A circle is a closed curved shape in which every point on the boundary is the same distance from one fixed point. That fixed point is called the centre. Circles appear in wheels, clocks, coins, plates and many other everyday objects.
Circle, circumference and circular region
The circle is the boundary itself. The distance around that boundary is called the circumference. The region inside the circle is sometimes called the circular region or disc. In everyday school mathematics, the word circle is often used for both the boundary and the inside region, so the meaning depends on the question.
Important parts of a circle
Centre
The centre is the fixed point in the middle of the circle. It is equally distant from every point on the circle.
Radius
A radius is a line segment from the centre to any point on the circle. All radii of the same circle have equal length. The plural of radius is radii.
Diameter
A diameter is a chord that passes through the centre. It joins two points on the circle. The diameter is twice the radius: d = 2r. Therefore, r = d/2.
Chord
A chord is a line segment joining any two points on the circle. The diameter is the longest chord.
Arc
An arc is a part of the circumference between two points. A minor arc is the shorter path, while a major arc is the longer path.
Sector
A sector is the region enclosed by two radii and the arc between them. It looks like a slice of pizza.
Segment
A segment is the region enclosed by a chord and its corresponding arc.
Semicircle and quadrant
A diameter divides a circle into two equal semicircles. Two perpendicular diameters divide it into four equal quadrants.
Tangent
A tangent is a line that touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent.
Secant
A secant is a line that passes through a circle and meets it at two points.
What is pi?
Pi, written as π, is the constant ratio of a circle’s circumference to its diameter. Its approximate value is 3.14159. For simple calculations, 3.14 or 22/7 is often used, but these are approximations.
Circumference of a circle
If the radius is r, circumference C = 2πr. If the diameter is d, circumference C = πd. Circumference is a length, so its answer uses ordinary units such as cm or m.
Example
A circle has radius 7 cm. Using π = 22/7, C = 2 × 22/7 × 7 = 44 cm.
Area of a circle
The area enclosed by a circle is A = πr². Area is measured in square units such as cm² or m².
Example
A circle has radius 5 cm. Using π ≈ 3.14, A = 3.14 × 5² = 3.14 × 25 = 78.5 cm².
Finding radius or diameter
When the diameter is given, divide it by 2 to get the radius. When the circumference is given, use r = C/(2π). When the area is given, first calculate A/π and then take its square root.
Example
A circle has diameter 18 cm. Its radius is 18 ÷ 2 = 9 cm. Its circumference is 18π cm, and its area is 81π cm².
Arc length and sector area
A full circle contains 360°. If a central angle is θ degrees, arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr².
Example
A sector has radius 6 cm and central angle 90°. It is one-quarter of a circle. Its arc length is 1/4 × 2π × 6 = 3π cm, and its area is 1/4 × π × 6² = 9π cm².
Useful circle facts
Equal radii form equal distances from the centre. The diameter is the longest chord. A perpendicular line from the centre to a chord bisects that chord. The angle around the centre is 360°. A radius and tangent meet at 90° at the point of contact.
Worked examples
Example 1: Find the diameter
The radius is 12 m. Diameter = 2 × 12 = 24 m.
Example 2: Find the circumference
The diameter is 10 cm. Circumference = π × 10 = 10π cm, which is approximately 31.4 cm.
Example 3: Find the radius from circumference
The circumference is 31.4 cm. Using π = 3.14, r = 31.4 ÷ (2 × 3.14) = 5 cm.
Common mistakes
Do not confuse radius with diameter. Use r² when calculating area, but do not square the radius when calculating circumference. Circumference uses length units; area uses square units. Keep π in the answer when an exact value is requested.
Practice questions
1. Find the diameter when r = 8 cm. 2. Find the circumference when d = 14 cm using π = 22/7. 3. Find the area when r = 4 cm using π = 3.14. 4. Name the longest chord. 5. What angle is formed between a radius and a tangent at the point of contact?
Answers
1. 16 cm. 2. 44 cm. 3. 50.24 cm². 4. The diameter. 5. 90°.
Continue learning
Continue with Mathematical Shapes, Geometry, Triangles and Length Measurement.