Grade 10 · Core Mathematics · Lesson 8 of 12

Area of Common Region Between Two Intersecting Circles

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of a Part of a Circle. In it, learners identify the common chord and the two circular segments forming the region of intersection between two circles.

Strand Measurements and GeometrySub-strand Area of a Part of a CircleLesson code 10CM-26-L840 minutes · 2 slides

KICD learning outcome (e)

Determine the area of common region between two intersecting circles.

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

Identify the common chord and the two circular segments forming the region of intersection between two circles.
Calculate the area of intersection between two circles by summing the areas of the respective segments.

Key vocabulary

Common Chord: A line segment connecting the two intersection points shared by two overlapping circles.; The common chord $A B$ divides the overlapping area into two distinct circular segments.
Area of Intersection: The total surface area of the common region shared between two overlapping shapes.; The area of intersection equals the sum of the two circular segments on $A B$ .

Lesson preview

Slide 1 · Understanding Intersecting Circles and Common ChordsModel · 18 min

Figure 1: Two intersecting circles showing a common chord that divides the shaded overlapping region into two segments.

When two circles with centres $O_{1}$ and $O_{2}$ intersect at points $A$ and $B$ , the line segment $A B$ is their common chord. This common chord divides the region of overlap into two circular segments: one belonging to the circle with centre $O_{1}$ , and the other to the circle with centre $O_{2}$ . To determine the area of the common region, we calculate the area of each circular segment independently using sector and triangle formulae, then add the two segment areas together.

Two intersecting circles with centres $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ .
The line segment connecting intersection points $A$ and $B$ is called their common chord.
The common region shared by both circles is split into two circular segments by the common chord.
Adding the area of the segment from the first circle to the area of the segment from the second circle gives the total area of intersection.

Worked example. Identify the common region as the sum of two identical circular segments on chord

A B

. Calculate the area of sector

A O_{1} B

\frac{90^{\circ}}{360^{\circ}} \times 3.142 \times 12^{2} = 0.25 \times 3.142 \times 144 = 113.112 {cm}^{2}

. Calculate the area of triangle

A O_{1} B

\frac{1}{2} \times 12 \times 12 \times \sin 90^{\circ} = 72 {cm}^{2}

. Find the area of segment 1:

113.112 - 72 = 41.112 {cm}^{2}

. Since the second circle is identical, segment 2 area is also

41.112 {cm}^{2}

. Sum the two segment areas:

41.112 + 41.112 = 82.224 {cm}^{2}

. Answer: $82.22 {cm}^{2}$ .

Check question: Two overlapping circular floral beds with centres $O_{1}$ and $O_{2}$ each have a radius of $10 m$ . The angle subtended by their common chord $A B$ at each centre is $90^{\circ}$ . Calculate the total area of the common region. (Take $π = 3.142$ )

$28.55 m^{2}$
$57.10 m^{2}$
$157.10 m^{2}$
$257.10 m^{2}$

Show answer

$57.10 m^{2}$ . For one floral bed, the sector area is $\frac{90^{\circ}}{360^{\circ}} \times 3.142 \times 10^{2} = 78.55 m^{2}$ . The area of the triangle is $\frac{1}{2} \times 10 \times 10 \times \sin 90^{\circ} = 50 m^{2}$ . The area of one circular segment is $78.55 - 50 = 28.55 m^{2}$ . Since the two floral beds are identical, the total area of the common region is $2 \times 28.55 = 57.10 m^{2}$ .

Slide 2 · Calculating the Area of Intersection for Non-Identical CirclesPractice · 22 min

Figure 2: Two non-identical intersecting circles with their radii, central angles, and a common chord dividing the shaded area.

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Homework and the other 11 lessons in this sub-strand

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Area of Common Region Between Two Intersecting Circles

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

Key vocabulary

Lesson preview

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All 12 lessons in Area of a Part of a Circle