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Area of Isosceles Triangle
The area of an isosceles triangle is the amount of space enclosed between the sides of the triangle. Besides the general area of the isosceles triangle formula, which is equal to half the product of the base and height of the triangle, different formulas are used to calculate the area of triangles, depending upon their classification based on sides. These different types based on sides are given below:
 Equilateral Triangle A triangle with all sides equal.
 Isosceles Triangle A triangle with any two sides/angles equal.
 Scalene Triangle A triangle with all unequal sides.
Let us understand the area of the isosceles triangle in detail in the following section.
What is Area of an Isosceles Triangle?
The area of an isosceles triangle is the total space or region covered between the sides of an isosceles triangle in twodimensional space. An isosceles triangle is defined as a triangle having two sides equal, which also means two equal angles. Here are some properties of an Isosceles triangle that distinguish it from other types of triangles:
 The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.
 The side opposite the vertex angle is called the base and base angles are equal.
 The perpendicular from the vertex angle bisects the base and it also bisects the vertex angle.
The area of an isosceles triangle is expressed in square units. Therefore, some units that can be used to represent the area of an isosceles triangle are m^{2}, cm^{2}, in^{2}, yd^{2}, etc.
Area of an Isosceles Triangle Formulas
The area of an isosceles triangle refers to the total space covered by the shape in 2D. The area of an isosceles triangle can be calculated in many ways based on the known elements of the isosceles triangle. The general basic formula that can be used to calculate the area of an isosceles triangle using height is given as, (1/2) × Base × Height
The following table summarizes different formulas that can be used to calculate the area of an isosceles triangle, for a different set of known parameters.
Known Parameters of Given Isosceles Triangle  Formula to Calculate Area (in square units) 

A = ½ × b × h 

A = ½[√(a^{2} − b^{2 }⁄4) × b] 

A = ½ × b × a × sin(α) 

A = [a^{2}×sin(β/2)×sin(α)] 

A = ½ × a^{2} 
where,
 b = base of the isosceles triangle
 a = measure of equal sides of the isosceles triangle
 α = measure of equal angles of the isosceles triangle
 β = measure of the angle opposite to the base
Area of Isosceles Triangle Using Sides
If the length of the equal sides and the base of an isosceles triangle are known, then the height or altitude of the triangle can be calculated. The formula to calculate the area of an Isosceles triangle using sides is given as,
Area of isosceles triangle using only sides = ½[√(a^{2}  b^{2}/4) × b]
where,
 b = base of the isosceles triangle
 h = height of the isosceles triangle
 a = length of the two equal sides
Derivation:
From the above figure, we know:
BD = DC = ½ BC = ½ b (perpendicular from the vertex angle bisects the base)
AB = AC = a (equal sides of an isosceles triangle)
Applying Pythagoras' theorem for ΔABD, we get:
a^{2 }= (b/2)^{2} + (AD)^{2}
AD = √(a^{2} − b^{2}/4)
The altitude of an isosceles triangle = √(a^{2} − b^{2}/4)
Also, we know the general area of the triangle formula is given as:
Area = ½ × b × h
Substituting value for height:
Area of isosceles triangle using only sides = ½[√(a^{2} − b^{2} /4) × b]
Isosceles Triangle Area Using Heron’s Formula
The area of an isosceles triangle formula can be easily derived using Heron’s formula as explained in the following steps. Heron's formula is used to find the area of a triangle when the measurements of its 3 sides are given.
Derivation:
The Heron's formula to find the area, A of a triangle whose sides are a,b, and c is:
A = √s(sa)(sb)(sc)
where,
 a, b, and c are the sides of the triangle.
 s is the semi perimeter of the triangle.
We know that the perimeter of a triangle with sides a, b, and c is a + b + c. Here, s is half of the perimeter of the triangle, and hence, it is called semiperimeter.
Thus, the semiperimeter is:
s = (a + b + c)/2
Now, for an isosceles triangle,
s = ½(a + a + b)
⇒ s = ½(2a + b)
or, s = a + (b/2)
Also,
Area = √[s(s−a)(s−b)(s−c)]
or, Area = √[s (s−a)^{2} (s−b)]
⇒ Area = (s−a)^{ }× √[s (s−b)]
Substituting the value for “s”
⇒ Area = (a + b/2 − a)^{ }× √[(a + b/2) × ((a + b/2) − b)]
⇒ Area = b/2 × √[(a + b/2) × (a − b/2)]
Area of isosceles triangle = b/2 × √(a^{2} − b^{2}/4) square units
where,
 b = base of the isosceles triangle
 a = length of the two equal sides
Isosceles Triangle Area Using Trigonometry(SAS and ASA)
The formula to find area of an isosceles triangle using length of 2 sides and angle between them or using 2 angles and length between them can be calculated using basic trigonometry concepts.
Using 2 sides and angle between them:
Area = ½ × b × a × sin(α) square units
where,
 b = base of the isosceles triangle
 a = length of the two equal sides
 α = angle between the unequal sides
Using 2 angles and length between them:
Area = [a^{2 }× sin(β/2) × sin(α)] square units
where,
 a = length of the two equal sides
 α, β = angles in an isosceles triangle
Area of an Isosceles Right Triangle
A right isosceles triangle is defined as the isosceles triangle which has one angle equal to 90°. The Formula to calculate the area for an isosceles right triangle can be expressed as,
Area = ½ × a^{2}
where a is the length of equal sides.
Derivation:
Let the equal sides of the right isosceles triangle be denoted as "a", as shown in the figure below:
The length of the hypotenuse, BC can be calculated using Pythagoras' Theorem,
BC^{2} = a^{2 }+ a^{2}
BC = √2 a
Area = ½ × base × height
Area = ½ × a × a = a^{2}/2 square units
Area of Isosceles Triangle Examples

Example 1: Find the area of an isosceles triangle given the length of the base is 10 cm and height is 17 cm?
Solution:
Base of the triangle (b) = 10 cm
Height of the triangle (h) = 17 cm
Area of Isosceles Triangle = (1/2) × b × h
= (1/2) × 10 × 17
= 5 × 17
= 85 cm^{2}
Answer: The area of the given isosceles triangle is 85 cm^{2}.

Example 2: Find the length of the base of an isosceles triangle whose area is 243 cm^{2}, and the altitude of the triangle is 9 cm.
Solution:
Area of the triangle, A = 243 cm^{2}
Height of the triangle (h) = 9 cm
The base of the triangle = b =?
Area of Isosceles Triangle = (1/2) × b × h
243 = (1/2) × b × 9
243 = (b × 9)/2
b = (243 × 2)/9
b = 54 cm
Answer: The altitude of the given isosceles triangle is 54 cm.

Example 3: Find the length of the equal sides of an isosceles triangle whose base is 24 cm and the area is 60 cm^{2}.
Solution:
We know that,
The base of the isosceles triangle = 24 cm
Area of the isosceles triangle = 60 cm^{2}
Area of isosceles triangle = b/2 × √(a^{2} − b^{2}/4)
Therefore,
60 = (24/2)√(a^{2 }− 24^{2}/4)
60 = 12√(a^{2 }− 144)
5 = √(a^{2}−144)
Squaring both sides, we get,
25 = a^{2}−144
a^{2 }= 169
⇒a = 13 cm
Answer: The length of the equal sides of the given isosceles triangle is 13 cm.
FAQs on Area of an Isosceles Triangle
What Does the Area of an Isosceles Triangle Mean?
The area of a figure is the region enclosed by the figure. Thus, the area of an isosceles triangle means the total space enclosed by an isosceles triangle.
What is an Isosceles Triangle?
An isosceles triangle is defined as a triangle having two sides equal, which also means two equal angles. Here are some properties of an Isosceles triangle that distinguish it from other types of triangles:
 The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.
 The side opposite the vertex angle is called the base and base angles are equal.
 The perpendicular from the vertex angle bisects the base and it also bisects the vertex angle.
What is the Formula for Area of Isosceles Triangle?
The area of an isosceles triangle refers to the total space covered by the shape in 2D. The area of an isosceles triangle can be calculated in many ways based on the known elements of the isosceles triangle.
 Using base and Height: Area = ½ × b × h
 Using all three sides: Area = ½[√(a^{2} − b^{2} ⁄4) × b]
 Using the length of 2 sides and an angle between them: Area = ½ × b × a × sin(α)
 Using two angles and length between them: A = [a^{2}×sin(β/2)×sin(α)]
 Area formula for an isosceles right triangle: Area = ½ × a^{2}
where,
b = base of the isosceles triangle
a = measure of equal sides of the isosceles triangle
α = measure of equal angles of the isosceles triangle
β = measure of the angle opposite to the base
How Do You Find the Height Using Area of an Isosceles Triangle?
The height or altitude of an Isosceles Triangle can be calculated by applying the Pythagorean Theorem for any two sides. The formula to calculate the height of an isosceles triangle is given as,
The altitude of an Isosceles Triangle = √(a^{2} − b^{2}/4) units
where,
b = base of the isosceles triangle
a = measure of equal sides of the isosceles triangle
What is the Perimeter and Area of an Isosceles Triangle?
The perimeter of an isosceles triangle is defined as the length of the boundary of an isosceles triangle. The formula for the perimeter of an isosceles triangle is given as, P = 2a + b units. While area is the total region covered by the isosceles triangle, given as, ½[√(a^{2} − b^{2} ⁄4) × b]
where,
 b = base of the isosceles triangle
 a = measure of equal sides of the isosceles triangle
How Do You Find the Area of an Isosceles Triangle Without Height?
The expression to calculate the area of an isosceles triangle without height can be calculated using Heron's formula. The formula to calculate the area of an isosceles triangle without height is given as,
Area of isosceles triangle = b/2 × √(a^{2} − b^{2}/4)
where,
 b = base of the isosceles triangle
 a = measure of equal sides of the isosceles triangle
How Do You Find the Area of an Isosceles Triangle Given Two Sides and an Angle?
The area of a triangle is half the product of the given two sides and sine of the included angle.
Area of Triangle with 2 Sides and Included Angle (SAS) formula is used to find the general formula for calculating the area of an isosceles triangle for SAS as,
Area = ½ × b × a × sin(α)
where,
 b = base of the isosceles triangle
 a = measure of equal sides of the isosceles triangle
 α = measure of equal angles of the isosceles triangle
How Do You Find the Area of an Isosceles Triangle With 3 Sides?
The area of an isosceles triangle with 3 sides can be calculated using Heron's formula, that is Area = \(\sqrt {s(s  a)(s  b)(s  c)} \). For an isosceles triangle, side c = side a. The general formula to calculate the area of an isosceles triangle is given as,
Area of isosceles triangle = b/2 × √(a^{2} − b^{2}/4)
where,
 b = base of the isosceles triangle
 a = measure of equal sides of the isosceles triangle
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