Grade 10 · Core Mathematics · Lesson 11 of 12

Exploring the Area of a Sector in Real-Life Contexts

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners derive the formula for the area of a circular sector using central angles and area ratios.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L1140 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

Derive the formula for the area of a circular sector using central angles and area ratios.
Calculate the area of circular sectors in practical real-life scenarios such as umbrella panels.

Key vocabulary

Sector: A region of a circle bounded by two radii and the connecting arc.; A sector with a central angle of 90° covers one-fourth of the total circle area.
Quadrant: A quarter of a circle formed by two perpendicular radii subtending a 90° angle.; Folding a circular paper into four equal quadrants creates sectors subtending 90° each.
Subtending: Forming or extending opposite to a specific central angle or arc in geometry.; Each fabric panel of the umbrella is subtending a central angle of 45 degrees.

Lesson preview

Slide 1 · Exploring Sector Areas Through Folding and Concentric CirclesOpening · 18 min

Figure 1: A circle divided into four equal quadrants, with one highlighted sector showing a radius of 14 cm and a central angle of 90 degrees.

A sector represents a fraction of a full circle defined by its central angle. When we construct a circle of a given radius around a central point, or draw concentric circles, we partition space systematically. Folding a circular cutout into four equal quadrants creates sectors subtending 90°. Since 90° out of 360° is the fraction 90°/360° = 1/4, the sector area is one-quarter of the total circle area. This proportional relationship holds for any central angle θ. Use Figure 1 to visualise the fraction of a circle occupied by a 90-degree sector.

Drawing a circle of specified radius establishes the base area for comparing fractional regions.
Folding a circular paper into four equal quadrants forms sectors with a central angle of 90°.
The fraction 90°/360° simplifies to 1/4, showing the sector takes one-fourth of the circle's area.
The formula for sector area uses the ratio of the sector's central angle to 360°.

Worked example. Identify the radius

r = 14 cm

and central angle

θ = 90^{\circ}

. Determine the fraction of the full circle:

\frac{θ}{360^{\circ}} = \frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}

. Calculate the area of the full circle:

A_{circle} = π r^{2} = \frac{22}{7} \times 14^{2} = 616 {cm}^{2}

. Multiply the fraction by the total area:

A_{sector} = \frac{1}{4} \times 616 {cm}^{2} = 154 {cm}^{2}

. Answer: The area of the $90^{\circ}$ sector is $154 {cm}^{2}$ ..

Check question: A circular paper cut-out of radius $21 cm$ is folded into four equal quadrants. What is the area of one quadrant? (Take $π = \frac{22}{7}$ )

$346.5 {cm}^{2}$
$693 {cm}^{2}$
$1386 {cm}^{2}$
$33 {cm}^{2}$

Show answer

$346.5 {cm}^{2}$ . A quadrant represents one-quarter of a circle, which means the central angle is $90^{\circ}$ . First, calculate the total area of the circle using $A = π r^{2}$ . The total area is $\frac{22}{7} \times 21^{2} = 1386 {cm}^{2}$ . Then, multiply the total area by $\frac{1}{4}$ (which is $\frac{90^{\circ}}{360^{\circ}}$ ) to find the area of one quadrant. The area is $\frac{1}{4} \times 1386 = 346.5 {cm}^{2}$ .

Slide 2 · Real-Life Applications of Circular Sector AreasModel · 22 min

Figure 2: A circle partitioned into eight equal sectors representing an umbrella canopy, with one panel showing a 42 cm radius and a 45-degree central angle.

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Homework and the other 11 lessons in this sub-strand

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Exploring the Area of a Sector in Real-Life 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