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Area of Isosceles Triangle

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Area of an isosceles triangle is the space enclosed by the sides of a triangle. The general formula for finding the area of the isosceles triangle is given by half the product of the base and height of the triangle. Other than this different formulas are used to find the area of triangles. Triangles are classified depending on their sides, different types of triangles based on sides are given below:

Equilateral Triangle: Triangle with all three sides equal.

Isosceles Triangle: Triangle with any two sides equal.

Scalene Triangle: Triangle with all sides unequal.

What is the Isosceles Triangle?

An isosceles triangle is one with two equal sides. The two angles opposing the two equal sides are also equal. Assume that in a triangle △ABC, if the sides AB and AC are equal, ABC is an isosceles triangle with ∠B = ∠C. The isosceles triangle is described by the theorem “If the two sides of a triangle are equal, then the angle opposite to them are likewise equal”.

Isosceles triangle

 

What is the Area of an Isosceles Triangle?

Total space covered inside the boundary of an isosceles triangle is termed its area. In an isosceles triangle, the area can be easily calculated if the height and the base of the triangle are given. The product of half with base and height of the isosceles triangle gives the area of the Isosceles triangle.

Isosceles Triangle Formula

Area of an isosceles triangle is given by the formula listed below:

Area = ½ × base × Height

Also,

Perimeter of isosceles triangle (P) = 2a + b
Altitude of isosceles triangle (h) = √(a2 − b2/4)

where, a, b are the sides of a isosceles triangle.

Area of Isosceles Triangle Formulas

Various formulas are used to find the Area of the Isosceles Triangle. Few of the most used formulas for the area of the isosceles triangle are listed below:

  • If base and height are given A = ½ × b × h
  • If all three sides are given A = ½[√(a2 − b2 ⁄4) × b]
  • If the length of 2 sides and an angle between them is given A = ½ × b × c × sin(α)
  • If two angles and the length between them is given A =
  • For an isosceles right triangle A = ½ × a2

Area of Isosceles Triangle Formula with Sides

When the length of equal sides and the length of the base of an isosceles triangle are given, then the height of the triangle can also be calculated by the given formula:

Altitude of an Isosceles Triangle = √(a2 − b2/4)

Area of Isosceles Triangle (if all sides are given) = ½[√(a2 − b2 /4) × b]

where,
b = base of the isosceles triangle
a = length of the two equal sides

How to Find the Area of an Isosceles Triangle?

To find the area of an Isosceles triangle follow these steps:

Step 1: Mark the length(l) and breadth(b) of the given triangle.

Step 2: Multiply the values obtained in step 1 and divide them by 2.

Step 3: The result obtained is the required area, it is measured in m2

Derivation for Area of Isosceles Triangle 

If the lengths of an isosceles triangle’s equal sides and base are known, the triangle’s height or altitude may be computed. The formula for calculating the area of an isosceles triangle with sides is as follows:

Isosceles triangle area = ½[√(a2 − b2 /4) × b]

where,

b = the isosceles triangle’s base
a = the length of two equal sides

Derivation for Isosceles Triangle Area

 

From the above figure, we have,

AB = AC = a (sides of equal length)

BD = DC = ½ BC = ½ b (Perpendicular from the vertex angle ∠A bisects the base BC)

Using Pythagoras theorem on ΔABD,

a2 = (b/2)2 + (AD)2

AD = [Tex]\sqrt{a^2 – \frac{b^2}{4}} [/Tex]

The altitude of an isosceles triangle = [Tex]\sqrt{a^2 – \frac{b^2}{4}} [/Tex]

It is known that the general formula of area of the triangle is, Area = ½ × b × h

Substituting value for height, we get

Area of isosceles triangle = ½[√(a2 − b2 /4) × b]

Area of Right Angled Isosceles Triangle

Area of an Isosceles Right Triangle is given by the formula 

Area of Isosceles Right Triangle Formula

 

Formula for Isosceles Right Triangle Area= ½ × a2

Derivation:

Area of an isosceles triangle (Area) = ½ ×base × height

Area = ½ × a × a = a2/2

Perimeter of Isosceles Right Triangle P = (2+√2)a

Derivation:

Perimeter of an isosceles right triangle is the sum of all the sides of an isosceles right triangle.

Let the two equal sides be a. By Pythagoras theorem the unequal side is a√2.

Perimeter of isosceles right triangle = a+a+a√2
                                                        = 2a+a√2
                                                        = a(2+√2)
                                                        = a(2+√2)

Area of Isosceles Triangle using Trigonometry

When the Length of the two Sides and the Angle between them are given,

A = ½ × b × c × sin(α)

where,
b, c are sides of a given triangle
α is the angle between them

When the two angles and sides between them are given,

A =

where,
c is sides of a given triangle
α, β is the angle associated with them

Solved Examples on Area of Isosceles Triangle

Example 1: Find the area of an isosceles triangle with an equal side of 13 cm and a base of 24 cm.

Solution:

We have, a = 13 and b = 24.

Area of isosceles triangle is given by, 

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{13^2 – \frac{24^2}{4}}\right) × 24 [/Tex]

= 1/2 × 5 × 24

= 60 cm2

Example 2: Find the area of an isosceles triangle with an equal side of 10 cm and a base of 12 cm.

Solution:

We have, a = 10 and b = 12.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{10^2 – \frac{12^2}{4}}\right) × 12 [/Tex]

= 1/2 × 8 × 12

= 48 cm2

Example 3: Find the area of an isosceles triangle with an equal side of 5 cm and a base of 6 cm.

Solution:

We have, a = 5 and b = 6.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{5^2 – \frac{6^2}{4}}\right) × 6 [/Tex]

= 1/2 × 4 × 6

= 12 cm2

Example 4: Find the area of an isosceles triangle with an equal side of 15 cm and a base of 24 cm.

Solution:

We have, a = 15 and b = 24.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{15^2 – \frac{24^2}{4}}\right) × 24 [/Tex]

= 1/2 × 9 × 24

= 108 cm2

Example 5: Find the area of an isosceles triangle with an equal side of 17 cm and a base of 30 cm.

Solution:

We have, a = 17 and b = 30.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{17^2 – \frac{30^2}{4}}\right) × 30 [/Tex]

= 1/2 × 8 × 30

= 120 cm2

Example 6: Find the area of an isosceles triangle with an equal side of 20 cm and a base of 24 cm.

Solution:

We have, a = 20 and b = 24.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{20^2 – \frac{24^2}{4}}\right) × 24 [/Tex]

= 1/2 × 16 × 24

= 192 cm2

Example 7: Find the area of an isosceles triangle with an equal side of 25 cm and a base of 30 cm.

Solution:

We have, a = 25 and b = 30.

Area of isosceles triangle is given by,

A = [Tex]\frac{1}{2} ×\left(\sqrt{a^2 – \frac{b^2}{4}}\right) × b [/Tex]

[Tex]\frac{1}{2} ×\left(\sqrt{25^2 – \frac{30^2}{4}}\right) × 30 [/Tex]

= 1/2 × 20 × 30

= 300 cm2

FAQs on Area of Isosceles Triangle

Question 1: What is the Area of an Isosceles Triangle?

Answer:

Area of a figure is the space enclosed by the boundaries of the figure. So, the area of an isosceles triangle can be defined as the space occupied by an isosceles triangle.

Question 2: What do you mean by an Isosceles Triangle?

Answer:

An isosceles triangle can be defined as a triangle that has two equal sides, also opposite angles are also equal in an Isosceles triangle. Some of the properties of an Isosceles triangle are:

  • Two equal sides of an isosceles triangle are equal and angle between them is termed as vertex angle or apex angle.
  • Side opposite to the vertex angle is termed as base and the base angles are also equal in an Isosceles Triangle.

Question 3: Write the Formula for finding the Area of an Isosceles Triangle.

Answer:

For calculating the area of an Isosceles triangle, the following formula is used:

A = ½ × b × h

where,

b is the base of Triangle,
h is the height of Triangle.

Question 4: Write the Formula for finding the Perimeter of an Isosceles Triangle.

Answer:

For calculating the perimeter of an isosceles triangle the following formula is used:

P = 2a + b

where,

a, b are sides of an isosceles triangle.

Question 5: Write the formula for the area of the isosceles right triangle.

Answer:

For calculating the area of a right angled Isosceles triangle, the following formula is used:

A = ½ × a2

where,

a is the side of the Triangle,

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Last Updated : 19 Mar, 2024
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