Circles, arcs and sectors

Table of Contents
1. CIE IGCSE Maths: Extended
2. Area & Circumference of Circles
3. Area & Circumference
4. How to Work with Circles
5. d = 2r
6. Working with Circle Formulae
7. Understanding Formulas for Area and Circumference
8. Semicircle Geometry Concepts
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CIE IGCSE Maths: Extended

Revision Notes

Syllabus Edition

First teaching 2023

First exams 2025

Area & Circumference of Circles

Area & Circumference

• Circles are unique shapes composed of points equidistant from a central point. Equidistant implies equal distances between all points.
equidistant
• Equidistant signifies uniform distances.
• The circumference of a circle represents its perimeter.
circumference
• The mathematical constant π (pi) (approximately 3.14159) correlates a circle's diameter with its circumference.
π (pi) diameter
• In calculations, you might need to provide the area rounded to a specific number of decimal places or significant figures. Alternatively, you could be requested to provide the precise value or "express your answer in terms of π," which could be relevant for non-calculator examinations.

• Alternatively you may be asked to give the exact value - or "give your answer in terms of π" - so this topic could crop up on the non-calculator paper!

How to Work with Circles

• You must understand the formulae for the area and circumference of a circle.

Formulae for Circles

• There are two versions for the circumference. It's crucial not to confuse the radius and diameter.
• Remember that d = 2r. You can remember the formulas by using different letters.

d = 2r

Working with Circle Formulae

• Working with circle formulae is similar to working with any other formula:

Steps:

• Write down what you know (what you want to know).
• Pick the correct formula.
• Substitute and solve.

Write Down

Formula

Substitute

Understanding Formulas for Area and Circumference

• If you're feeling pressured and can't recall which formula is for what, remember that area is always measured in square units (cm², m², etc.). Therefore, the formula containing r² pertains to area.
• The circumference, on the other hand, is simply a length. Hence, its units will be the same as those for length (cm, m, etc).

Calculating Area and Perimeter of a Semicircle

Find the area and perimeter of the semicircle illustrated in the diagram below:

Given answers should be in terms of π.

The area of a semicircle amounts to half the area of a full circle sharing the same diameter. Therefore, start by determining the area of the complete circle.

Discover the radius by dividing the diameter by 2.

Substitute this value into the circle area formula. Present your answer in terms of π (do not multiply by π).

Semicircle Geometry Concepts

• Area of a Semicircle:

  Calculate the area by dividing the full area of a circle by 2.

  Example: If the area is 32π cm², the area of the semicircle is 32π cm².

• Perimeter of a Semicircle:

  The perimeter consists of the arc of the circle (half of the circumference) and the diameter of the semicircle.

  Find the circumference by substituting the radius value into the formula.

  Example: If the radius is 8 cm, calculate the circumference accordingly.

• Length of the Arc:

  Determine the length of the arc by dividing the full area by 2.

  Example: Find the curved part of the perimeter by dividing the total area by 2.

Visual Representations

Area & Circumference of Circles

• Perimeter Calculation:Calculate the full perimeter by adding it to the diameter's length of the circle.
• Perimeter Formula:The perimeter is given by the formula: Perimeter = 8π + 16 cm.
• Testing Your Knowledge:Challenge yourself with exercises to reinforce your understanding.
• Exploring Further:Delve into the next topic for a deeper understanding of circle concepts.
• Additional Resources:Download supplementary notes on the Area & Circumference of Circles for more insights.