuring 6 inches. So the two legs will be the congruent sides of the triangle.
To calculate the perimeter we must first find the measure of the legs.
Let each leg be \( x\) inches in length.
Apply the Pythagorean theorem:
\( (Hypotenuse)^2~=~(leg)^2~+~(leg)^2\) Write Pythagorean theorem
\( 6^2~=~x^2~+~x^2\) Substitute \( x\) for the legs and 6 for hypotenuse
\( 36~=~2x^2\) Simplify
\( \frac{36}{2}~=~\frac{2x^2}{2}\) Divide each side by 2
\( 18~=~x^2\) Simplify
\( \sqrt{18}~=~\sqrt{x^2}\) Take positive square root on each side
\( 3\sqrt{2}~=~x\) Simplify
So, the measure of the legs is \(3 \sqrt{2}\) inches each.
Now let us find the perimeter of this triangle:
\( P~=~2a~+~b\) Write the formula for the perimeter
\( P~=~2~\times~3\sqrt{2}~+~6\) Substitute \( 3\sqrt{2}\) for a and 6 for b
\( P~=~6\sqrt{2}~+~6\) Multiply
\( P~=~6(\sqrt{2}~+~1)\)
Therefore the perimeter of the isosceles right triangle is \( 6(\sqrt{2}~+~1)\) inches.
Example 4: Hazel made a triangular poster as a part of an art project. She decides to border the poster with coloured tape. Find the length of tape required if the dimensions of the poster are 2 feet, 2.5 feet and 2 feet.
Solution :
Dimensions of the poster are 2 feet, 2.5 feet and 2 feet.
The triangular poster has two congruent sides, so it is an isosceles triangle. To find the length of tape required for the border we have to calculate the perimeter of the poster.
\( P~=~2a~+~b\) Write the formula for the perimeter
\( P~=~2~\times~2~+~2.5\) Substitute 2 for a and 2.5 for b
\( P~=~6.5\) Simplify
So, Hazel needs 6.5 feet colored tape to border the poster.
A triangle with any two congruent sides is called an isosceles triangle.
There are two congruent sides in an isosceles triangle.
The perimeter of a triangle is calculated by adding the length of all three sides of the triangle.
A triangle with two sides of equal length and one interior angle equal to 90° is called a right isosceles triangle.
The angles opposite to the congruent sides of an isosceles triangle are equal in measure. So, there are two equal angles in an isosceles triangle.