Area of an Annular Sector

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

1. Define an annular sector and identify its constituent geometric parts.
2. Derive and apply the formula for the area of an annular sector given inner and outer radii and central angle.
3. Solve practical word problems involving annular sector shapes such as racing tracks, brake pads, and flower gardens.

Key vocabulary

Annular sector

A portion of an annulus bounded by two radii and two concentric arcs.

The inner radius and outer radius form an annular sector with central angle 60°.

Concentric circles

Circles that lie in the same plane and share the exact same centre point.

Drawing two concentric circles creates the inner and outer boundaries of an annulus.

Annulus

The ring-shaped region enclosed between two concentric circles of different radii.

An annular sector represents a fractional section cut out from a full annulus.

Lesson preview

Slide 1 · Understanding and Calculating Annular Sector AreaModel · 18 min

An annular sector is the region bounded by two concentric circular arcs and two radial lines extending from the common centre. To determine its area, we find the difference between the area of the larger circular sector and the area of the smaller circular sector. Factoring out the common fractional multiplier yields the standard formula $\text{Area}=\frac{\theta }{{360}^{\circ }}\pi (R^2-r^2)$, where $R$ is the outer radius, $r$ is the inner radius, and $\theta$ is the central angle in degrees.

• Concentric circles share the same centre point but have different radius lengths.
• An annular sector is formed by slicing an annulus between two boundary radiuses.
• Subtracting the smaller sector area from the larger sector area gives the region's area.
• The formula for annular sector area is $\text{Area}=\frac{\theta }{{360}^{\circ }}\pi (R^2-r^2)$.

Worked example. Identify inner radius $r=5\text{ cm}$, outer radius $R=9\text{ cm}$, and central angle $\theta ={60}^{\circ }$. Calculate the difference between squared radii: $R^2-r^2=9^2-5^2=81-25=56{\text{ cm}}^2$. Apply the formula: $\text{Area}=\frac{\theta }{{360}^{\circ }}\pi (R^2-r^2)=\frac{{60}^{\circ }}{{360}^{\circ }}\times 3.142\times 56$. Simplify the calculation: $\text{Area}=\frac{1}{6}\times 175.952=\mathrm{29.32533...}{\text{ cm}}^2$. Answer: $29.33{\text{ cm}}^2$ (to 2 decimal places).

Check question: An annular sector has an inner radius of $6\text{ cm}$, an outer radius of $10\text{ cm}$, and a central angle of ${45}^{\circ }$. Taking $\pi =3.142$, calculate its area.

1. $6.28{\text{ cm}}^2$
2. $25.14{\text{ cm}}^2$
3. $39.28{\text{ cm}}^2$
4. $53.41{\text{ cm}}^2$

Show answer

$25.14{\text{ cm}}^2$. Using the formula $\text{Area}=\frac{\theta }{{360}^{\circ }}\times \pi \times (R^2-r^2)$: $\text{Area}=\frac{{45}^{\circ }}{{360}^{\circ }}\times 3.142\times ({10}^2-6^2)$ $\text{Area}=\frac{1}{8}\times 3.142\times (100-36)$ $\text{Area}=\frac{1}{8}\times 3.142\times 64=8\times 3.142=25.136{\text{ cm}}^2$ Rounding to two decimal places gives $25.14{\text{ cm}}^2$.

Slide 2 · Applying Annular Sector Area in Contextual ProblemsPractice · 22 min

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• 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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All 12 lessons in Area of a Part of a Circle

1. 1Area of an AnnulusDetermine the area of an annulus in different situations
2. 2Area of an Annulus in Applied ContextsDetermine the area of an annulus in different situations
3. 3Area of a Sector of a Circle in Real-Life SituationsWork out the area of a sector of a circle in real-life situations
4. 4Area of a Sector of a Circle in Real-Life SituationsWork out the area of a sector of a circle in real-life situations
5. 5Area of an Annular SectorDetermine the area of an annular sector in different situations
6. 6Area of a Segment of a CircleWork out the area of a segment of a circle
7. 7Area of a Segment of a Circle in ContextWork out the area of a segment of a circle
8. 8Area of Common Region Between Two Intersecting CirclesDetermine the area of common region between two intersecting circles
9. 9Applications of Area of Parts of a CircleApply the area of a part of a circle in real-life situations
10. 10Applying Area of Parts of a Circle in Real-Life SituationsApply the area of a part of a circle in real-life situations
11. 11Exploring the Area of a Sector in Real-Life ContextsExplore the use of the area of a part of a circle in real-life situations
12. 12Applications of Area of a Part of a CircleExplore the use of the area of a part of a circle in real-life situations