Area of Isosceles Triangle

What is an Isosceles Triangle?

A triangle is a three-sided polygon with three angles. When two sides of a triangle are equal in length and their opposite angles are also equal, it is called an isosceles triangle. This symmetry gives the triangle special properties which make calculations easier in geometry.

Terminology:

• The two equal sides are called the "legs" of the triangle.

• The third side, which is not equal to the other two, is called the "base."

• The angles opposite the equal sides are equal.

• The vertex angle is the angle between the two equal sides.

• The base angles are the angles opposite the equal sides and are always equal.

 

Table of Content

• Key Properties of an Isosceles Triangle
• Basic Formula for Area of a Triangle
• Finding the Height Using Pythagorean Theorem
• Using Heron’s Formula
• Real-Life Applications of Isosceles Triangles
• Practice Problems
• Conclusion
• FAQs on Area of Isosceles Triangle

 

Key Properties of an Isosceles Triangle

• Exactly two sides are of equal length.

• Exactly two angles (base angles) are equal.

• The altitude (or height) drawn from the vertex to the base:

• Bisects the base into two equal parts.

• Forms two congruent right-angled triangles.

• Acts as a median (divides the base into two equal segments).

• Acts as an angle bisector (divides the vertex angle into two equal parts).

 

Basic Formula for Area of a Triangle

The area of any triangle, including an isosceles triangle, is given by:

Area = ½ × base × height

To use this formula for an isosceles triangle, you need to know the length of the base and the perpendicular height (altitude) from the vertex to the base.

 

Finding the Height Using Pythagorean Theorem

Often in isosceles triangles, only the lengths of the sides are known, not the height. You can use the Pythagorean Theorem to find the height.

Let:

a = length of each equal side (leg)

b = length of the base

h = height (from vertex to base)

Since the altitude divides the base into two equal halves, each half = b/2.

 

Using the Pythagorean Theorem:

a² = h² + (b/2)²

⇒ h² = a² - (b² / 4)

⇒ h = √(a² - (b² / 4))

Then substitute into the area formula:

Area = ½ × b × √(a² - (b² / 4))

 

Worked Example Using Height Formula

Given: a = 5 cm, b = 6 cm

Step 1: Calculate height

h = √(5² - (6²/4)) = √(25 - 9) = √16 = 4 cm

Step 2: Use area formula

Area = ½ × 6 × 4 = 12 cm²

 

Using Heron’s Formula

When all three sides of a triangle are known (especially when the height is not easily found), use Heron’s Formula.

Steps:

1. Let the three sides be: a, a, and b (isosceles triangle)

2. Calculate semi-perimeter (s): s = (a + a + b)/2 = (2a + b)/2

3. Use Heron’s formula:

Area = √[s(s - a)(s - a)(s - b)]

 

Worked Example Using Heron’s Formula

Let a = 7 cm, b = 6 cm

Step 1: Find semi-perimeter

s = (7 + 7 + 6)/2 = 20/2 = 10

Step 2: Apply Heron’s Formula

Area = √[10(10 - 7)(10 - 7)(10 - 6)]  

= √[10 × 3 × 3 × 4]  

= √360 ≈ 18.97 cm²

 

Real-Life Applications of Isosceles Triangles

• Architecture: Used in truss designs, roof supports, and bridges due to their symmetry and stability.
  
  
• Art and Craft: Helpful in creating symmetrical designs, folds, and paper models.
  
  
• Engineering: Load distribution in towers and cranes often relies on isosceles triangle structures.
  
  
• Pyramids and Monuments: The sides of pyramids form isosceles triangles.
  
  
• Traffic and Road Signs: Warning signs are often triangular and symmetrical in shape.

Height, Median, and Angle Bisector

 

One of the key reasons the isosceles triangle is so useful is the overlapping nature of height, median, and angle bisector. In an isosceles triangle:

The height from the vertex not only measures altitude but also:

• Bisects the base into two equal parts (acts as a median).

• Splits the vertex angle into two equal angles (acts as an angle bisector).

• Forms two right-angled triangles, enabling use of trigonometry or Pythagoras.

This symmetry allows for simplified calculations and helps solve complex geometry problems with ease.

 

Practice Problems

1. Find the area of an isosceles triangle with equal sides of 10 cm and a base of 12 cm.

2. Use Heron’s formula to find the area of an isosceles triangle with sides 13 cm, 13 cm, and 10 cm.

3. Prove using geometry that the altitude bisects the base of an isosceles triangle.

4. Derive the height formula using the Pythagorean Theorem for an isosceles triangle.

5. Given base = 8 cm and height = 6 cm, find the area of the triangle.

6. If the area of an isosceles triangle is 24 cm² and base is 6 cm, find the height.

 

Conclusion

An isosceles triangle has two equal sides and two equal angles.

The height from the vertex bisects the base and splits the triangle into two equal right-angled triangles.

Area formula using height: Area = ½ × base × height

Height can be calculated using: h = √(a² - (b² / 4))

Use Heron’s formula when all sides are given: Area = √[s(s - a)(s - a)(s - b)]

Isosceles triangles are used in construction, design, and pattern-making for their symmetry and strength.

 

Related Links

Area of Trapezium - Master the formula and real-life applications of the area of a trapezium with step-by-step examples and practice questions.

Area of a Triangle - Understand how to calculate the area of any triangle using base-height, Heron’s formula, and more. Includes visuals, formulas, and solved problems.

 

Frequently Asked Questions on Area of Isosceles Triangle

1. What is the formula for the area of an isosceles triangle?

Answer: The basic formula is:
Area = ½ × base × height

If height is not known, use side lengths with:
Area = ½ × b × √(a² − (b² / 4))
Where:

• a = equal side length

• b = base

• √(a² − (b² / 4)) gives the height

 

2. How to find the area of an isosceles triangle without a base?

Answer: If the base is unknown but other sides or angles are given, use either:

• Heron’s formula if all 3 sides are known:
  Area = √[s(s - a)(s - a)(s - b)]
  where s = (2a + b)/2

• Or trigonometry, if angle is given between equal sides:
  Area = ½ × a × a × sin(θ)
  where θ is the angle between the equal sides.

 

3. How to find an area of a triangle with two equal sides?

Answer: This is an isosceles triangle. Use:

• If base and equal sides (a) are known:
  Area = ½ × base × height,
  with height = √(a² − (b² / 4))

• If all sides are known:
  Use Heron’s formula as shown above.

 

4. What is the area of the isosceles right triangle class 9?

Answer: An isosceles right triangle has two equal sides and a right angle between them.

If each equal side is a, then:
Area = ½ × a × a = a² / 2

This is because the two equal sides act as the perpendicular base and height.

 

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