Welcome to Circles, Arcs, and Sectors!
Hi there! This chapter is all about understanding curved shapes, which falls under the big topic of Mensuration (measuring shapes). Circles are everywhere—from car tyres and clocks to pizzas and Ferris wheels—so knowing how to measure their size is a crucial life skill and an essential part of your IGCSE Maths exam.
Don't worry if geometry feels challenging. We'll break down the formulas into simple, easy-to-remember parts, focusing on the angle fraction that makes all the calculations work!
Section 1: The Anatomy of a Circle and Full Circle Calculations
1.1 Essential Circle Vocabulary
Before we start measuring, let’s quickly review the key parts of a circle you need to know:
- Centre: The middle point, equidistant from all points on the boundary.
- Radius (\(r\)): The distance from the centre to the circumference.
- Diameter (\(d\)): The distance across the circle through the centre. (Remember: \(d = 2r\))
- Circumference (\(C\)): The perimeter, or the total distance around the outside of the circle.
- Arc: A part of the circumference.
- Sector: A region bounded by two radii and an arc (like a slice of pizza).
- Chord: A straight line segment connecting two points on the circumference (it does not have to pass through the centre).
1.2 Core Formulas for the Whole Circle
For any calculation involving arcs or sectors, you MUST know the formulas for the full circle first. These formulas will be given in the exam paper, but learning them makes life much faster!
Tip: The Magic Number \(\pi\) (Pi)
The constant \(\pi\) is approximately 3.14159... It represents the ratio of a circle's circumference to its diameter. In calculations, always use the \(\pi\) button on your calculator for accuracy, or follow the question instructions (e.g., "Use \(\pi = 3.14\)" or "Give your answer in terms of \(\pi\)").
A. Circumference (Perimeter)
This is the distance *around* the circle. Think of it as the length of the boundary line.
$$\text{Circumference } (C) = 2 \pi r$$ or $$\text{Circumference } (C) = \pi d$$
B. Area
This is the amount of 2D space *inside* the circle.
$$\text{Area } (A) = \pi r^2$$
When you find Area, the units are squared (e.g., \(cm^2\)). Therefore, the formula for Area must involve $r$ squared: \(\pi r^2\).
When you find Circumference (a length), the units are not squared, so the formula is \(2\pi r\).
Section 2: Arcs and Arc Length (Parts of the Circumference)
An arc is just a curved portion of the circle’s perimeter. To find its length, we figure out what fraction of the full circle (\(360^\circ\)) the arc represents, based on the angle at the centre.
2.1 The Arc Length Formula
If the central angle of the arc is \(\theta\) (in degrees), the arc length is calculated using the fraction \(\frac{\theta}{360}\).
$$\text{Arc Length } (L) = \frac{\theta}{360} \times 2 \pi r$$
Did you know? If your angle \(\theta\) is \(180^\circ\), you are finding half the circumference, which is exactly what \(\frac{180}{360} \times 2\pi r\) calculates!
2.2 Step-by-Step: Calculating Arc Length
Let's say a circle has a radius of \(5\) cm, and the arc is subtended by a central angle of \(72^\circ\).
- Identify knowns: \(r = 5\) cm, \(\theta = 72^\circ\). The full circumference formula is \(2\pi r\).
- Set up the fraction: The angle fraction is \(\frac{72}{360}\).
- Substitute into the formula: $$\text{Arc Length } = \frac{72}{360} \times 2 \times \pi \times 5$$
- Calculate (in terms of \(\pi\)): $$\text{Arc Length } = \frac{1}{5} \times 10\pi = 2\pi \text{ cm}$$
- Calculate (as decimal to 3 s.f.): $$\text{Arc Length } = 2 \times \pi \approx 6.28 \text{ cm}$$
2.3 Perimeter of a Sector
When asked for the Perimeter of a Sector, remember that the perimeter includes all the edges of the shape. A sector has the curved arc length PLUS the two straight radii that go back to the centre.
$$\text{Perimeter of Sector} = \text{Arc Length} + 2r$$
Example: Using the values above (\(L = 2\pi\) and \(r = 5\)):
$$\text{Perimeter} = 2\pi + 5 + 5 = 2\pi + 10$$
$$\text{Perimeter} \approx 6.283 + 10 = 16.283 \approx 16.3 \text{ cm (3 s.f.)}$$
Do not confuse the Arc Length (the curved part only) with the Perimeter of the Sector (curved part + two radii). Always read the question carefully!
Section 3: Sectors and Sector Area (Parts of the Area)
A sector is a region like a wedge or a slice. Just like with arc length, we calculate the area by finding the fraction of the total circle area that the sector occupies.
3.1 The Sector Area Formula
If the central angle of the sector is \(\theta\) (in degrees), the sector area is calculated using the same angle fraction: \(\frac{\theta}{360}\).
$$\text{Sector Area } (A) = \frac{\theta}{360} \times \pi r^2$$
3.2 Step-by-Step: Calculating Sector Area
Let's find the area of the sector with \(r = 5\) cm and \(\theta = 72^\circ\).
- Identify knowns: \(r = 5\) cm, \(\theta = 72^\circ\). The full area formula is \(\pi r^2\).
- Set up the fraction: The angle fraction is \(\frac{72}{360}\).
- Substitute into the formula: $$\text{Sector Area} = \frac{72}{360} \times \pi \times 5^2$$
- Calculate (in terms of \(\pi\)): $$\text{Sector Area} = \frac{1}{5} \times 25\pi = 5\pi \text{ cm}^2$$
- Calculate (as decimal to 3 s.f.): $$\text{Sector Area} = 5 \times \pi \approx 15.7 \text{ cm}^2 \text{ (3 s.f.)}$$
3.3 Working with Major and Minor Sectors
The total angle in a circle is \(360^\circ\). If a question specifies a minor arc/sector, use the smaller angle \(\theta\). If it specifies the major arc/sector, use the reflex angle (\(360^\circ - \theta\)).
Example: If the minor sector angle is \(120^\circ\), the major sector angle is \(360^\circ - 120^\circ = 240^\circ\).
Section 4: Key Calculation Techniques and Exam Tips
4.1 Calculations in Terms of \(\pi\)
Sometimes, questions ask for an answer "in terms of \(\pi\)". This means you leave the symbol \(\pi\) in your answer, treating it like a variable. This provides the mathematically "exact" answer.
Step 1: Calculate the numerical and fractional parts of the equation first.
Step 2: Multiply everything, but leave \(\pi\) at the end.
Example: Find the arc length for a \(30^\circ\) sector with \(r = 6\).
$$\text{Arc Length} = \frac{30}{360} \times 2 \pi (6)$$
$$\text{Arc Length} = \frac{1}{12} \times 12 \pi$$
$$\text{Arc Length} = 1 \pi$$
$$\text{Answer: } \pi$$
4.2 Using Given Information
In harder questions, you might be given the area or circumference and asked to find the radius or angle. You just need to rearrange the formulas!
Example: A circle has a circumference of \(40\pi\) cm. Find its radius.
- Write down the formula: \(C = 2\pi r\)
- Substitute known values: \(40\pi = 2\pi r\)
- Solve for \(r\): Divide both sides by \(2\pi\).
$$\frac{40\pi}{2\pi} = r$$
$$r = 20 \text{ cm}$$
Don't worry if this seems tricky at first. It's just a linear equation, but using \(\pi\) instead of numbers can look intimidating!
Quick Review Summary
Formulas (Angle \(\theta\) in degrees)
- Full Circle Circumference: \(C = 2\pi r\)
- Full Circle Area: \(A = \pi r^2\)
- Arc Length: \(L = \frac{\theta}{360} \times 2\pi r\)
- Sector Area: \(A_s = \frac{\theta}{360} \times \pi r^2\)
- Perimeter of Sector: \(L + 2r\)
Key Takeaway: All arc and sector calculations rely on finding the fraction of the circle using the central angle \(\frac{\theta}{360}\), and then multiplying that fraction by the appropriate full circle formula (circumference for lengths, area for areas).