Isosceles Triangle Perimeter Formula: Definition, Formulas, Problems

You have landed on the right page to learn about Isosceles Triangle Perimeter Formula. A triangle is the smallest polygon with three sides. We can classify the triangle into three types based on its sides as equilateral, isosceles and scalene. An isosceles triangle has any two sides equal. The perimeter of an isosceles triangle means the sum of the lengths of three sides of the triangle.

We find the perimeter when putting up lights around the house or fencing the backyard garden, planning the construction of the building, building a swimming pool, etc. Let us dive into the article to understand the concept of the perimeter of the isosceles triangle. Continue reading to know more.

What is a Triangle?

A triangle is a polygon that has three sides. We can define it as a figure bounded or enclosed by three line segments. It is the smallest polygon.

A triangle will have three sides and three vertices. Based on the measure of the length of the sides, we can classify the triangle as an equilateral triangle, isosceles triangle and scalene triangle. An equilateral triangle has all sides equal. An isosceles triangle has any two sides equal, and a scalene triangle has all three sides different. Let us learn about the isosceles triangle in detail.

What is an Isosceles Triangle?

If any two of the three sides of a triangle are equal to each other, then the triangle is called an isosceles triangle.

In an isosceles triangle, the two angles opposite to the two equal sides are congruent.

Perimeter

The word perimeter is taken from the Greek word ‘peri’, which means around, and ‘metron’, which means measure.

The total length of the edges of any shape is known as the perimeter. For any polygon, the perimeter is the sum of the side length of it. For any closed figure except polygon, perimeter means the length of its boundary or the outer line.

Perimeter of a Triangle

We can compute the perimeter of a triangle by adding the length of its three sides

If a triangle has \(3\) sides \(a,\,b\) and \(c,\) then the perimeter of the triangle is:
\(P = \left( {a + b + c} \right)\,{\rm{units}}\)

Perimeter of an Isosceles Triangle

The perimeter of an isosceles triangle is the sum of the length of all the sides. Since an isosceles triangle has two identical sides, the perimeter is equal to the sum of double the length of equal sides plus the length of the other side. We can measure perimeter in units such as millimetres \(\left( {{\rm{mm}}} \right){\rm{,}}\) inches \(\left( {{\rm{in}}} \right){\rm{,}}\) yards \(\left( {{\rm{yd}}} \right){\rm{,}}\) centimetres \(\left( {{\rm{cm}}} \right){\rm{,}}\) and meters \(\left( {\rm{m}} \right)\) and so on. 

Formula for Perimeter of an Isosceles Triangle

If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter is:
\(P = a + a + b = \left( {2a + b} \right)\,{\rm{units}}\)

Perimeter of an Isosceles Right Triangle

A triangle with one right-angle is called a right-angled triangle. We know that the right angle means \({90^{\rm{o}}}.\) In a triangle, two right-angle are not possible as the sum of all three angles is \({180^{\rm{o}}}.\) The side opposite to the \({90^{\rm{o}}}\) angle is the longest side of the triangle, known as the hypotenuse. The other two angles measure less than \({90^{\rm{o}}}\) each, and their sum will be equal to \({90^{\rm{o}}}.\)

In an isosceles right-angled triangle, two sides are identical, and the third side is the hypotenuse. The two equal angles of an isosceles right triangle measure \({45^{\rm{o}}}.\) We can obtain the perimeter of an isosceles right-angled triangle by finding the sum of the length of all sides. If the lengths of the two equal sides are \(l\) and the length of the hypotenuse is \(h\,{\rm{units}},\) the perimeter of an isosceles right triangle is:

\(P=h+l+l\)

As described above, the perimeter of an isosceles right triangle \(△PQR\) is 

\(P = PQ + QR + RP = h + 2l\)

One of the most important formulas associated with any right triangle is the Pythagoras theorem.
Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. In an isosceles right triangle, the other two sides are congruent.

Now, if we apply the Pythagoras theorem, we have,
\(h = \sqrt {{l^2} + {l^2}} = \sqrt 2 \times l\)
\( \Rightarrow h = \sqrt 2 l\)
\( \Rightarrow l = \frac{h}{{\sqrt 2 }}\)
We can substitute this value to find the perimeter if one of them is unknown.
Suppose the length of equal sides \((l)\) is given, then the perimeter of an isosceles right triangle will be
\(P = h + 2l \Rightarrow \sqrt 2 l + 2l = l(\sqrt 2 + 2)\)
Similarly, if the length of the hypotenuse \((h)\) is given, then the perimeter of an isosceles right triangle will be
\(P = h + 2l \Rightarrow h + 2 \times \frac{h}{{\sqrt 2 }} = h + \sqrt 2 h = h(1 + \sqrt 2 )\)

Solved Examples- Isosceles Triangle Perimeter Formula

Q.1. Identify the type of the triangle given below.

Ans: In the given figure, two angles are of equal measurement. Hence, the opposite sides to these angles will be equal. Since the two sides are equal, this is an isosceles triangle.

Q.2. Calculate the perimeter of an isosceles triangle whose base is \({\rm{3}}\,{\rm{cm}}{\rm{,}}\) and congruent sides are \({\rm{5}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given that congruent sides are \({\rm{5}}\,{\rm{cm}}{\rm{,}}\) and the base is \({\rm{3}}\,{\rm{cm}}{\rm{.}}\)
If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, then:
\(P = a + a + b = (2a + b){\rm{ units}}{\rm{.}}\)
Therefore, the perimeter of the given triangle\( = 2 \times 5 + 3 = 13\;{\rm{cm}}\)
Hence, the perimeter of the given isosceles triangle is \({\rm{13}}\,{\rm{cm}}{\rm{.}}\)

Q.3. Calculate the base of an isosceles triangle whose perimeter is \({\rm{30}}\,{\rm{cm}}{\rm{,}}\) and congruent sides are \({\rm{8}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given that congruent sides are \({\rm{8}}\,{\rm{cm}}{\rm{,}}\) and the perimeter is \({\rm{30}}\,{\rm{cm}}{\rm{.}}\)
If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter
\(P = a + a + b = (2a + b){\rm{ units}}\)
Now, substituting \(a=8\) in the above we have,
\((2a + b) = 2 \times 8 + b = 16 + b\)
According to the statement, \(16 + b = 30\)
\( \Rightarrow b = 30 – 16 = 14\)
Therefore, the base of the isosceles triangle is \({\rm{14}}\,{\rm{cm}}{\rm{.}}\)

Q.4. The base of an isosceles triangle is \({\rm{10}}\,{\rm{cm}}{\rm{,}}\) and the perimeter is \({\rm{40}}\,{\rm{cm}}{\rm{.}}\) Find the other two sides.
Ans: It is given that the base is \({\rm{10}}\,{\rm{cm}}\) and the perimeter is \({\rm{40}}\,{\rm{cm}}{\rm{.}}\)
If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an Isosceles triangle, its perimeter \(P = a + a + b = (2a + b){\rm{ units}}\)
Now, substituting \(b=10\) in the above we have,
\((2a + b) = 2a + 10\)
According to the statement, \(2a + 10 = 40\)
\( \Rightarrow 2a = 40 – 10 = 30\)
\( \Rightarrow a = \frac{{30}}{2} = 15\)
Therefore, the other two sides of the isosceles triangle are \({\rm{15}}\,{\rm{cm}}{\rm{.}}\)

Q.5. The hypotenuse of an isosceles triangle is \({\rm{5}}\,{\rm{cm}}{\rm{.}}\) Then, find the perimeter.
Ans: Given that the hypotenuse is \({\rm{5}}\,{\rm{cm}}{\rm{.}}\)
We know, if the length of the hypotenuse \((h)\) is given, then the perimeter of an isosceles right triangle will be
\(P = h + 2l\)
\( \Rightarrow h + 2 \times \frac{h}{{\sqrt 2 }} = h + \sqrt 2 h = h(1 + \sqrt 2 )\)
Hence, the perimeter of the given triangle\( = 5(1 + \sqrt 2 )\,{\rm{cm}}\)

Summary

In this article, we learned about the perimeter and the perimeter of an isosceles triangle. We learned about the properties and deduced formulas to find the perimeter of an isosceles right triangle. We also solved some problems related to the concept.

Frequently Asked Questions

Q.1. What is the formula for the perimeter of an isosceles triangle?
Ans: An isosceles triangle is a triangle with two equal sides. If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter is given by \(P = a + a + b = (2a + b)\,{\rm{units}}{\rm{.}}\)

 Q.2. How do you find the perimeter from the side of an isosceles triangle?
Ans: The perimeter of an isosceles triangle is the sum of the length of all the sides. Since an isosceles triangle has two identical sides, the perimeter is double the length of the equal sides plus the length of the other side. We can measure perimeter in units such as millimetres \(\left( {{\rm{mm}}} \right){\rm{,}}\) inches \(\left( {{\rm{in}}} \right){\rm{,}}\) yards \(\left( {{\rm{yd}}} \right){\rm{,}}\) centimetres \(\left( {{\rm{cm}}} \right){\rm{,}}\) and meters \(\left( {\rm{m}} \right)\) and so on. 

 Q.3. How do you find the third side of an isosceles triangle if the perimeter is given?
Ans: An isosceles triangle has two identical sides. Hence, its perimeter is double the length of the equal sides plus the length of the other side.  If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter
\(P = a + a + b = (2a + b)\,{\rm{units}}\)
Substituting the values of \(P\) and \(a\) in the above equation, we can find the third side of the isosceles triangle.

Question-4: Can an isosceles triangle be a right triangle?
Answer: Yes, an isosceles triangle can be a right triangle. The perimeter of an isosceles right-angled triangle is obtained by finding the sum of the length of all sides. If the lengths of the other two sides are \(‘l’\) and the length of the hypotenuse is \(‘h’\,{\rm{units,}}\) and then the perimeter of an isosceles right triangle is
\(P=h+l+l\)

Q.5. How do you find the measure of equal sides of an isosceles triangle if its perimeter and the measurement of the third side is given?
Ans: If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter is given by \(P = a + a + b = (2a + b)\,{\rm{units}}{\rm{.}}\)
If the perimeter is known and the measure of the third side is known, we can find the equal side by substituting the values in the above expression.

Now you are provided with all the necessary information on the isosceles triangle perimeter formula and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.