What is the area of an isosceles triangle with base 8 cm and lateral side 5 cm?

The area of an isosceles triangle is the amount of region enclosed by it in a two-dimensional space. The general formula for the area of triangle is equal to half the product of the base and height of the triangle. Here, a detailed explanation of the isosceles triangle area, its formula and derivation are given along with a few solved example questions to make it easier to have a deeper understanding of this concept.

Table of Contents Show

What is the Formula for Area of Isosceles Triangle?
Area of Isosceles Triangle Formula
List of Formulas to Find Isosceles Triangle Area
How to Calculate Area if Only Sides of an Isosceles Triangle are Known?
Derivation for Isosceles Triangle Area Using Heron’s Formula
Area of Isosceles Right Triangle Formula
Perimeter of Isosceles Right Triangle Formula
Area of Isosceles Triangle Using Trigonometry
Solved Examples
Practice Questions

Check more mathematics formulas here.

What is the Formula for Area of Isosceles Triangle?

The total area covered by an isosceles triangle is known as its area. For an isosceles triangle, the area can be easily calculated if the height (i.e. the altitude) and the base are known. Multiplying the height with the base and dividing it by 2, results in the area of the isosceles triangle.

What is an isosceles triangle?

An isosceles triangle is a triangle that has any of its two sides equal in length. This property is equivalent to two angles of the triangle being equal. An isosceles triangle has two equal sides and two equal angles. The name derives from the Greek iso (same) and Skelos (leg). An equilateral triangle is a special case of the isosceles triangle, where all three sides and angles of the triangle are equal.

An isosceles triangle has two equal side lengths and two equal angles, the corners at which these sides meet the third side is symmetrical in shape. If a perpendicular line is drawn from the point of intersection of two equal sides to the base of the unequal side, then two right-angle triangles are generated.

Area of Isosceles Triangle Formula

The area of an isosceles triangle is given by the following formula:

Also,

The perimeter of the isosceles triangle	P = 2a + b
The altitude of the isosceles triangle	h = √(a2 − b2/4)

List of Formulas to Find Isosceles Triangle Area

Formulas to Find Area of Isosceles Triangle
Using base and Height	A = ½ × b × h where b = base and h = height
Using all three sides	A = ½[√(a2 − b2 ⁄4) × b] a is the measure of equal sides b is the base of triangle
Using the length of 2 sides and an angle between them	A = ½ × a × b × sin(α) a is the measure of equal sides b is the base of triangle
Using two angles and length between them	A = [a2×sin(β)×sin(α)/ 2×sin(2π−α−β)] a is the measure of equal sides b is the base of triangle α is the measure of equal angles β is the angle opposite to the base
Area formula for an isosceles right triangle	A = ½ × a2 a is the measure of equal sides

How to Calculate Area if Only Sides of an Isosceles Triangle are Known?

If the length of the equal sides and the length of the base of an isosceles triangle are known, then the height or altitude of the triangle is to be calculated using the following formula:

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

Thus,

Area of Isosceles Triangle Using Only Sides = ½[√(a2 − b2 /4) × b]

Here,

b = base of the isosceles triangle
h = height of the isosceles triangle
a = length of the two equal sides

Derivation for Isosceles Triangle Area Using Heron’s Formula

The area of an isosceles triangle can be easily derived using Heron’s formula as explained below.

According to Heron’s formula,

Area = √[s(s−a)(s−b)(s−c)]

Where, s = ½(a + b + c)

Now, for an isosceles triangle,

s = ½(a + a + b)

⇒ s = ½(2a + b)

Or, s = a + (b/2)

Now,

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 of “s”

⇒ Area = (a + b/2 − a) × √[(a + b/2) × ((a + b/2) − b)]

⇒ Area = b/2 × √[(a + b/2) × (a − b/2)]

Or, area of isosceles triangle = b/2 × √(a2 − b2/4)

Area of Isosceles Right Triangle Formula

The formula for Isosceles Right Triangle Area= ½ × a2

Derivation:

Area = ½ ×base × height

area = ½ × a × a = a2/2

Perimeter of Isosceles Right Triangle Formula

Derivation:

The perimeter of an isosceles right triangle is the sum of all the sides of an isosceles right triangle.

Suppose the two equal sides are a. Using Pythagoras theorem the unequal side is found to be a√2.

Hence, perimeter of isosceles right triangle = a+a+a√2

= 2a+a√2

= a(2+√2)

Area of Isosceles Triangle Using Trigonometry

Using Length of 2 Sides and Angle Between Them

A = ½ × b × c × sin(α)

Using 2 Angles and Length Between Them

A = [c2×sin(β)×sin(α)/ 2×sin(2π−α−β)]

Solved Examples

Example 1:

Find the area of an isosceles triangle given b = 12 cm and h = 17 cm?
Solution:

Base of the triangle (b) = 12 cm

Height of the triangle (h) = 17 cm

Area of Isosceles Triangle = (1/2) × b × h

= (1/2) × 12 × 17

= 6 × 17

= 102 cm2

Example 2:

Find the length of the base of an isosceles triangle whose area is 243 cm2, and the altitude of the triangle is 27 cm.

Solution:

Area of the triangle = A = 243 cm2

Height of the triangle (h) = 27 cm

The base of the triangle = b =?

Area of Isosceles Triangle = (1/2) × b × h

243 = (1/2) × b × 27

243 = (b×27)/2

b = (243×2)/27

b = 18 cm

Thus, the base of the triangle is 18 cm.

Question 3:

Find the area, altitude and perimeter of an isosceles triangle given a = 5 cm (length of two equal sides), b = 9 cm (base).

Solution:

Given, a = 5 cm

b = 9 cm

Perimeter of an isosceles triangle

= 2a + b

= 2(5) + 9 cm

= 10 + 9 cm

= 19 cm

Altitude of an isosceles triangle

h = √(a2 − b2/4)

= √(52 − 92/4)

= √(25 − 81/4) cm

= √(25–81/4) cm

= √(25−20.25) cm

= √4.75 cm

h = 2.179 cm

Area of an isosceles triangle

= (b×h)/2

= (9×2.179)/2 cm²

= 19.611/2 cm²

A = 9.81 cm²

Question 4:

Find the area, altitude and perimeter of an isosceles triangle given a = 12 cm, b = 7 cm.

Solution:

Given,

a = 12 cm

b = 7 cm

Perimeter of an isosceles triangle

= 2a + b

= 2(12) + 7 cm

= 24 + 7 cm

P = 31 cm

Altitude of an isosceles triangle

= √(a2 − b2⁄4)

= √(122−72/4) cm

= √(144−49/4) cm

= √(144−12.25) cm

= √131.75 cm

h = 11.478 cm

Area of an isosceles triangle

= (b×h)/2

= (7×11.478)/2 cm²

= 80.346/2 cm²

= 40.173 cm²

Practice Questions

Find the altitude of the triangle if the length of its base is 25 cm and the area enclosed is 375 cm2?
The length of the base of an isosceles triangle is half of its altitude. If the altitude of the triangle is 14cm, find the area enclosed by it?
Find the area of an isosceles triangle, whose length of two equal sides is 5 cm and the length of the third side is 6 cm?
Find the length of each side of a right isosceles triangle whose area is 112.5 cm2.

An isosceles triangle can be defined as a special type of triangle whose 2 sides are equal in measure. For an isosceles triangle, along with two sides, two angles are also equal in measure.

The area of an isosceles triangle is defined as the amount of space occupied by the isosceles triangle in the two-dimensional plane.

To calculate the area of an isosceles triangle, the following formula is used:

A = ½ × b × h

The formula to calculate the perimeter of an isosceles triangle is:

P = 2a + b

The isosceles triangle calculator is the best choice if you are looking for a quick solution to your geometry problems. Find the isosceles triangle area, its perimeter, inradius, circumradius, heights, and angles - all in one place. If you want to build a kennel, find out the area of the Greek temple isosceles pediment, or simply do your maths homework, this tool is here for you. Experiment with the calculator or keep reading to find out more about the isosceles triangle formulas and the isosceles triangle theorem.

An isosceles triangle is a triangle with two sides of equal length, called legs. The third side of the triangle is called the base. The vertex angle is the angle between the legs. The angles with the base as one of their sides are called the base angles.

Here are the most important properties of isosceles triangles:

It has an axis of symmetry along its vertex height;
The two angles opposite the legs are equal; and
The isosceles triangle can be acute, right, or obtuse, but it depends only on the vertex angle (base angles are always acute)

The equilateral triangle is a special case of an isosceles triangle. You can learn about all the possible types of triangles in the classifying triangles calculator. Additionally, if you wish to delve further into the characteristics of an equilateral triangle, check out equilateral triangle calculator

To calculate the isosceles triangle area, you can use many different formulas. The most popular ones are the equations:

Given leg a and base b:

area = (1/4) × b × √( 4 × a² - b² )
Given h height from apex and base b or h2 height from the other two vertices and leg a:

area = 0.5 × h × b = 0.5 × h2 × a
Given any angle and leg or base

area = (1/2) × a × b × sin(base_angle) = (1/2) × a² × sin(vertex_angle)

Also, you can check our triangle area calculator to find other equations, which work for every type of triangle, not only for the isosceles one.

To calculate the isosceles triangle perimeter, simply add all the sides of the triangle:
perimeter = a + a + b = 2 × a + b

Isosceles triangle theorem, also known as the base angles theorem, claims that if two sides of a triangle are congruent, the angles opposite to these sides are congruent.

Also, the converse theorem exists, stating that if two angles of a triangle are congruent, then the sides opposite those angles are congruent.

A golden triangle, which is also called a sublime triangle, is an isosceles triangle in which the leg is in the golden ratio to the base:

a / b = φ ~ 1.618

The golden triangle has some unusual properties:

It's the only triangle with three angles in 2:2:1 proportions
It's the shape of the triangles found in the points of pentagrams
It's used to form a logarithmic spiral

Let's find out how to use this tool with a simple example. Have a look at this step-by-step solution:

Determine what is your first given value. Assume we want to check the properties of the golden triangle. Type 1.681 inches into the leg box.
Enter second known parameter. For example, take a base equal to 1 in.
All the other parameters are calculated in the blink of an eye! We checked, for instance, that the isosceles triangle perimeter is 4.236 in and that the angles in the golden triangle are equal to 72° and 36° - the ratio is equal to 2:2:1, indeed.

You can use this calculator to determine different parameters than in the example, but remember that there are generally two distinct isosceles triangles with a given area and other parameters, e.g., leg length. Our calculator will show one possible solution.

To compute the area of an isosceles triangle with leg a and base b, follow these steps:

Apply the Pythagorean theorem to find the height: √( a² - b²/4 ).
Apply the standard triangle area formula, i.e., multiply base b by the height found in Step 1 and then divide by 2.
That's it. The final formula we have is:

area = ½ × b × √( a² - b²/4 ).

We compute the perimeter of an isosceles triangle with leg a and base b with the help of the formula perimeter = 2 × a + b. This formula makes use of the fact that the two legs of an isosceles triangle are of equal length.

The answer is 6.93. To derive it, we can use the formula area = ½ × b × √( a² - b²/4 ) with a = b = 4.

Alternatively, we can notice that we have here an equilateral triangle: the area formula simplifies to area = a² × √3 / 4 with a = 4.