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Arcs, Sectors and Segments

Neil Trivedi

Teacher

Neil Trivedi

Contents

Arc Lengths

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Areas of Sectors

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Area of Segments

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Challenging Question

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Arc Lengths

Definition:

The arc length of a circle is the portion of the circumference subtended by a given central angle.

The image shows a circle with a radius drawn from the centre to the circumference, forming an angle at the centre used to illustrate circular measure or sector geometry.

Formula:

In degrees:

In radians:


And some extra terminology:

The diagram shows a circle divided into a major sector and a minor sector, illustrating how a central angle determines the corresponding arc lengths and sector areas.

Example 1:

A sector of a circle with radius cm has an arc length of cm. Find the angle subtended at the centre of the circle.

Single Step: Substitute the values given into the arc length formula in radians and solve it.

The diagram shows a circular sector with central angle 𝜃, radius 𝑟, and labelled arc length, illustrating the relationship between angle, radius, and arc length.

No answer provided.

Areas of Sectors

Definition:

A sector is a portion of a circle bounded by two radii and an arc.

The diagram shows a circle with a shaded sector of radius 𝑟 and central angle 𝜃, illustrating how a sector is defined by two radii and the intercepted arc.

Formula:

In degrees:

In radians:

Example 2:

A sector of a circle has radius cm and the angle subtended at the centre of the circle is . What is the area of the sector?

Single Step: Substitute the values given into the area of sector formula in radians.

The diagram shows a circular sector with radius 𝑟, central angle 𝜃, and arc length 𝑙, illustrating the relationship between angle, radius, and arc length.

cm

No answer provided.

Area of Segments

Definition:

A segment is the region of a circle bounded by an arc and a chord.

The diagram shows a circular sector of radius 𝑟 and angle 𝜃 with the shaded region representing the circular segment formed by subtracting the triangle from the sector.

Formula:

In radians only:

The formula is derived from subtracting the area of a triangle from the area of the sector.

Area of the Sector

Area of the Triangle

Area of Segment Area of Sector Area of Triangle

Area of Segment

Factorising out ,

Area of Segment

Example 3:

Find the area of the shaded segment.

The diagram shows a circular sector with centre 𝑂, radius 𝑟, and central angle 𝜃=120^∘, highlighting the shaded circular segment between the arc and the chord.

Step 1: Convert the angle from degrees to radians.

Step 2: Substitute the values given into the area of segment formula.

Note: Ensure your calculator is in radians mode when computing these numbers.

Then expanding the brackets:

cm

No answer provided.

Example 4:

In the diagram,  is the diameter of a circle of radius cm, and radians. Given that the area of  is four times that of the shaded segment, show that

The diagram shows a semicircle with centre 𝑂, a chord forming a central angle 𝜃, and the shaded region representing the circular segment between the arc and the chord.

Step 1: Find the area of the segment and .

We know that , and  due to symmetry of the graph at (to do with primary and secondary values), so

Area of segment

Area of segment 

Step 2: Write the area of  as four times of the area of the segment and rearrange.

Moving the and over to the other side,

No answer provided.

Challenging Question

Practice Questions

Introduction to Radians Small Angle Approximations