Grade 10 · Core Mathematics · Lesson 10 of 12

Applying Area of Parts of a Circle in Real-Life Situations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners calculate the surface area of annular sectors in engineering and architecture contexts such as racing tracks, brake pads, and town drainage systems.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L1040 minutes · 2 slides

KICD learning outcome (f)

Apply 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 surface area of annular sectors in engineering and architecture contexts such as racing tracks, brake pads, and town drainage systems.
Determine the area of circular segments and overlapping circular regions in practical applications including helicopter surveillance zones, dais carpets, and intersecting circular ponds.

Key vocabulary

Annular sector: A region within a ring enclosed by two concentric arcs and two radii.; The surface area of a curved racing track bend is calculated as an annular sector.
Concentric arcs: Arcs of circles that share the same central point but have different radii.; The outer and inner edges of the horse shoe brake pad form concentric arcs.
Common chord: A line segment connecting the two intersection points shared by two overlapping circles.; The common chord divides the overlapping circular ponds into two separate segment areas.
Intersection area: The shared region where two overlapping circular figures or structures cross each other.; We sum two circular segment areas to determine the total intersection area of two discs.

Lesson preview

Slide 1 · Real-World Applications of Annular SectorsModel · 18 min

Figure 1: An annular sector representing a racing track bend, featuring an inner radius of 68 m, an outer radius of 80 m, and a 120-degree central angle.

In practical engineering and architecture, circular pathways, town drainage systems, horseshoe brake pads, and racing track bends form annular sectors bounded by concentric circular arcs and central angles. To calculate the surface area of an annular sector, we find the difference between the areas of the outer sector and the inner sector. Alternatively, we multiply the central angle fraction $\frac{θ}{360^{\circ}}$ by the annulus area $π (R^{2} - r^{2})$ , where $R$ is the outer radius and $r$ is the inner radius.

Annular sectors represent ring segments bounded by inner radius $r$ , outer radius $R$ , and central angle $θ$ .
The area formula for an annular sector is $Area = \frac{θ}{360^{\circ}} \times π (R^{2} - r^{2})$ .
For racing tracks with width $w$ , outer radius $R$ is calculated as $r + w$ .
Vehicle horseshoe brake pads and town drainage channels frequently use annular sector surface area calculations.

Worked example. Determine the inner radius:

r = R - w = 80 - 12 = 68 m

. Calculate

R^{2} - r^{2}

80^{2} - 68^{2} = 6400 - 4624 = 1776 m^{2}

. Apply the annular sector area formula:

Area = \frac{120^{\circ}}{360^{\circ}} \times 3.142 \times 1776

. Simplify the angle fraction and multiply:

Area = \frac{1}{3} \times 5580.192 = 1860.064 m^{2}

. Round to 2 decimal places:

1860.06 m^{2}

. Answer: $1860.06 m^{2}$ .

Check question: A vehicle brake pad is in the shape of an annular sector with inner radius $r = 20 cm$ , outer radius $R = 25 cm$ , and central angle $θ = 72^{\circ}$ . Taking $π = 3.142$ , calculate its surface area.

$15.71 {cm}^{2}$
$141.39 {cm}^{2}$
$644.11 {cm}^{2}$
$706.95 {cm}^{2}$

Show answer

$141.39 {cm}^{2}$ . The fraction of the circle is $\frac{72^{\circ}}{360^{\circ}} = 0.2$ . The difference between the squares of the radii is $25^{2} - 20^{2} = 625 - 400 = 225 {cm}^{2}$ . The surface area of the annular sector is $0.2 \times 3.142 \times 225 = 141.39 {cm}^{2}$ .

Slide 2 · Real-World Applications of Circular Segments and Intersection RegionsPractice · 22 min

Figure 2: Two intersecting circular ponds with radii of 12 m and 6√2 m, overlapping to form a common chord measuring 12 m in length.

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Applying Area of Parts of a Circle in Real-Life Situations

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