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Chapter Analysis
Intermediate8 pages • EnglishQuick Summary
This chapter focuses on exploring the area of circles and related geometrical figures such as sectors and segments. The key idea is to understand the formulae for calculating the areas of these sections of a circle, based on their radius and central angles. The chapter also covers practical applications of these concepts, such as in designing objects with circular shapes and using circle segments for real-world calculations.
Key Topics
- •Areas of sectors of circles
- •Areas of segments of circles
- •Length of arcs
- •Sector area calculation using central angles
- •Real-world applications of circles
- •Unitary Method in circle geometry
- •Major and minor sectors
- •Major and minor segments
Learning Objectives
- ✓Understand the concept of sectors and segments in circles
- ✓Calculate the area of a sector given its angle and radius
- ✓Apply the formula for areas of circles to practical problems
- ✓Differentiate between major and minor sectors and segments
- ✓Compute the length of an arc using central angles
- ✓Solve real-world problems involving circle geometry
Questions in Chapter
Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
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Find the area of a quadrant of a circle whose circumference is 22 cm.
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The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
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A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector.
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In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord.
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Additional Practice Questions
Explain how the area of a sector is derived from the area of a circle.
mediumAnswer: The area of a sector is derived by taking a fraction of the total area of a circle based on the central angle of the sector. The formula used is (θ/360) * π * r² where θ is the angle in degrees and r is the radius.
If the radius of a circle is doubled, how does that affect the area of a sector with a fixed angle?
easyAnswer: Doubling the radius of a circle quadruples the area of any sector because the area formula (θ/360) * π * r² involves squaring the radius.
Calculate the length of an arc in a circle with radius 15 cm, given that the central angle is 75°.
hardAnswer: The length of the arc can be calculated using the formula (θ/360) * 2 * π * r. Substituting the values, (75/360) * 2 * π * 15 ≈ 19.634 cm.
A circular garden with diameter 70 meters has a path of width 10 meters running around it. What is the area of the path?
mediumAnswer: The area of the path is found by calculating the area of the larger circle and subtracting the area of the garden. The radius of the larger circle is 70/2 + 10 = 45 meters, and its area is π * 45². The garden area is π * 35². The path area = π * 45² - π * 35² = 1256.64 square meters.
Describe the difference between a major sector and a minor sector of a circle.
mediumAnswer: A minor sector is the smaller region of a circle created when a circle is divided by a chord or arc, while the major sector represents the larger region. In terms of central angles, the minor sector's angle is less than 180°, whereas the major sector's angle is more than 180°.