A circular tray of radius 10 cm has a chord subtending a central angle of 90°. Calculate the area of the minor segment. (Take π = 3.142).

28.55 cm^2. First, find the sector area: (90°)/(360°) × 3.142 × 10^2 = 78.55 cm^2. Next, find the triangle area: (1)/(2) × 10 × 10 × 90° = 50 cm^2. Finally, subtract the triangle area from the sector area: 78.55 - 50 = 28.55 cm^2.

Grade 10 · Core Mathematics · Lesson 6 of 12

Area of a Segment of a Circle

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners identify minor and major segments created by a chord in a circle.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L640 minutes · 3 slides

KICD learning outcome (d)

Work out the area of a segment of a circle.

By the end of this lesson, learners should be able to

Identify minor and major segments created by a chord in a circle.
Calculate the area of a minor segment by subtracting triangle area from sector area.
Determine the area of a major segment using the total circular area and minor segment area.

Key vocabulary

Segment: A region of a circle bounded by a chord and an arc.; The chord divides the circle into a minor segment and a major segment.
Chord: A straight line segment joining two points on a circle boundary.; A chord of length $10 cm$ subtends an angle at the centre.
Major segment: The larger region of a circle bounded by a chord and major arc.; Subtracting the minor segment area from total circular area gives the major segment area.
Subtend: To form an angle at the centre of a circle from an arc or chord.; The chord subtends an angle of $120^{\circ}$ at the centre of the stage.

Lesson preview

Slide 1 · Understanding Circular Segments and Minor Segment AreaOpening · 12 min

Figure 1: A circle with a fifteen-centimetre radius displaying a shaded minor segment formed by a chord and a sixty-degree central angle.

A chord divides a circle into two regions called segments. The smaller region is the minor segment, while the larger region is the major segment. To calculate the area of a minor segment, we find the area of the circular sector subtended by the central angle and subtract the area of the isosceles triangle formed by the two radii and the chord.

A chord splits a circle into a minor segment and a major segment.
A circular sector contains both the minor segment and a central triangle.
Subtracting the triangle area from the sector area yields the minor segment area.
The formula for triangle area is $\frac{1}{2} r^{2} \sin θ$ where $θ$ is the central angle.

Worked example. Identify sector angle

θ = 60^{\circ}

and radius

r = 15 cm

. Calculate sector area:

{Area}_{sector} = \frac{60^{\circ}}{360^{\circ}} \times 3.142 \times 15^{2} = \frac{1}{6} \times 3.142 \times 225 = 117.825 {cm}^{2}

. Calculate triangle area:

{Area}_{triangle} = \frac{1}{2} \times 15 \times 15 \times 0.8660 = 97.425 {cm}^{2}

. Subtract triangle area from sector area:

{Area}_{segment} = 117.825 - 97.425 = 20.4 {cm}^{2}

. Answer: The area of the minor segment is $20.4 {cm}^{2}$ ..

Check question: A circular tray of radius $10 cm$ has a chord subtending a central angle of $90^{\circ}$ . Calculate the area of the minor segment. (Take $π = 3.142$ ).

$28.55 {cm}^{2}$
$50 {cm}^{2}$
$78.55 {cm}^{2}$
$128.55 {cm}^{2}$

Show answer

$28.55 {cm}^{2}$ . First, find the sector area: $\frac{90^{\circ}}{360^{\circ}} \times 3.142 \times 10^{2} = 78.55 {cm}^{2}$ . Next, find the triangle area: $\frac{1}{2} \times 10 \times 10 \times \sin 90^{\circ} = 50 {cm}^{2}$ . Finally, subtract the triangle area from the sector area: $78.55 - 50 = 28.55 {cm}^{2}$ .

Slide 2 · Calculating Minor and Major Segment AreasModel · 14 min

Figure 2: A circle split into minor and major segments by a chord, demonstrating that their combined areas form the entire circle.

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Homework and the other 11 lessons in this sub-strand

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Area of a Segment of a Circle

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 12 lessons in Area of a Part of a Circle