Grade 10 · Core Mathematics · Lesson 2 of 12

Area of an Annulus in Applied Contexts

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners determine annular surface measurements when given the inner diameter and border width.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L240 minutes · 2 slides

KICD learning outcome (a)

Determine the area of an annulus in different situations.

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

Determine annular surface measurements when given the inner diameter and border width.
Find the surface space of ring-shaped regions given both inner and outer diameters in real-life contexts.

Key vocabulary

Annulus: The region bounded between two concentric circles having different radii.; The tiled border around the fish pond forms an annulus of area $12018.15 {cm}^{2}$ .
Radius: The straight distance from the centre of a circle to its boundary.; Halving the $240 cm$ pond diameter gives an inner radius of $120 cm$ .
Concentric: Circles of different sizes that share the exact same central point.; The inner and outer edges of the steel washer form concentric circles.

Lesson preview

Slide 1 · Annulus Area from Inner Diameter and Border WidthModel · 18 min

Figure 1: A diagram of a circular pond with a border, showing the inner diameter, inner radius, border width, and outer radius.

When solving practical problems involving concentric circular structures, measurements are often provided as diameters or border widths rather than direct radii. To calculate the area of the surrounding annular region, first halve any given diameter to obtain the inner radius $r$ . Next, add the surrounding border width to the inner radius to determine the outer radius $R$ . Finally, calculate both circular areas using $π R^{2}$ and $π r^{2}$ , subtracting the inner area from the outer area.

Always divide a given diameter by two to determine the corresponding circular radius accurately.
Add the width of a surrounding border to the inner radius to obtain the outer radius.
Calculate the outer circle area $π R^{2}$ and inner circle area $π r^{2}$ separately.
Subtract the inner circular area from the outer circular area to find the annulus area.

Worked example. Determine the inner radius

r

from the pond diameter:

r = \frac{240 cm}{2} = 120 cm

. Determine the outer radius

R

by adding the border width:

R = 120 cm + 15 cm = 135 cm

. Calculate inner circular area:

{Area}_{inner} = 3.142 \times 120^{2} = 3.142 \times 14400 = 45244.8 {cm}^{2}

. Calculate outer circular area:

{Area}_{outer} = 3.142 \times 135^{2} = 3.142 \times 18225 = 57262.95 {cm}^{2}

. Subtract inner area from outer area:

{Area}_{border} = 57262.95 {cm}^{2} - 45244.8 {cm}^{2} = 12018.15 {cm}^{2}

. Answer: The area of the tiled border is $12018.15 {cm}^{2}$ ..

Check question: A circular water tank top of diameter $160 cm$ has a central circular access opening of diameter $40 cm$ . Calculate the surface area of the solid tank top surrounding the opening taking $π = 3.142$ .

$18852 {cm}^{2}$
$75408 {cm}^{2}$
$20108.8 {cm}^{2}$
$21365.6 {cm}^{2}$

Show answer

$18852 {cm}^{2}$ . First, find the radii by halving the diameters. Inner radius $r = 40 cm \div 2 = 20 cm$ and outer radius $R = 160 cm \div 2 = 80 cm$ . The area is $3.142 \times (80^{2} - 20^{2}) = 3.142 \times (6400 - 400) = 3.142 \times 6000 = 18852 {cm}^{2}$ .

Slide 2 · Annulus Area with Given Diameters and Everyday ObjectsPractice · 22 min

Figure 2: A diagram of a steel washer that indicates its inner and outer diameters alongside the shaded ring surface.

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Homework and the other 11 lessons in this sub-strand

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Area of an Annulus in Applied Contexts

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