A circular garden of radius 7 m has a straight path along chord AB subtending a central angle of 90°. Calculate the area of the minor segment bounded by chord AB and the arc. (Take π = 3.142)

13.99 m^2. Sector area = (90°)/(360°) × 3.142 × 7^2 = 38.4895 m^2. Triangle area = (1)/(2) × 7 × 7 × (90°) = 24.5 m^2. Minor segment area = Sector area - Triangle area = 38.4895 - 24.5 = 13.9895 ≈ 13.99 m^2.

Grade 10 · Core Mathematics · Lesson 12 of 12

Applications of Area of a Part of a Circle

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners calculate the area of an annular sector formed by concentric arcs in real-life architectural contexts.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L1240 minutes · 2 slides

KICD learning outcome (g)

Explore the use of the area of a part of a circle in real-life situations.

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

Calculate the area of an annular sector formed by concentric arcs in real-life architectural contexts.
Calculate the area of a circular segment by finding the difference between sector and triangle areas.
Determine the total area of intersection between two overlapping circular regions.

Key vocabulary

Annular sector: A region bounded by two concentric circular arcs and two radial lines.; The red tiled pathway around the fountain forms an annular sector of area $3.40 m^{2}$ .
Circular segment: A portion of a circle enclosed between a chord and its corresponding arc.; Subtracting the triangle area from the sector area gives the circular segment of $13.99 m^{2}$ .
Intersection area: The common region shared by two overlapping geometric figures or circular boundaries.; The intersection area of the two circular ponds equals the sum of both segments.

Lesson preview

Slide 1 · Annular Sectors and Circular Segments in DesignModel · 18 min

Figure 1: Two diagrams showing an annular sector formed by concentric circles and a minor segment bounded by a chord.

Calculating areas of complex circular shapes in real life involves breaking them into standard components. An annular sector is bounded by two concentric circular arcs and two radii, formed by taking the difference between two sectors of the same central angle. A circular segment is formed by subtracting the area of a central triangle from the area of its corresponding sector. These methods allow architects and engineers to calculate materials for curved structures accurately.

An annular sector is formed between two concentric circular arcs of different radii sharing a common central angle.
Subtracting the smaller sector area from the larger sector area yields the exact area of an annular sector.
A circular segment area equals the sector area minus the triangle area formed by the two radii and chord.
Real-life architectural layouts use annular sectors and circular segments for efficient spatial planning and tile estimations.

Worked example. Calculate outer sector area:

A_{1} = \frac{θ}{360^{\circ}} \times π R^{2} = \frac{40^{\circ}}{360^{\circ}} \times 3.142 \times {4.0}^{2} = \frac{1}{9} \times 50.272 \approx 5.586 m^{2}

. Calculate inner sector area:

A_{2} = \frac{θ}{360^{\circ}} \times π r^{2} = \frac{40^{\circ}}{360^{\circ}} \times 3.142 \times {2.5}^{2} = \frac{1}{9} \times 19.6375 \approx 2.182 m^{2}

. Subtract inner sector area from outer sector area:

Area = A_{1} - A_{2} = 5.586 - 2.182 = 3.404 m^{2}

. Answer: $3.40 m^{2}$ .

Check question: A circular garden of radius $7 m$ has a straight path along chord $A B$ subtending a central angle of $90^{\circ}$ . Calculate the area of the minor segment bounded by chord $A B$ and the arc. (Take $π = 3.142$ )

$13.99 m^{2}$
$24.50 m^{2}$
$38.49 m^{2}$
$62.99 m^{2}$

Show answer

$13.99 m^{2}$ . Sector area $= \frac{90^{\circ}}{360^{\circ}} \times 3.142 \times 7^{2} = 38.4895 m^{2}$ . Triangle area $= \frac{1}{2} \times 7 \times 7 \times \sin (90^{\circ}) = 24.5 m^{2}$ . Minor segment area $= Sector area - Triangle area = 38.4895 - 24.5 = 13.9895 \approx 13.99 m^{2}$ .

Slide 2 · Area of Intersecting Circular Regions in Real LifePractice · 22 min

Figure 2: Two intersecting circles showing the lens-shaped overlapping region divided into two minor segments by a common chord.

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Applications of Area of a Part 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