Circles can be divided into parts, such as arcs, sectors and segments. Being able to find the length of an arc and the area of sectors and segments is essential when working with circular shapes.

 

 

Length of a Circular Arc

A circular arc is part of the circumference of a circle.

 

The length of an arc depends on:
• the radius of the circle
• the size of the angle at the centre

 

The full circumference of a circle is:
$$
2\pi r
$$

 

If the angle at the centre is given in degrees, the length of the arc is found using:
$$
arc\ length = \frac{\theta}{360^\circ} \times 2\pi r
$$

 

where \( \theta \) is the angle at the centre.

 

This formula finds the fraction of the full circumference.

 

The angle must be the angle at the centre of the circle

 

 

Area of a Sector

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

 

The area of a full circle is:
$$
\pi r^2
$$

 

The area of a sector is a fraction of the full circle area.

 

If the angle at the centre is \( \theta \) degrees, the area of the sector is:
$$
area = \frac{\theta}{360^\circ} \times \pi r^2
$$

 

Larger angles give larger sector areas.

 

 

Area of a Segment

A segment is the region between a chord and the arc of a circle.

 

To find the area of a segment:
• find the area of the sector
• find the area of the triangle formed by the two radii and the chord
• subtract the triangle area from the sector area

 

Area of the triangle is found using:
$$
area = \frac{1}{2}ab\sin\theta
$$

 

where \( a \) and \( b \) are the radii and \( \theta \) is the angle between them.

 

This method works when the angle at the centre is known.

 

The segment area is always smaller than the sector area

 

 

Using Circular Measures Correctly

When solving problems involving arcs, sectors or segments:

• check that the angle is at the centre
• use the correct formula
• keep units consistent
• include units in the final answer

 

Angles are usually given in degrees unless stated otherwise.

 

 

Key Points to Remember

An arc is part of a circle’s circumference.
Arc length is a fraction of the full circumference.
A sector is a fraction of the circle’s area.
A segment is found by subtracting a triangle from a sector.
The central angle determines both arc length and sector area.

 

Understanding circular arcs, sectors and segments allows you to solve a wide range of geometric problems involving circles accurately and confidently.