Radian measure, arc length and area of sector

Radian Measure, Arc Length, and Area of Sector

Welcome to this chapter on radian measure! If you have ever wondered why mathematicians use radians instead of degrees, you are about to find out. While degrees divide a circle into \(360\) arbitrary slices (an idea dating back to ancient Babylon), radians measure angles using the circle's own geometry: its radius and arc length. This makes geometric formulas remarkably simple and elegant.

In your Pearson Edexcel International GCSE Further Pure Mathematics examinations (Papers 1 and 2), questions on radians often form the starting steps of larger coordinate geometry, trigonometry, or calculus problems. Don't worry if this feels unfamiliar at first—once you grasp the core relationships, these formulas are among the easiest marks to secure on the exam paper.

1. What Is a Radian?

Imagine bending a piece of string equal in length to the radius \(r\) of a circle along the circumference. The angle subtended at the centre by this curved arc is defined as exactly \(1\text{ radian}\) (often abbreviated as \(1\text{ rad}\) or written with a superscript \(\text{c}\), such as \(1^{\text{c}}\)).

Because the full circumference of a circle is \(2\pi r\), the complete angle for one full turn (\(360^\circ\)) contains exactly \(2\pi\) radiuses along its edge. This gives us the fundamental equivalence:

\(2\pi\text{ radians} = 360^\circ\)

Dividing both sides by \(2\), we obtain the most useful conversion identity:

\(\pi\text{ radians} = 180^\circ\)

Converting Between Degrees and Radians

To convert between the two systems:

• From degrees to radians: multiply by \(\frac{\pi}{180^\circ}\)
• From radians to degrees: multiply by \(\frac{180^\circ}{\pi}\)

Example: Convert \(60^\circ\) to radians.
\(60^\circ \times \frac{\pi}{180^\circ} = \frac{60\pi}{180} = \frac{\pi}{3}\text{ radians}\)

Example: Convert \(\frac{3\pi}{4}\text{ radians}\) to degrees.
\(\frac{3\pi}{4} \times \frac{180^\circ}{\pi} = \frac{3 \times 180^\circ}{4} = 135^\circ\)

Essential Angle Equivalences You Must Recognise Instantly

• \(30^\circ = \frac{\pi}{6}\text{ rad}\)
• \(45^\circ = \frac{\pi}{4}\text{ rad}\)
• \(60^\circ = \frac{\pi}{3}\text{ rad}\)
• \(90^\circ = \frac{\pi}{2}\text{ rad}\)
• \(180^\circ = \pi\text{ rad}\)
• \(270^\circ = \frac{3\pi}{2}\text{ rad}\)
• \(360^\circ = 2\pi\text{ rad}\)

Key Takeaway: \(\pi\text{ radians} = 180^\circ\). Keep this relationship in your head at all times.

2. Arc Length of a Circle

An arc is a continuous portion of the circumference of a circle. When an angle \(\theta\) at the centre is measured in radians, the relationship between arc length \(s\) and radius \(r\) is extraordinarily direct.

Recall from standard mathematics that in degrees, \(\text{arc length} = \frac{\theta^\circ}{360^\circ} \times 2\pi r\).
Replacing \(360^\circ\) with \(2\pi\) radians gives:

\(s = \frac{\theta}{2\pi} \times 2\pi r = r\theta\)

Arc Length Formula:
\(s = r\theta\)
where \(r\) is the radius and \(\theta\) is the subtended angle in radians.

Crucial Syllabus Note: This formula is not given on the examination formula sheet (Appendix 4). You are required to know and recall it from memory (Appendix 5).

Perimeter of a Sector

A sector is shaped like a slice of pizza. Its boundary consists of two straight edges (each of length \(r\)) and one curved edge (the arc \(s\)).

\(\text{Perimeter of sector} = 2r + s = 2r + r\theta = r(2 + \theta)\)

Worked Example 1: Calculating Arc Length and Perimeter

A circle has a radius of \(6\text{ cm}\). A sector has a central angle of \(\frac{2\pi}{3}\text{ radians}\).
Find:
(a) The arc length of the sector, giving your answer as an exact multiple of \(\pi\).
(b) The total perimeter of the sector, correct to \(3\) significant figures.

Solution:
(a) Using \(s = r\theta\):
\(s = 6 \times \frac{2\pi}{3} = 4\pi\text{ cm}\)
(b) Using \(\text{Perimeter} = 2r + s\):
\(\text{Perimeter} = 2(6) + 4\pi = 12 + 4\pi \approx 12 + 12.566 = 24.6\text{ cm}\) (to \(3\text{ s.f.}\))

Key Takeaway: Arc length is simply \(s = r\theta\). If a question asks for the perimeter of a sector, remember to add both radii (\(2r\)) to the arc length.

3. Area of a Sector

A sector is the region enclosed by two radii and the arc between them. Just like arc length, measuring \(\theta\) in radians simplifies the area formula.

Using the proportion of a full circle (\(2\pi\)):
\(\text{Area} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta\)

Sector Area Formula:
\(A = \frac{1}{2}r^2\theta\)
where \(r\) is the radius and \(\theta\) is the central angle in radians.

Crucial Syllabus Note: This formula is also not provided on the examination formula sheet. You must commit it to memory.

Alternative Form for Area

Notice that because \(s = r\theta\), we can substitute \(s\) into the area formula:

\(A = \frac{1}{2}r(r\theta) = \frac{1}{2}rs\)

This is useful if you are given the arc length and the radius directly without needing to calculate \(\theta\) first!

Worked Example 2: Finding an Unknown Angle

A sector of a circle of radius \(8\text{ cm}\) has an area of \(28\text{ cm}^2\).
Find the angle \(\theta\) of the sector in radians, correct to \(3\) significant figures.

Solution:
\(A = \frac{1}{2}r^2\theta\)
\(28 = \frac{1}{2}(8)^2\theta\)
\(28 = \frac{1}{2}(64)\theta\)
\(28 = 32\theta\)
\(\theta = \frac{28}{32} = \frac{7}{8} = 0.875\text{ radians}\)

Key Takeaway: The area of a sector is \(A = \frac{1}{2}r^2\theta\). Always ensure \(\theta\) is expressed in radians before multiplying.

4. Segments of a Circle

A segment is the region bounded by a chord and the arc connecting its endpoints. A chord divides a circle into two segments: a minor segment (smaller area) and a major segment (larger area).

Area of a Minor Segment

To find the area of the minor segment, subtract the area of the isosceles triangle formed by the centre and the chord from the area of the entire sector:

\(\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}\)

Recall the triangle area formula from section 10D: \(\text{Area} = \frac{1}{2}ab\sin C\).
Here, the two sides are both radii (\(a = r, b = r\)) and the included angle is \(\theta\):

\(\text{Area of triangle} = \frac{1}{2}r^2\sin\theta\)

Putting these together:

\(\text{Area of segment} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta)\)

Perimeter of a Segment

The boundary of a segment consists of the curved arc and the straight chord.
• Curved arc: \(s = r\theta\)
• Straight chord: Using the cosine rule from section 10D (\(c^2 = a^2 + b^2 - 2ab\cos C\)):
\(\text{chord}^2 = r^2 + r^2 - 2(r)(r)\cos\theta = 2r^2 - 2r^2\cos\theta = 2r^2(1 - \cos\theta)\)
Alternatively, splitting the isosceles triangle into two right-angled triangles gives \(\text{chord} = 2r\sin\left(\frac{\theta}{2}\right)\).

\(\text{Perimeter of segment} = r\theta + 2r\sin\left(\frac{\theta}{2}\right)\)

Worked Example 3: Full Examination-Style Problem

A circle with centre \(O\) has radius \(r = 5\text{ cm}\). Points \(A\) and \(B\) lie on the circumference such that \(\angle AOB = 1.2\text{ radians}\).
Calculate:
(a) The length of chord \(AB\), to \(3\) significant figures.
(b) The area of the minor segment enclosed by the chord \(AB\) and the arc \(AB\), to \(3\) significant figures.

Solution:
(a) Using the cosine rule on \(\triangle AOB\):
\(AB^2 = 5^2 + 5^2 - 2(5)(5)\cos(1.2)\)
Ensure your calculator is in RAD mode! \(\cos(1.2) \approx 0.362358\)
\(AB^2 = 25 + 25 - 50(0.362358) = 50 - 18.1179 = 31.882\)
\(AB = \sqrt{31.882} \approx 5.65\text{ cm}\)

(b) Using the segment area formula:
\(\text{Area} = \frac{1}{2}r^2(\theta - \sin\theta)\)
\(\text{Area} = \frac{1}{2}(5^2)(1.2 - \sin(1.2))\)
With calculator in radian mode: \(\sin(1.2) \approx 0.932039\)
\(\text{Area} = \frac{25}{2}(1.2 - 0.932039) = 12.5 \times 0.267961 \approx 3.35\text{ cm}^2\)

Key Takeaway: Segment area is \(\frac{1}{2}r^2(\theta - \sin\theta)\). Never compute this with your calculator set to degrees!

5. Common Traps and Exam Advice

• Calculator Mode: This is the single most common cause of lost marks in Further Pure Mathematics. When evaluating \(\sin\theta\) or \(\cos\theta\) where \(\theta\) is in radians, check that your calculator display shows an R (radians), not a D (degrees).
• Perimeter vs. Arc Length: Read the question wording carefully. If asked for the perimeter of the sector, students often calculate only the arc length \(r\theta\) and forget to add the two straight sides (\(+ 2r\)).
• Exact vs. Rounded Values: Edexcel papers frequently request answers in terms of \(\pi\) (for example, \(4\pi\text{ cm}\)). Unless specified otherwise, give exact multiples of \(\pi\) or round non-exact answers to \(3\) significant figures.
• Angles Given in Degrees: If an examination problem provides an angle in degrees, you must convert it to radians before using \(s = r\theta\) or \(A = \frac{1}{2}r^2\theta\).

6. Quick Reference Summary

Conversion: \(\pi\text{ rad} = 180^\circ \implies 1\text{ rad} = \frac{180^\circ}{\pi}\)

Arc Length: \(s = r\theta\)

Perimeter of Sector: \(P = 2r + r\theta\)

Area of Sector: \(A = \frac{1}{2}r^2\theta\)

Area of Triangle in Sector: \(A_{\text{triangle}} = \frac{1}{2}r^2\sin\theta\)

Area of Minor Segment: \(A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)\)