Mathematics · Class 10 · Mensuration and Surface Areas · Term 2

Area of a Segment of a Circle

Students will calculate the area of a segment of a circle by subtracting the area of a triangle from a sector.

• CBSE Learning Outcomes
• NCERT: Areas Related to Circles - Class 10

About This Topic

The area of a segment of a circle is the region enclosed by a chord and the arc it subtends. In Class 10, students compute the minor segment area by subtracting the triangular portion's area from the sector's area. They use the formula (r²/2)(θ - sin θ), where θ is the central angle in radians, after converting degrees if needed. This builds on prior knowledge of circle areas, sectors, and triangles, with practice problems requiring precise steps for radius and angle values.

Positioned in the Mensuration and Surface Areas unit, this topic strengthens geometric reasoning and calculation accuracy, vital for CBSE board exams. Students analyse how segment area increases non-linearly with central angles, from small θ near zero to larger values approaching semicircle limits. Constructing problems involving both sectors and segments reinforces application skills.

Active learning benefits this topic greatly, as physical models clarify the subtraction concept. When students draw, cut, and measure paper segments for varying angles, they visualise the geometry intuitively, correct misconceptions through comparison, and connect formulas to real shapes, improving retention and problem-solving confidence.

Key Questions

1. Explain the logical steps involved in calculating the area of a minor segment.
2. Analyze how the area of a segment changes with different central angles.
3. Construct a problem that requires finding the area of both a sector and a segment.

Learning Objectives

• Calculate the area of a minor segment of a circle given the radius and central angle.
• Explain the geometric steps involved in deriving the formula for the area of a segment.
• Analyze the relationship between the central angle and the area of a circular segment.
• Construct a word problem requiring the calculation of a circular segment's area.

Before You Start

Area of a Circle

Why: Students must know how to calculate the total area of a circle to then find the area of a sector.

Area of a Sector of a Circle

Why: This topic directly builds on finding the area of a sector, which is a component of the segment area calculation.

Area of a Triangle (including isosceles triangles)

Why: Calculating the area of the triangle formed by the radii and the chord is essential for the subtraction method.

Key Vocabulary

Circular Segment	The region of a circle bounded by a chord and the arc subtended by the chord.
Circular Sector	The part of a circle enclosed by two radii and the arc between them, like a slice of pizza.
Chord	A straight line segment connecting two points on the circumference of a circle.
Central Angle	An angle whose vertex is the center of the circle and whose sides are radii intersecting the circle at two points.

Watch Out for These Misconceptions

Common Misconception

Segment area equals sector area minus chord length.

What to Teach Instead

The correct subtraction uses the triangle area formed by the two radii and chord, not the chord length itself. Pairs constructing and cutting paper models reveal the exact region, helping students see why triangle area fits perfectly inside the sector.

Common Misconception

Formula works directly with degrees, no conversion needed.

What to Teach Instead

θ must be in radians for the formula; degrees require multiplication by π/180. Small group angle charts with conversion practice build fluency, as students match calculated areas to measured ones from models.

Common Misconception

Major segment area uses the same minor formula without adjustment.

What to Teach Instead

Major segment requires the reflex angle or full circle minus minor segment. Whole class relays with mixed problems clarify this, as teams debate and verify steps collaboratively.

Active Learning Ideas

Pairs Construction: Paper Segments

In pairs, students draw circles of fixed radius using a compass, mark central angles of 60°, 90°, and 120°, cut out sectors, fold along radii to form triangles, and subtract areas by weighing paper pieces or tracing. They calculate using the formula and compare results. Discuss discrepancies as a pair.

35 min·Pairs

Small Groups: Angle Variation Charts

Groups draw segments for angles 30° to 150° in 30° steps, compute areas in a table, and plot graphs of area versus θ. Use compasses and protractors for accuracy. Share graphs to identify patterns like rapid growth beyond 90°.

40 min·Small Groups

Whole Class: Problem Relay

Divide class into teams. Team 1 creates a segment problem with given r and θ, passes to Team 2 for sector area, Team 3 for triangle area, and Team 4 for segment area and verification. Rotate roles twice.

30 min·Whole Class

Individual: Custom Problems

Each student invents two problems: one for minor segment, one comparing segments with different angles but same r. Solve, then swap with a neighbour for checking calculations and units.

25 min·Individual

Real-World Connections

• Architects and civil engineers use segment calculations when designing curved structures, such as the cross-section of tunnels or the shape of window panes in historical buildings.
• Cartographers might use principles related to segments when calculating the area of irregularly shaped landmasses or water bodies shown on maps, especially those with curved boundaries.
• Jewellery designers sometimes create circular or semi-circular patterns where understanding the area of segments is useful for estimating material needed for decorative inlays or gemstone settings.

Assessment Ideas

Quick Check

Provide students with a diagram of a circle with radius 10 cm and a central angle of 60 degrees. Ask them to calculate the area of the minor segment, showing each step: area of sector, area of triangle, and final subtraction.

Discussion Prompt

Pose this question: 'Imagine two segments in the same circle. One has a central angle of 30 degrees, and the other has 90 degrees. How does the area of the second segment compare to the first, and why?' Guide students to discuss the non-linear relationship.

Exit Ticket

On a small slip of paper, ask students to write down the formula for the area of a minor segment and identify one situation where calculating this area might be practically useful.

Frequently Asked Questions

How to calculate area of a minor segment of a circle class 10?

First, find sector area as (θ/360) × πr², with θ in degrees. Subtract triangle area (1/2 r² sin θ), converting θ to radians for sin. Or use direct formula (r²/2)(θ - sin θ) in radians. Practice with r=7 cm, θ=60° yields about 8.4 cm², building step-wise precision for exams.

What is the difference between sector and segment area?

Sector is the pie-shaped region from centre to arc, area (θ/360)πr². Segment is sector minus the isosceles triangle, the curved triangular bit. Visual models show sector includes the triangle, while segment excludes it, key for problems asking both.

How does central angle affect segment area?

As θ increases from 0° to 180°, segment area grows non-linearly: small for acute angles, peaks near semicircle. For fixed r, doubling θ more than doubles area due to arc bulge. Graphs from group activities confirm this, analysing patterns like slower growth past 90°.

How can active learning help with area of segments?

Hands-on paper cutting and measuring for different angles makes the sector-triangle subtraction concrete, countering rote errors. Group charting reveals angle effects visually, while relays build collaborative verification. These methods boost understanding over worksheets, with students retaining formulas through tangible exploration and peer discussion.