CIE IGCSE Mathematics (0580) Circles, arcs and sectors Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Circles, arcs and sectors Study Notes
LEARNING OBJECTIVE
- Circles
Key Concepts:
- Circles and its Arc Length and Sector Area
Circles
Circles
A circle is a round shape with all points on its boundary equidistant from the center.
Parts of a Circle:
- Radius (r): Distance from the center to any point on the circle.
- Diameter (d): Distance across the circle through the center. \( d = 2r \)
- Circumference: The perimeter (distance around) of the circle.
- Area: The space enclosed by the circle.
Key Formulas:
- Circumference:
\( C = \pi d \) or \( C = 2\pi r \) - Area:
\( A = \pi r^2 \) - Use \( \pi \approx 3.14 \) or calculator’s π button for accuracy.
Important Notes:
- Use radius if not told otherwise.
- Always include correct units (e.g. cm² for area, cm for circumference).
- Round your final answer appropriately (usually to 2 decimal places unless told otherwise).
Example:
Find the circumference of a circle with radius 7 cm.
▶️ Answer/Explanation
Using \( C = 2\pi r \):
\( C = 2 \times \pi \times 7 = 14\pi \approx 43.98 \ \text{cm} \)
Answer: 43.98 cm
Example:
Find the area of a circle with a diameter of 10 cm.
▶️ Answer/Explanation
Radius = \( \frac{10}{2} = 5 \ \text{cm} \)
Using \( A = \pi r^2 \):
\( A = \pi \times 5^2 = 25\pi \approx 78.54 \ \text{cm}^2 \)
Answer: 78.54 cm²
Example:
A circular garden has a path of width 1.5 m running all around it. The radius of the garden without the path is 7 m. Find:
- the area of the path
- the total length of fencing required to go around the outer edge of the path
▶️ Answer/Explanation
outer radius (including path):
\( R = 7 + 1.5 = 8.5 \ \text{m} \)
\( A_{\text{outer}} = \pi R^2 = \pi (8.5)^2 = \pi \times 72.25 \approx 227.00 \ \text{m}^2 \)
\( A_{\text{inner}} = \pi r^2 = \pi (7)^2 = \pi \times 49 \approx 153.94 \ \text{m}^2 \)
Area of the path = outer area − inner area:
\( A_{\text{path}} = 227.00 – 153.94 = 73.06 \ \text{m}^2 \)
Circumference of the outer edge for fencing:
\( C = 2\pi R = 2\pi \times 8.5 \approx 2 \times 3.1416 \times 8.5 \approx 53.41 \ \text{m} \)
Arc Length and Sector Area
Arc Length and Sector Area
Arc length is the distance along the curved line of a circle (or any curve). In the context of a circle, it’s a portion of the circle’s circumference between two points on the curve.
- Arc Length:
\( \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \)
A sector is a portion of a circle bounded by two radii and the arc between them. A full circle has an angle of 360°.
- Sector Area:
\( \text{Sector Area} = \frac{\theta}{360} \times \pi r^2 \) - \( \theta \) is the angle of the sector in degrees.
Important Notes:
- If given the diameter, halve it to find radius.
- Make sure the angle is in degrees, and use accurate π values if allowed.
- The arc is part of the circle’s circumference, the sector is part of its area.
Example:
Find the length of an arc of a circle with radius 6 cm and angle 90°.
▶️ Answer/Explanation
Using \( \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \):
\( \frac{90}{360} \times 2\pi \times 6 = \frac{1}{4} \times 12\pi = 3\pi \approx 9.42 \ \text{cm} \)
Answer: 9.42 cm
Example:
The angle of a sector is 135°, and the area of the sector is 81.4 cm². Find the arc length, giving your answer to 2 decimal places.
▶️ Answer/Explanation
We are given sector area and angle, and we must find arc length. First, find the radius using:
\( \text{Sector Area} = \frac{\theta}{360} \times \pi r^2 \)
\( 81.4 = \frac{135}{360} \times \pi r^2 \Rightarrow 81.4 = \frac{3}{8} \pi r^2 \)
\( r^2 = \frac{81.4 \times 8}{3\pi} = \frac{651.2}{3\pi} \approx \frac{651.2}{9.4248} \approx 69.09 \)
\( r \approx \sqrt{69.09} \approx 8.31 \ \text{cm} \)
Now use arc length formula: \( \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \)
\( = \frac{135}{360} \times 2\pi \times 8.31 = \frac{3}{8} \times 2\pi \times 8.31 \)
\( = \frac{3}{4} \times \pi \times 8.31 \approx 0.75 \times 3.1416 \times 8.31 \approx 19.58 \ \text{cm} \)
Answer: 19.58 cm
Example:
In a circle with a radius of 14 cm, a sector has an arc length of 16 cm. Find the area of the sector.
▶️ Answer/Explanation
First, use arc length to find the angle:
\( \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \)
\( 16 = \frac{\theta}{360} \times 2\pi \times 14 \Rightarrow 16 = \frac{\theta}{360} \times 28\pi \)
\( \frac{16}{28\pi} = \frac{\theta}{360} \Rightarrow \theta = \frac{16 \times 360}{28\pi} \approx \frac{5760}{87.9646} \approx 65.47^\circ \)
Now use sector area formula:
\( \text{Area} = \frac{\theta}{360} \times \pi r^2 = \frac{65.47}{360} \times \pi \times 14^2 \)
\( \approx 0.18186 \times \pi \times 196 \approx 0.18186 \times 615.752 \approx 112.01 \ \text{cm}^2 \)
Answer: 112.01 cm²