Welcome to Arc Length, Sectors, and Mensuration!
Have you ever wondered how to calculate the crust length on a single slice of pizza, or how much icing you need to cover a fan-shaped cake? That is exactly what this chapter is all about! In this guide, we will explore parts of circles—known as arcs and sectors—and learn how to solve real-world measurement (mensuration) problems with confidence.
Don't worry if geometry formulas have felt overwhelming in the past. Once you learn the simple secret of "fractions of a circle," every single formula in this topic becomes easy to understand and remember!
Quick Review: Circle Basics
Before diving into parts of circles, let's make sure our foundation is rock solid.
• Radius (\(r\)): The distance from the exact centre of the circle to any point on the outer edge.
• Diameter (\(d\)): The straight line distance right across the circle passing through the centre. Remember: \(d = 2r\).
• Circumference (\(C\)): The total perimeter (distance all the way around) of a full circle: \(C = 2\pi r\) or \(C = \pi d\).
• Area of a Full Circle (\(A\)): The total flat space inside the circle: \(A = \pi r^2\).
The Master Key: The "Fraction of a Circle" Concept
Think about a full turn. A complete circle has an angle of \(360^\circ\) at its centre.
If you take a slice of a circle with an angle of \(\theta\) (the Greek letter theta, used for angles) at the centre, that slice represents a specific fraction of the entire circle:
Fraction of circle \(= \frac{\theta}{360^\circ}\)
Analogy: If a pizza has a central angle of \(90^\circ\), the fraction is \(\frac{90^\circ}{360^\circ} = \frac{1}{4}\). You have a quarter of the pizza! If the angle is \(180^\circ\), you have \(\frac{180^\circ}{360^\circ} = \frac{1}{2}\) (a semicircle).
Key Takeaway: Whenever you want to find the length or area of a circular slice, you simply take the standard full-circle formula and multiply it by \(\frac{\theta}{360^\circ}\)!
1. Arc Length
What is an Arc?
An arc is a curved portion of the circumference of a circle. Think of it as just the curved crust of your pizza slice.
• Minor Arc: The shorter curved path between two points (central angle \(\theta < 180^\circ\)).
• Major Arc: The longer curved path around the outside (central angle \(\theta > 180^\circ\)). If a question asks for the major arc, use \(360^\circ - \theta\).
The Arc Length Formula
\(\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r\) (or \(\frac{\theta}{360^\circ} \times \pi d\))
Step-by-Step Example: Calculating Arc Length
Example: A circle has a radius of \(6\text{ cm}\). Calculate the length of a minor arc with a central angle of \(60^\circ\). Give your answer to \(1\) decimal place.
Step 1: Identify what you know: \(\theta = 60^\circ\) and \(r = 6\text{ cm}\).
Step 2: Write the formula: \(\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r\)
Step 3: Substitute the values: \(\text{Arc Length} = \frac{60^\circ}{360^\circ} \times 2 \times \pi \times 6\)
Step 4: Simplify: \(\text{Arc Length} = \frac{1}{6} \times 12\pi = 2\pi \approx 6.283...\text{ cm}\)
Step 5: Round as requested: \(\text{Arc Length} \approx 6.3\text{ cm}\).
Crucial Difference: Arc Length vs. Perimeter of a Sector
Watch out! One of the most common exam traps is confusing arc length with the perimeter of a sector.
• Arc length: Just the curved boundary line.
• Perimeter of a sector: The entire perimeter, which includes the curved arc plus the two straight radius edges!
\(\text{Perimeter of a Sector} = \text{Arc Length} + 2r = \left(\frac{\theta}{360^\circ} \times 2\pi r\right) + 2r\)
Key Takeaway: If an exam asks for the perimeter of a slice, always remember to add the two straight sides (\(+ 2r\)) to the curved arc!
2. Area of a Sector
What is a Sector?
A sector is the region enclosed by two radii and the arc connecting them. It is the actual slice of the circle (the flat 2D shape).
• Minor Sector: The smaller slice (angle \(\theta < 180^\circ\)).
• Major Sector: The larger remaining portion of the circle (angle \(> 180^\circ\)).
The Sector Area Formula
\(\text{Area of a Sector} = \frac{\theta}{360^\circ} \times \pi r^2\)
Step-by-Step Example: Calculating Sector Area
Example: Find the area of a sector of a circle with radius \(8\text{ cm}\) and a central angle of \(45^\circ\). Give your answer to \(3\) significant figures.
Step 1: Identify the values: \(\theta = 45^\circ\), \(r = 8\text{ cm}\).
Step 2: Write the formula: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\)
Step 3: Substitute values: \(\text{Area} = \frac{45^\circ}{360^\circ} \times \pi \times 8^2 = \frac{1}{8} \times \pi \times 64 = 8\pi\)
Step 4: Calculate and round: \(8\pi \approx 25.132... \approx 25.1\text{ cm}^2\).
Working Backwards: Finding the Angle \(\theta\) or Radius \(r\)
Higher-tier questions often give you the area or arc length and ask you to find the unknown angle or radius.
Example: A sector has a radius of \(5\text{ cm}\) and an area of \(20\text{ cm}^2\). Find the central angle \(\theta\) to \(1\) decimal place.
Step 1: Set up the equation: \(20 = \frac{\theta}{360^\circ} \times \pi \times 5^2\)
Step 2: Simplify the right side: \(20 = \frac{\theta}{360^\circ} \times 25\pi\)
Step 3: Multiply both sides by \(360^\circ\): \(7200 = \theta \times 25\pi\)
Step 4: Divide by \(25\pi\): \(\theta = \frac{7200}{25\pi} \approx \frac{7200}{78.5398...} \approx 91.673...\)
Step 5: Final answer: \(\theta \approx 91.7^\circ\).
Key Takeaway: To find the area of a sector, simply multiply the full circle's area (\(\pi r^2\)) by the angle fraction \(\frac{\theta}{360^\circ}\).
3. Area of a Segment
What is a Segment?
A segment is the region between a chord (a straight line joining two points on a circle) and the arc above it. Think of it as cutting off just the crust of a pizza with a straight knife cut.
The Formula for the Area of a Segment
To find the area of a segment, you take the area of the entire sector and subtract the triangle formed by the two radii and the chord:
\(\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}\)
Recall the formula for the area of any triangle when you know two sides and the included angle: \(\text{Area} = \frac{1}{2}ab\sin C\). Because both sides are radii (\(r\)), the triangle area inside a circle is:
\(\text{Area of Triangle} = \frac{1}{2}r^2\sin\theta\)
Therefore:
\(\text{Area of Segment} = \left(\frac{\theta}{360^\circ} \times \pi r^2\right) - \left(\frac{1}{2}r^2\sin\theta\right)\)
Step-by-Step Example: Segment Area
Example: A circle has radius \(10\text{ cm}\). A chord connects the ends of an arc that subtends an angle of \(120^\circ\) at the centre. Find the area of the minor segment to \(2\) decimal places.
Step 1: Calculate the sector area:
\(\text{Area of Sector} = \frac{120^\circ}{360^\circ} \times \pi \times 10^2 = \frac{1}{3} \times 100\pi \approx 104.720\text{ cm}^2\)
Step 2: Calculate the triangle area (make sure your calculator is in DEGREE mode!):
\(\text{Area of Triangle} = \frac{1}{2} \times 10 \times 10 \times \sin(120^\circ) = 50 \times \frac{\sqrt{3}}{2} \approx 43.301\text{ cm}^2\)
Step 3: Subtract the triangle from the sector:
\(\text{Area of Segment} = 104.720 - 43.301 = 61.419\text{ cm}^2\)
Step 4: Round: \(\text{Area} \approx 61.42\text{ cm}^2\).
Key Takeaway: \(\text{Segment} = \text{Sector} - \text{Triangle}\). Always compute both parts separately first to keep your work clear and avoid mistakes.
4. Mensuration Problems & 3D Connections
Composite Shapes
In examination questions, sectors are frequently combined with other 2D shapes (such as squares, rectangles, or triangles). The golden rule for composite shapes is:
1. Break the shape into simple individual shapes you recognise.
2. Calculate each area or length separately.
3. Add or subtract the pieces as required.
The Cone Connection: Unfolding a Sector
Did you know? If you take a circular sector made of paper and join its two straight edges together, it forms the curved surface of a 3D cone!
• The radius of the original sector (\(R\)) becomes the slant height (\(l\)) of the cone.
• The arc length of the sector becomes the circumference of the cone's base (\(2\pi r_{\text{base}}\)).
• The curved surface area of a cone formula is \(\text{CSA} = \pi r l\), which directly comes from the area of that unrolled sector!
5. Common Mistakes to Avoid
• Angle Mode Error: Ensure your calculator is set to Degrees (D) and not Radians (R) when calculating \(\sin\theta\).
• Radius vs. Diameter: Always double-check if the question gives you the diameter or the radius. If given the diameter, divide by \(2\) immediately!
• Sector Perimeter Trap: Do not write down just the arc length when asked for the perimeter of a sector. Always add \(+ 2r\).
• Premature Rounding: Keep full calculator values or store them in memory until your final answer to avoid rounding inaccuracy.
Quick Review Summary Table
• Arc Length: \(\frac{\theta}{360^\circ} \times 2\pi r\)
• Perimeter of Sector: \(\left(\frac{\theta}{360^\circ} \times 2\pi r\right) + 2r\)
• Area of Sector: \(\frac{\theta}{360^\circ} \times \pi r^2\)
• Area of Segment: \(\left(\frac{\theta}{360^\circ} \times \pi r^2\right) - \left(\frac{1}{2}r^2\sin\theta\right)\)