Area of Isosceles Triangle Formula

An Isosceles Triangle is one in which two sides are equal in length. By this definition , an equilateral Triangle is also an Isosceles Triangle. Let us consider an Isosceles Triangle as shown in the following diagram (whose sides are known, say a, a and b).

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As the altitude of an Isosceles Triangle drawn from its vertical angle is also its angle bisector and the median to the base (which can be proved using congruence of Triangles), we have two right Triangles as shown in the figure above.

Using the Pythagorean theorem, we have the following result.

\[AC^2=AD^2+DC^2\Rightarrow h^2=a^2-(\frac{b}{2})^2\Rightarrow h=\sqrt{a^2-\frac{b^2}{4}}\]

So the area of the Isosceles can be calculated as follows.

\[area=\frac{1}{2}bh=\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}\]

The perimeter of the Isosceles Triangle is relatively simple to calculate, as shown below.

\[perimeter=2a+b\]

Also note that the area of the Isosceles Triangle can be calculated using Heron’s formula.

Area = \[\sqrt{s(s-a)(s-a)(s-b)}\]......(1)

s=\[\frac{a+a+a}{2}=a+\frac{b}{2}\]

Placing the value of ‘s’ in eq (i)

Area=\[\sqrt{(a+\frac{b}{2})(a+\frac{b}{2}-a)(a+\frac{b}{2}-a)(a+\frac{b}{2}-b)}\]

\[\sqrt{(a+\frac{b}{2})(\frac{b}{2})(\frac{b}{2})(\frac{2a-2b+b}{2})}\]

\[\sqrt{(\frac{2a+b}{2})(\frac{b^2}{4})(\frac{2a-b}{2})}\]

\[\frac{b}{2}\sqrt\frac{4a^2-b^2}{4}\]=\[\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}\]

Trigonometry can also be used in the case of Isosceles Triangles more easily because of the congruent right Triangles. 

Let’s look at an example to see how to use these formulas.

Question: If the base and the area of an Isosceles Triangle are respectively

8cm and 12cm2, then find its perimeter.

Solution:

\[b=4cm, area=12cm^2\]

\[area=\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}\]

\[12=\frac{8}{2}\sqrt{a^2-\frac{8^2}{4}}\]

\[3=\sqrt{a^2-16}=a^2=25\Rightarrow a=5cm\]

Thus the perimeter of the Isosceles Triangle is calculated as follows.

\[perimeter=2a+b=2\times 5+8=18cm\]

Why don’t you try solving the following sum to see if you have mastered using these formulas?

Question: Calculate the area of an Isosceles Triangle whose sides are 13 cm, 13 cm and 24 cm.

Options:

1. 60cm2

2. 45cm2

3. 30cm2

4. none of these

Answer: (a)

Solution:

\[a=13cm,b=24cm\]

\[Area=\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}\]

\[Area=\frac{24}{2}\sqrt{13^2-\frac{24^2}{4}}=12=\sqrt{169-144}=12\times 5=60cm^2\]

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Basic Rules about Isosceles Triangles

Triangle is constituted of three sides. But to identify a Triangle as an Isosceles Triangle it has to have some definite characteristics. One of the main characteristics of an Isosceles Triangle is that the two legs of the Triangle must be equal in length. Apart from these two, there will be the base of the Triangle that is not equal to the length of the two sides. There are also some other theorems that consider another feature to be equally important in order to identify a particular Triangle as an Isosceles Triangle. According to this theorem, the angles that will be opposite to the sides of the Triangle that are equal in length will also be equal.

Purpose of Learning the Concepts of Isosceles Triangles

The students should know the basic and foundational concepts of Geometry. The concept of an Isosceles Triangle is included in the syllabus of the students so that they can develop their idea of angles and the lengths of a Triangle. Questions from each of the topics of Geometry should be included in the question papers. If the students want to score good marks in Mathematics then they should understand the concept of every chapter that is included in the syllabus.

The foundational knowledge of Geometry will be particularly helpful for those students who want to pursue their academics or want to undertake research projects in the field of Mathematics. They may have to solve questions regarding this particular chapter in order to qualify for various other entrance Examinations and engineering Examinations. 

Understanding the concept of an Isosceles Triangle is important because, in order to pursue their career in the engineering field of studies, the students need to find out the values of unknown angles and they should be very good at determining the shapes and lengths of various objects.

Simple Methods to Prove that a Particular Triangle is Isosceles

There are some simple facts that the students need to remember in order to prove or identify a Triangle as an Isosceles Triangle. The first rule is to check if the two sides of the Triangle are equal in length or not. If the length of the two sides of a Triangle is equal in length, then you have to check whether the base angles of the Triangle are equal or not. The base angles signify those two angles that are formed between the base of the Triangle and the two sides of the Triangle that are equal in length. If you find that the third angle, that is the angle between the two sides of the Triangle that are equal in length, is 90 degrees, then you can conclude that this particular Triangle is a Right Isosceles Triangle.

Can Isosceles Triangles be considered as Equilateral Triangles?

To find the answer to this particular question, you need to understand the concept of equilateral Triangles. Equilateral Triangles are those Triangles that are constituted by three sides that are equal in length. Since all the three sides of an equilateral Triangle are of similar length, it is quite predictable that the angles formed between these three sides of the Triangle are also equal.

Isosceles Triangles are formed by three sides. Among these three, any two sides will be equal in length and the angles formed at the opposite of the sides will also be equal. Because of this special characteristic of Isosceles Triangles, it can be considered that every equilateral Triangle can also be an Isosceles Triangle. But every Isosceles Triangle cannot be considered an equilateral Triangle because all the three angles of the Isosceles Triangle will not be equal.