In the study of geometry, understanding the vocabulary related to circles is fundamental. This section delves into the essential terms such as centre, radius, diameter, and more, which are pivotal in solving geometric problems. Through examples and focused explanations, we aim to demystify these concepts for IGCSE students, facilitating a deeper comprehension and application in various mathematical contexts.

Centre

The centre of a circle is its geometric middle point. Every point on the circle's boundary is equidistant from the centre. This fundamental concept is the starting point for understanding other terms related to circles.

Radius (Radii)

The radius (plural: radii) is a line segment from the centre of the circle to any point on its boundary. It's a crucial measure that determines the size of the circle.

Example:

If a circle has a radius of 3 cm, its diameter would be twice as much, which is 6 cm.

Diameter

The diameter is a line segment that passes through the centre of the circle, connecting two points on its boundary. It is exactly twice the length of the radius.

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Example:

Given a circle with a radius of 5 cm, the diameter would be $2 \times 5 = 10$ cm.

Circumference

The circumference is the perimeter or the total distance around the circle. It is calculated using the formula $C = 2\pi r$ or $C = \pi d$.

Example:

Calculate the circumference of a circle with a diameter of 4 cm.

Solution:

$C = \pi d = \pi \times 4 \approx 12.57$ cm

Semicircle

A semicircle is half of a circle, formed by dividing the circle along its diameter. It has an arc that measures 180 degrees.

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Chord

A chord is a line segment whose endpoints lie on the circle's circumference. Unlike diameters, chords do not necessarily pass through the centre of the circle.

Tangent

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is the only common point between the circle and the tangent.

Example:

If a tangent at point A on the circle is perpendicular to the radius, then the angle formed between the radius and the tangent is 90 degrees.

Major/Minor Arc

An arc refers to a portion of the circle's circumference. A major arc is more than half of the circle, while a minor arc is less than half.

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Sector

A sector is a 'pie-shaped' part of a circle, enclosed by two radii and an arc. It resembles a slice of pie or pizza.

Example:

The area of a sector with a central angle of 60 degrees in a circle with a radius of 4 cm is calculated using the formula $Area = \frac{\theta}{360} \times \pi r^2$, where $\theta$ is the central angle in degrees.

Solution:

$A = \frac{60}{360} \times \pi \times (4)^2$

$A = \frac{1}{6} \times \pi \times 16$

$A = \frac{16}{6} \times \pi$

$A = 8.3776 \, \text{cm}^2$

Segment

A segment is the area enclosed by a chord and the arc lying between the chord's endpoints. Like sectors, segments can be major or minor, depending on the size of the enclosed arc.

Example:

Finding the area of a segment involves subtracting the area of the triangular part (formed by the chord and the two radii) from the area of the corresponding sector.

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Practice Questions

Question 1: Circumference Calculation

Given a circle with a radius of 7 cm, calculate its circumference.

Solution:

$C = 2\pi r$

$C = 2 \times \pi \times 7$

$C \approx 43.98 \text{ cm }$

Question 2: Area of a Sector

A circle has a radius of 10 cm. Calculate the area of a sector formed by a central angle of 30 degrees.

Solution:

$Area = \frac{\theta}{360} \times \pi \times r^2$, where $\theta$ is in degrees.

To use radians, which is needed for a more precise calculation, $\theta = \frac{30 \times \pi}{180}$.

$Area = \frac{30 \times \pi}{360} \times \pi \times 10^2$

$Area \approx 26.18 \text{ cm}^2$

Question 3: Length of a Tangent

From a point 8 cm away from the centre of a circle, a tangent is drawn to the circle of radius 6 cm. Calculate the length of the tangent from the point outside the circle to the point of tangency.

Solution:

Using Pythagoras' theorem, where $d$ is the distance from the point to the centre and $r$ is the radius of the circle:

$\text{Tangent}^2 = d^2 - r^2$

$\text{Tangent}^2 = 8^2 - 6^2$

$\text{Tangent} = \sqrt{64 - 36}$

$\text{Tangent} \approx 5.29 \text{ cm }$