Grade 10 · Core Mathematics · Lesson 1 of 12

Area of an Annulus

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners define concentric circles and the annulus formed between them.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L140 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

Define concentric circles and the annulus formed between them.
Calculate the area of an annulus given inner and outer radii or diameters with path widths.

Key vocabulary

Concentric circles: Circles of different radii that share the exact same centre point.; The circular pond and surrounding walking path form concentric circles on the ground.
Annulus: The ring-shaped region bounded between two concentric circles of different radii.; We calculate the area of the annulus by subtracting inner area from outer area.
Annular path: A ring-shaped walkway or border created between two concentric circular boundaries.; The paved annular path around the garden has a width of two metres.

Lesson preview

Slide 1 · Concentric Circles and the AnnulusOpening · 18 min

Figure 1: Two concentric circles with a common centre O, showing an inner radius of 8 metres and an outer radius of 13 metres forming a shaded annulus.

When two circles share the exact same centre but have different radii, they are called concentric circles. The ring-shaped space enclosed between the inner circle boundary and the outer circle boundary is defined as an annulus. To calculate the area of an annulus, we compute the area of the larger outer circle using $A_{outer} = π R^{2}$ and subtract the area of the smaller inner circle $A_{inner} = π r^{2}$ . Thus, Area $= π R^{2} - π r^{2}$ . Look at Figure 1 to see how inner and outer radii define an annulus.

Concentric circles share a common centre point but have different radii values.
An annulus is the flat ring-shaped region lying between two concentric circles.
The area of an annulus equals the outer circle area minus inner circle area.
Using algebraic notation, the area formula for an annulus is $A = π R^{2} - π r^{2}$ .

Worked example. Find the inner area using

A_{inner} = π r^{2} = 3.142 \times 8^{2} = 3.142 \times 64 = 201.088 m^{2}

. Find the outer area using

A_{outer} = π R^{2} = 3.142 \times 13^{2} = 3.142 \times 169 = 530.998 m^{2}

. Subtract inner area from outer area:

530.998 - 201.088 = 329.91 m^{2}

. Answer: $329.91 m^{2}$ .

Check question: A ring target has an inner circle of radius 4 cm and an outer concentric circle of radius 7 cm. Using $π = 3.142$ , what is the area of the ring between them?

$28.278 {cm}^{2}$
$103.686 {cm}^{2}$
$153.958 {cm}^{2}$
$204.23 {cm}^{2}$

Show answer

$103.686 {cm}^{2}$ . The outer circle area is $3.142 \times 7^{2} = 153.958 {cm}^{2}$ . The inner circle area is $3.142 \times 4^{2} = 50.272 {cm}^{2}$ . Subtracting gives $153.958 - 50.272 = 103.686 {cm}^{2}$ .

Slide 2 · Calculating Annulus Area from Diameters and Border WidthsModel · 22 min

Figure 2: A circular fountain surrounded by a uniform path, demonstrating how to find the outer radius by adding the path width to the inner radius.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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