How to prove that area of circular sector | prove area of sector | math class9 new book exercise 6.2

How to Prove the Area of a Circular Sector – Explained

Understanding the area of a circular sector is an important concept in geometry, especially for Class 9 students. This blog post will explain the derivation of the sector area formula, step-by-step, and provide examples for better understanding.

What is a Circular Sector?

A sector of a circle is the region enclosed by two radii and the corresponding arc. It looks like a slice of pizza or a piece of pie from a circular dish. The size of a sector depends on the central angle (θ) it subtends at the center of the circle.

Deriving the Area Formula for a Sector

The formula for the area of a circular sector is:

$A = \frac{1}{2} r^2 \theta$

where:

• A = Area of the sector
• r = Radius of the circle
• θ = Central angle (in radians)

Step-by-Step Derivation:

1. Start with the Area of a Full Circle:
   • The total area of a circle is given by:
     $A_{circle} = \pi r^2$ 
2. Consider the Sector’s Proportion to the Full Circle:
   • A full circle corresponds to a central angle of
     $2\pi$ 

     radians.

   • A sector with angle
     $heta$ 

     is a fraction of the full circle.

   • The fraction of the circle covered by the sector is:
     $\frac{\theta}{2\pi}$ 

3. Multiply This Fraction by the Total Area of the Circle:

   $A_{sector} = \frac{\theta}{2\pi} \times \pi r^2$

   • Simplifying:
     $A_{sector} = \frac{1}{2} r^2 \theta$ 

Applying the Formula – Example Calculation

Let’s apply this formula with an example:

Example: Find the area of a sector in a circle with a radius of 7 cm and a central angle of 1.5 radians.

Solution: Using the formula:

$A = \frac{1}{2} r^2 \theta$ $A = \frac{1}{2} \times 7^2 \times 1.5$ $A = \frac{1}{2} \times 49 \times 1.5$ $A = 36.75 \text{ cm}^2$

Thus, the area of the sector is 36.75 cm².

Why This Formula is Important?

• It helps in solving real-life geometry problems related to circular objects.
• It is useful in engineering, physics, and architecture for designing curved structures.
• It forms a fundamental part of trigonometry and calculus concepts in higher studies.

Final Thoughts

Understanding the derivation of the sector area formula makes it easier to remember and apply. By practicing problems using this method, students can confidently solve related questions in their exams.

If you have any questions or need further explanations, feel free to ask in the comments below! 📚✍️