What is Area Related to Circle Class 10: Complete Guide with Formulas

Understanding What is Area Related to Circle in Class 10

The chapter on Areas Related to Circles in Class 10 NCERT Mathematics focuses on calculating areas of different parts of a circle and shapes involving circles. The 'area related to circle' means finding the surface enclosed by the circle or its parts such as sectors and segments.

Key concepts include:

• Area of a full circle
• Area of sectors (a 'slice' of the circle)
• Area of segments (part of a circle cut off by a chord)
• Areas of combined shapes involving circles

This chapter builds on your earlier knowledge of circles and introduces formulas to calculate these areas accurately.

Formula for Area of a Circle and Its Derivation

The fundamental formula for the area of a circle is:

$$\text{Area} = \pi r^2$$

where $r$ is the radius of the circle.

Derivation in brief:

• Imagine cutting the circle into many small sectors and rearranging them to approximate a parallelogram.
• The base of this parallelogram is half the circumference, i.e., $\pi r$.
• The height is the radius $r$.
• Area of parallelogram = base × height = $\pi r \times r = \pi r^2$.

This formula is the foundation for all area calculations related to circles in Class 10.

Calculating Area of a Sector in Class 10 NCERT Maths

A sector is a portion of a circle enclosed by two radii and the arc between them. To find its area, use the formula:

$$\text{Area of sector} = \frac{\theta}{360} \times \pi r^2$$

where:

• $\theta$ = central angle in degrees
• $r$ = radius of the circle

Example: Find the area of a sector with radius 7 cm and central angle 60°.

Solution:

$$\text{Area} = \frac{60}{360} \times \pi \times 7^2 = \frac{1}{6} \times \pi \times 49 = \frac{49\pi}{6} \approx 25.6 \text{ cm}^2$$

This formula helps solve problems involving parts of circles in your Class 10 exams.

Area of a Segment: Combining Sector and Triangle Areas

A segment is the region between a chord and the corresponding arc of a circle. To find its area:

$$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}$$

• First, calculate the sector area using the central angle.
• Then find the area of the triangle formed by the two radii and the chord.

Formula for triangle area when two sides and included angle are known:

$$\text{Area} = \frac{1}{2} r^2 \sin \theta$$

where $\theta$ is in degrees.

Example: Find the area of a segment with radius 14 cm and central angle 60°.

Solution:

• Sector area = $\frac{60}{360} \times \pi \times 14^2 = \frac{1}{6} \times \pi \times 196 = \frac{196\pi}{6} \approx 102.67$ cm²
• Triangle area = $\frac{1}{2} \times 14^2 \times \sin 60^\circ = 0.5 \times 196 \times 0.866 = 84.9$ cm²
• Segment area = $102.67 - 84.9 = 17.77$ cm²

This method is essential for solving segment problems in Class 10.

Comparing Areas: Circle, Sector, and Segment

Understanding the difference between these areas is key for Class 10 students. Here's a quick comparison:

This table helps you quickly identify which formula to use based on the problem.

Tips to Master Areas Related to Circles for Class 10 Exams

• Understand formulas, don’t just memorize: Know why formulas work.
• Draw neat diagrams: Visualizing helps in solving problems.
• Practice NCERT examples: They cover typical exam questions.
• Use approximate values of $\pi$ carefully: Either 3.14 or 22/7 depending on the question.
• Check units: Always write area in square units (cm², m²).
• Revise regularly: This chapter appears frequently in CBSE exams.

Following these tips will boost your confidence and accuracy in the Areas Related to Circles chapter.