Area Of Isosceles Triangle

5/5 - (1 bình chọn)

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.

Mục Lục

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 two-dimensional 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 legsand the angle between them is called the vertex angleor 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 m2, cm2, in2, yd2, etc.

Area of an Isosceles Triangle Formulas

The area of an isosceles triangle refers to the total space covered by the shape in 2-D. 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 TriangleFormula to Calculate Area (in square units)
Base and HeightA = ½ × b × h
All three sidesA = ½[√(a2 − b2 ⁄4) × b]
Length of 2 sides and an angle between themA = ½ × b × a × sin(α)
Two angles and length between themA = [a2×sin(β/2)×sin(α)]
Isosceles right triangleA = ½ × a2

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[√(a2 – b2/4) × 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:
a2 = (b/2)2 + (AD)2
AD = √(a2 − b2/4)

The altitude of an isosceles triangle = √(a2 − b2/4)

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[√(a2 − b2 /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(s-a)(s-b)(s-c)

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 semi-perimeter.

Thus, the semi-perimeter 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 × √(a2 − b2/4) 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 = [a2 × 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 × a2

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,

BC2 = a2 + a2

BC = √2 a

Area = ½ × base × height

Area = ½ × a × a = a2/2 square units

Area of Isosceles Triangle Examples

Example 2: Find the length of the base of an isosceles triangle whose area is 243 cm2, and the altitude of the triangle is 9 cm.

Solution:

Area of the triangle, A = 243 cm2

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 cm2.

Solution:

We know that,

The base of the isosceles triangle = 24 cm

Area of the isosceles triangle = 60 cm2

Area of isosceles triangle = b/2 × √(a2 − b2/4)

Therefore,

60 = (24/2)√(a2 − 242/4)

60 = 12√(a2 − 144)

5 = √(a2−144)

Squaring both sides, we get,

25 = a2−144

a2 = 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 2-D. 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[√(a2 − b2 ⁄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 = [a2×sin(β/2)×sin(α)]
  • Area formula for an isosceles right triangle: Area = ½ × a2

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 = √(a2 − b2/4) 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[√(a2 − b2 ⁄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 × √(a2 − b2/4)
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

where,

  • b = base of the isosceles triangle
  • a = measure of equal sides of the isosceles triangle

Area of Isosceles Triangle Formula

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

Area = ½ × base × Height

Also,

The perimeter of the isosceles triangleP = 2a + b
The altitude of the isosceles triangleh = √(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
Using all three sides A = ½[√(a2 − b2 ⁄4) × b]
Using the length of 2 sides and an angle between them A = ½ × b × c × sin(α)
Using two angles and length between them A = [c2×sin(β)×sin(α)/ 2×sin(2π−α−β)]
Area formula for an isosceles right triangle A = ½ × a2

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:

The 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

P = a(2+√2)

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)

= 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)]

Example Questions

Question 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

Question 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

Frequently Asked Questions

What is an Isosceles Triangle?

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.

What does the Area of an Isosceles Triangle Mean?

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

What is the Formula for Area of Isosceles Triangle?

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

A = ½ × b × h

What is the Formula for Perimeter of Isosceles Triangle?

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

P = 2a + b

Area of Isosceles Triangle Formula with Solved Examples

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).

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.

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

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=12cm2

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

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.

FAQs (Frequently Asked Questions)

1.What is an Isosceles Triangle?

An Isosceles Triangle can be defined as a special type of Triangle whose at least 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 the amount of region enclosed by it in a two-dimensional space. The general formula for the area of the Triangle is equal to half of the product of the base and height of the Triangle.

The area of an Isosceles Triangle is defined as the amount of space occupied by the Isosceles Triangle in the two-dimensional area. To calculate the area of an equilateral Triangle, the following formula is used:

  • A = ½ × b × h

The formula to calculate the perimeter of an equilateral Triangle is:

  • P = 2a + b

2.Why is Vedantu trusted, an education partner?

Vedantu makes subject wise study elements for students of all types to make their learning method easy and understandable. Study modules on all topics given by us support uncomplicated access so that learners can read concepts clearly without confusion.

Students will benefit from the support we bring to score high as well as develop a strong conceptual understanding. We have covered all the topics and sub-topics of all the subjects and they are created in a step by step way to make the students’ work easy and simple. Our main maxim is to make the learning process simple and improve a higher retention rate. We always think from the examination point of view before preparing these answers.

3.What is Heron’s Formula Area Triangle?

Heron’s formula is a formula that can be used to find the area of a Triangle when given its three side lengths. It could be applied to all shapes of the Triangle, as long as we know its lengths of three sides. The formula is as follows: The area of a Triangle whose side lengths are a, b, (a, b), (a,b), and c c c is given by.

Example to find the area of a Triangle, multiply the base by the height, and then divide by 2. The division by 2 actually comes from the certainty that a parallelogram can be divided into 2 Triangles.

4.Explain the concept of similarity of Triangles.

Here, in Maths concept of similarity of Triangles concept, you will be learning what are the corresponding angles of two Triangles and what are corresponding sides of two Triangles. If the angle of the two Triangles is the same then, it will be called an equiangular Triangle. 

Criteria for the Similarity of Triangles: 

In this concept, you will be learning about various criteria to find out the similarity of the given Triangles. You will be introduced mainly about AAA (Angle-Angle-Angle) criteria of similarity, SSS (Side-Side-Side) criteria of similarity and SAS (Side-Angle-Side) criteria of similarity. 

  • Areas of Similar Triangles:

Here, you will be taught about how corresponding sides will be equal to the ratio of the given Triangle area.

5.What are the different types of IsosWhat is an Isosceles TriangleWhat is an Isosceles Triangleceles Triangles?

Triangles can be of different types depending on the angle that is made by the two sides of the Triangle that are equal in length. The different types of Isosceles Triangles are named the Isosceles acute Triangle, Isosceles obtuse Triangle and Isosceles Right Triangle.

If the Triangle is constituted of three angles that are less than 90 degrees and if any two of the three angles are equal, then this particular Triangle will be an Isosceles acute Triangle.

If a particular Triangle is constituted of three angles and any two of them are less than 90° and one of the angles is more than 90 degrees, then that particular Triangle will be known as an Isosceles obtuse Triangle.

An Isosceles Right Triangle will be constructed by two sides that are equal in length and the base angles will be equal in measurement. Also, the angle that is at the opposite of the base of the Triangle will be 90 degrees. Download the free study materials from the website of Vedantu to get a detailed understanding of the various types of Triangles and their characteristics.

Area of Isosceles Triangle Using Sides

The area of an isosceles triangle can be found by calculating the height or altitude of the isosceles triangle if the lengths of legs (equal sides) and base are given.
The formula used to calculate the area of the isosceles triangle by using the lengths of the equals sides and base are given as follows:

Here,
a− length of legs (equal sides of the isosceles triangle)
b− length of unequal side or base of the isosceles triangle.
Derivation:
Let the isosceles triangle with the length of the equal sides or legs is “a” units, and the length of the base of the isosceles triangle is “b” units.

We know that in an isosceles triangle, the altitude drawn divides the base into two equal parts.
In the given isosceles triangle ABC,
AB=AC=a units and BC=b units, AD=h, BD=DC=b/2 units.
In the right-angled triangle ADB, by using the Pythagoras theorem,
Hyp2=side2+side2
AB2=AD2+BD2

Area of Isosceles Triangle Using Heron’s Formula

Heron’s formula is used to calculate the area of a triangle when the length of three sides is given. The area of the isosceles triangle using Heron’s formula is given below:

Consider an isosceles triangle with sides “a” and base “b”,

Area of Isosceles Triangle Using Trigonometric Ratios

The area of an isosceles triangle when the length of two sides and angle between them is given or when two angles and the length of side included between them is given can be found by using trigonometric ratios as follows:

When two sides and angle between them is given:

Area of an Isosceles Right Triangle

A triangle in which one angle is a right angle (90) and with two equal sides other than hypotenuse is called the isosceles right triangle.
The area of the isosceles right triangle is

Area of Isosceles Triangle Formulas

The various formulas to be used to calculate the area of the Isosceles triangle is given below:

Parameters are given for the isosceles triangleFormula to be used to calculate the area of the Isosceles Triangle

Solved Examples on Area of Isosceles Triangle

Q.1. Find the triangle area with the length of the sides is 3cm, 3cm, 4cm.
Ans: Given, length of the sides of the triangle is 3cm, 3cm, 4cm.

Hence, the area of the given right isosceles triangle is 8cm2

.Summary

In this article, we have studied the definition of the isosceles triangle and the unique properties of the isosceles triangle. We have discussed the area of the isosceles triangle, which tells the amount of surface or space enclosed between the sides of the isosceles triangle.

The general formula for calculating the area of isosceles triangle is

FAQs on Area of Isosceles Triangle

Q.1. What is the formula of the area of an isosceles triangle?
Ans: The general formula used to calculate the area of an isosceles

Q.5. What does the area of the isosceles triangle mean?
Ans: The area of an isosceles triangle is the amount of surface or space enclosed between the sides of the isosceles triangle.

How to Find the Area of an Isosceles Triangle

An isosceles triangle is a triangle with two sides of the same length. These two equal sides always join at the same angle to the base (the third side), and meet directly above the midpoint of the base.[1] You can test this yourself with a ruler and two pencils of equal length: if you try to tilt the triangle to one direction or the other, you cannot get the tips of the pencils to meet. These special properties of the isosceles triangle allow you to calculate the area from just a couple pieces of information.

Method 1 Finding the Area from the Side Lengths

1 Review the area of a parallelogram. Squares and rectangles are parallelograms, as is any four-sided shape with two sets of parallel sides. All parallelograms have a simple area formula: area equals base multiplied by the height, or A = bh. If you place the parallelogram flat on a horizontal surface, the base is the length of the side it is standing on. The height (as you would expect) is how high it is off the ground: the distance from the base to the opposite side. Always measure the height at a right (90 degree) angle to the base.

  • In squares and rectangles, the height is equal to the length of a vertical side, since these sides are at a right angle to the ground.

2 Compare triangles and parallelograms. There’s a simple relationship between these two shapes. Cut any parallelogram in half along the diagonal, and it splits into two equal triangles. Similarly, if you have two identical triangles, you can always tape them together to make a parallelogram. This means that the area of any triangle can be written as A = ½bh, exactly half the size of a corresponding parallelogram.

3 Find the isosceles triangle’s base. Now you have the formula, but what exactly do “base” and “height” mean in an isosceles triangle? The base is the easy part: just use the third, unequal side of the isosceles.

  • For example, if your isosceles triangle has sides of 5 centimeters, 5 cm, and 6 cm, use 6 cm as the base.
  • If your triangle has three equal sides (equilateral), you can pick any one to be the base. An equilateral triangle is a special type of isosceles, but you can find its area the same way.

4 Draw a line between the base to the opposite vertex. Make sure the line hits the base at a right angle. The length of this line is the height of your triangle, so label it h. Once you calculate the value of h, you’ll be able to find the area.

  • In an isosceles triangle, this line will always hit the base at its exact midpoint.

5 Look at one half of your isosceles triangle. Notice that the height line divided your isosceles triangle into two identical right triangles. Look at one of them and identify the three sides:

  • One of the short sides is equal to half the base: b/2.
  • The other short side is the height, h.
  • The hypotenuse of the right triangle is one of the two equal sides of the isosceles. Let’s call it s.

7 Solve for h. Remember, the area formula uses b and h, but you don’t know the value of h yet. Rearrange the formula to solve for h:

8 Plug in the values for your triangle to find h. Now that you know this formula, you can use it for any isosceles triangle where you know the sides. Just plug in the length of the base for b and the length of one of the equal sides for s, then calculate the value of h.

  • For example, you have an isosceles triangle with sides 5 cm, 5 cm, and 6 cm. b = 6 and s = 5.
  • Substitute these into your formula:

9 Plug the base and height into your area formula. Now you have what you need to use the formula from the start of this section: Area = ½bh. Just plug the values you found for b and h into this formula and calculate the answer. Remember to write your answer in terms of square units.

  • To continue the example, the 5-5-6 triangle had a base of 6 cm and a height of 4 cm.
  • A = ½bh
    A = ½(6cm)(4cm)
    A = 12cm2.

10 Try a more difficult example. Most isosceles triangles are more difficult to work with than the last example. The height often contains a square root that doesn’t simplify to an integer. If this happens, leave the height as a square root in simplest form. Here’s an example:

  • What is the area of a triangle with sides 8 cm, 8 cm, and 4 cm?
  • Let the unequal side, 4 cm, be the base b.

Method 2 Using Trigonometry

1 Start with a side and an angle. If you know some trigonometry, you can find the area of an isosceles triangle even if you don’t know the length of one of its side. Here’s an example problem where you only know the following:

  • The length s of the two equal sides is 10 cm.
  • The angle θ between the two equal sides is 120 degrees.

2 Divide the isosceles into two right triangles. Draw a line down from the vertex between the two equal sides, that hits the base at a right angle. You now have two equal right triangles.

  • This line divides θ perfectly in half. Each right triangle has an angle of ½θ, or in this case (½)(120) = 60 degrees.

3 Use trigonometry to find the value of h. Now that you have a right triangle, you can use the trigonometric functions sine, cosine, and tangent. In the example problem, you know the hypotenuse, and you want to find the value of h, the side adjacent to the known angle. Use the fact that cosine = adjacent / hypotenuse to solve for h:

  • cos(θ/2) = h / s
  • cos(60º) = h / 10
  • h = 10cos(60º)

4 Find the value of the remaining side. There is one remaining unknown side of the right triangle, which you can call x. Solve for this using the definition sine = opposite / hypotenuse:

  • sin(θ/2) = x / s
  • sin(60º) = x / 10
  • x = 10sin(60º)

5 Relate x to the base of the isosceles triangle. You can now “zoom out” to the main isosceles triangle. Its total base b is equal to 2x, since it was divided into two segments each with a length of x.

6 Plug your values for h and b into the basic area formula. Now that you know the base and height, you can rely on the standard formula A = ½bh:

7 Turn this into a universal formula. Now that you know how this is solved, you can rely on the general formula without going through the full process every time. Here’s what you end up with if you repeat this process without using any specific values (and simplifying using properties of trigonometry):

MATHS Related Links

Area of Regular Polygon Formula

Area Formulas

Area of a Square Formula

Area Of Isosceles Triangle ⭐️⭐️⭐️⭐️⭐️

Formula for area of triangle

Area of a Trapezoid Formula

Area Of An Octagon Formula ⭐️⭐️⭐️⭐️⭐

Area Of An Octagon Formula ⭐️⭐️⭐️⭐️⭐

Area of a Pentagon Formula ⭐️⭐️⭐️⭐️⭐

Surface Area Formulas

Surface Area of a Cube Formula

Surface Area of a Prism Formula

Surface Area of a Rectangular Prism Formula

Surface Area of a Square Pyramid Formula

Surface Area of a Rectangle Formula

Hãy bình luận đầu tiên

Để lại một phản hồi

Thư điện tử của bạn sẽ không được hiện thị công khai.


*