A circular cardboard disc is divided into sectors. One sector has a central angle of 54°. What fraction of the entire disc surface does this sector represent?

(3)/(20). Divide the central angle 54° by 360° to get 54°360°, which simplifies to (3)/(20) by dividing both the numerator and the denominator by 18.

Grade 10 · Core Mathematics · Lesson 3 of 12

Area of a Sector 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 determine the fraction of a circle represented by a sector given its central angle.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L340 minutes · 2 slides

KICD learning outcome (b)

Work out the area of a sector of a circle in real-life situations.

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

Determine the fraction of a circle represented by a sector given its central angle.
Calculate the area of a sector of a circle using the formula Area = (theta / 360) * pi * r^2 in practical contexts.

Key vocabulary

Sector: A region enclosed by two radii and an arc of a circle.; The rose garden forms a circular sector with a central angle of $55^{\circ}$ .
Central angle: An angle whose vertex lies at the centre of a circle.; The sector subtends a central angle of $160^{\circ}$ at the circular tray centre.

Lesson preview

Slide 1 · Understanding Sectors as Fractions of a CircleOpening · 18 min

Figure 1: A shaded sector with a central angle of 55 degrees, demonstrating how its size is expressed as a fraction of a complete circle.

A sector is a portion of a circular area bounded by two radii and an arc. Because a complete turn around the centre of a circle equals $360^{\circ}$ , any central angle $θ$ subtended by a sector represents a specific fractional part of the complete circle. We express this relationship as the ratio $\frac{θ}{360^{\circ}}$ . Comparing this fraction to the whole circle allows us to analyse real-life circular objects such as divided gardens or circular craft cutouts. Use Figure 1 to see how a central angle defines a sector's fractional part.

A sector is a region bounded by two radii and an enclosing circular arc.
The total angle at the centre of any complete circle is always $360^{\circ}$ .
A sector with central angle $θ$ represents the fraction $\frac{θ}{360^{\circ}}$ of the entire circle.
Simplifying the fraction $\frac{θ}{360^{\circ}}$ gives the exact proportion of the total circular region.

Worked example. State the formula for the fraction of a circle represented by a sector:

Fraction = \frac{θ}{360^{\circ}}

. Substitute the given central angle

θ = 55^{\circ}

into the expression:

Fraction = \frac{55^{\circ}}{360^{\circ}}

. Simplify the fraction by dividing both the numerator and the denominator by 5:

\frac{55}{360} = \frac{11}{72}

. Answer: $\frac{11}{72}$ .

Check question: A circular cardboard disc is divided into sectors. One sector has a central angle of $54^{\circ}$ . What fraction of the entire disc surface does this sector represent?

$\frac{3}{20}$
$\frac{3}{10}$
$\frac{17}{20}$
$\frac{27}{50}$

Show answer

$\frac{3}{20}$ . Divide the central angle $54^{\circ}$ by $360^{\circ}$ to get $\frac{54^{\circ}}{360^{\circ}}$ , which simplifies to $\frac{3}{20}$ by dividing both the numerator and the denominator by 18.

Slide 2 · Evaluating Area of a Circular SectorModel · 22 min

Figure 2: A circle showing a shaded sector of 160 degrees with a radius of 21 centimetres, alongside the steps to calculate its area.

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Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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Area of a Sector 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