In geometry, an isosceles triangle is a triangle having two sides of equal length. The two angles opposite to the equal sides are equal and are always acute. Various formulas forisosceles triangles are explained below. The two important formulas for isosceles triangles are thearea of a triangle and the perimeter of a triangle.

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What Are the Isosceles Triangles Formulas?

An isosceles triangle hastwo sides of equal length and two equal sidesjoin at the same angle to the base i.e. the third side. Thus, in an isosceles triangle, thealtitude is perpendicular from the vertex which is common to the equal sides. Suchspecial properties of the isosceles trianglehelp us to calculate its area as well as its altitude with the help of the isosceles triangleformulas.

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Isosceles Triangle Formulas

Area of an Isosceles Triangle:It is the space occupied by thetriangle. Here we have three formulas to find the area of a triangle, based on the given parameters.

Area = 1/2× Base× Height

Area = \(\frac{b}{2} \sqrt{a^{2}-\frac{b^{2}}{4}}\)(Here a is the equal side, and b is the base of the triangle.)

Area = 1/2×abSinα(Here a and b are the lengths of two sides andα is the angle between these sides.)

Perimeter of an Isosceles Triangle: In an isosceles triangle, there are three sides:two equalsides and one base. In order to calculate the perimeter of an isosceles triangle, the expression 2a+ bis used,

P = 2a+ b

(Here, the length of the equal side is aand the length of the base is b)

Altitude of an Isosceles Triangle: In an isosceles triangle, its height is the perpendicular distance from itsvertex to its base. In order to calculate the heightof an isosceles triangle, the expressionh = √(a2–b2/4) is used,

h = \(\sqrt{a^{2}-\frac{b^{2}}{4}}\)

Let us check a few examples to more clearly understand the use of formulas for isosceles triangles.


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Examples Using Formulas for Isosceles Triangles

Example 1:Determine the area of an isosceles triangle that has a base 'b' of 8 unitsand the lateralside 'a' of 5 units?

Solution:Applying Pythagoras'theorem:

a2= (b/2)2+ h2

h2= a2- (b/2)2= 52- 42which gives h = 3

Area 'A' = (1/2) ×b ×h = (1/2) 8 × 3 = 12 unit2

Answer: The area of an isosceles triangle is12 unit2.​​​​

Example 2:Find the lateral side of an isosceles triangle with an area of 20 unit2and a base of 10 units?

Solution:Using theformula of area of an isosceles triangle:

A = (1/2) b h = 20

Given b = 10,

To find: lateral side

h = 40 / 10 = 4

Applying Pythagora's theorem:

a2= (b/2)2+ h2= √ ( 52+ 42) = √41

Answer: The lateral side of an isosceles triangle is√41.

Example 3:Calculate the area, altitude, and perimeter of an isosceles triangle if its two equal sides are of length 6 unitsand the third side is 8 units.

Solution:

Givena = b = 6 units, c= 8 units

To find:area, altitude, and perimeter of an isosceles triangle

Perimeter of the isosceles triangle,

P = 2×a + b

P = 2×6 + 8

= 20 units

Altitude of the isosceles triangle,

h = √(a2–b2/4)

h =√(62–82/4)

h =√(36−16)

h =√20units

Area of the isosceles triangle,

A =1/2×b×h

=1/2×8×√20

=√20/4 square units

Answer:


FAQs onFormulas for Isosceles Triangles

What Is Isosceles Triangle Formula in Geometry?

In geometry, the isosceles triangle formulas are defined as the formulas for calculatingthe area and perimeter of an isoscelestriangle.

Area = 1/2× Base× HeightArea = \(\frac{b}{2} \sqrt{a^{2}-\frac{b^{2}}{4}}\)Area = 1/2×abSinα

(Here a and b are the lengths of two sides andαis the angle between these sides.)

How To UseIsosceles Triangle Formula?

We can use the isoscelestriangle formulas as follows:

Step 1: Check for the parameter(area, perimeter, or height)to be derived or calculated.Step 2: Identify the side of the isoscelestriangle and put the value in the required formula - area, perimeter, or height.

In case,area, perimeter, or altitude of the isoscelestriangle are given, you can find the measure of the side of the triangle by equating the given values to the respective isoscelestriangle formula.

What Is 'a' in IsoscelesTriangle Formula?

In an isosceles triangle formula, be it area, perimeter, or altitude, 'a' refers to the measure of the equal sides of the isoscelestriangle.

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Area = 1/2× Base× HeightArea = \(\frac{b}{2} \sqrt{a^{2}-\frac{b^{2}}{4}}\)Area = 1/2×abSinα

(Here a and b are the lengths of two sides andαis the angle between these sides.)

How To Find Perimeterof Triangle Using Isosceles Triangle Formula?

We know that the perimeter of any figureis the sum of all its sidesthus,

Step 1: Identify the sides of the isosceles triangle - two equal sides a and base b.Step 2: Put the values in the perimeter formula, P = 2a+ bStep 3: Write the value so obtained with an appropriate unit.