Why circles pop up in everyday life
Ever wondered why wheels spin so smoothly? That's the magic of circles! From bike tyres to pizza slices, circles are everywhere, and the exam loves them too.
💡 In Simple Words: A circle is a line of points that are all the same distance from a centre point. Its parts—radius, diameter, chord, tangent—follow easy rules that you can picture with a bike wheel.
Key terms you need to know
- Radius: the distance from the centre to any point on the circle. Think of a spoke on a wheel.
- Diameter: a line that passes through the centre and touches the circle at both ends. It’s just two radii stuck together.
- Chord: a straight line joining any two points on the circle. Imagine a shortcut across a park.
- Tangent: a line that touches the circle at exactly one point, like a pencil just grazing a ball.
- Secant: a line that cuts the circle at two points.
- Arc: a part of the circle’s edge between two points.
- Sector: the “pizza slice” shape bounded by two radii and the arc between them.
- Segment: the region between a chord and the arc it cuts off.
Radius and diameter
The radius (r) is the basic building block. If you double it, you get the diameter (d). In formula form, d = 2r. So whenever you know one, you instantly know the other.
Chord, diameter and the perpendicular bisector
Draw any chord. If you drop a line from the centre to the middle of that chord, two things happen:
- The line meets the chord at a right angle (90°).
- That line also splits the chord into two equal halves.
In other words, the radius that meets a chord at its midpoint is always perpendicular to the chord. This fact helps you find missing lengths fast.
Tangent and its special property
A tangent kisses the circle at just one point—call it the point of contact. The magic rule: the radius drawn to that point is always perpendicular to the tangent. It’s like a flagpole (the radius) standing straight up where the flag (the tangent) flutters away.
Arc, sector and segment
An arc is simply a slice of the circle’s edge. When you connect the two ends of an arc with two radii, you get a sector—think of a slice of pizza. If you cut off a piece of the circle with a chord instead of two radii, the shape you keep is a segment.
Important formulas you’ll use
- Circumference (the distance around) = 2πr = πd. π (pi) is about 3.14.
- Area (the space inside) = πr².
- Length of a chord (c) when you know the distance (d) from the centre to the chord: c = 2√(r² – d²).
Worked example
Problem: In a circle of radius 10 cm, the distance from the centre to a chord is 6 cm. Find the length of the chord.
Solution: Use the chord formula c = 2√(r² – d²). Plug in r = 10 and d = 6.
First, square the radius: 10² = 100.
Next, square the distance: 6² = 36.
Subtract: 100 – 36 = 64.
Take the square root: √64 = 8.
Finally, double it: 2 × 8 = 16 cm. So the chord is 16 cm long.
Notice how the perpendicular from the centre to the chord made a right‑angled triangle. That’s the same idea we used in the formula.
Quick reference table
| Term | Definition | Key property |
|---|---|---|
| Radius (r) | Distance from centre to any point on the circle | Half of the diameter |
| Diameter (d) | Line through centre touching circle at two points | d = 2r |
| Chord | Line joining two points on the circle | Perpendicular from centre bisects it |
| Tangent | Line touching circle at exactly one point | Radius at point of contact ⟂ tangent |
| Arc | Part of the circumference between two points | Measured in degrees or length |
| Sector | Region bounded by two radii and the intervening arc | Area = (θ/360)·πr² where θ is the central angle |
| Segment | Region between a chord and its arc | Area = sector area – triangle area |
📝 Likely Exam Questions
- Find the diameter of a circle whose circumference is 31.4 cm.
Solution: C = πd ⇒ d = C/π = 31.4/3.14 = 10 cm. - In a circle of radius 7 cm, a tangent meets a radius at a right angle. What is the length of the tangent segment from the point of contact to a point 5 cm away from the centre along the tangent?
Solution: Use Pythagoras in the right‑triangle formed: (tangent)² + r² = (5)² ⇒ t² = 25 – 49 = negative, so the point cannot be inside the circle. Hence the question expects the tangent length from the point of contact to the external point, which is √(5² – 7²) = √(25 – 49) – not possible. The correct interpretation is that the external point must be farther than the radius; if the distance were 10 cm, then t = √(10² – 7²) = √(100 – 49) = √51 ≈ 7.14 cm. - Calculate the length of a chord that is 8 cm from the centre of a circle with radius 13 cm.
Solution: c = 2√(13² – 8²) = 2√(169 – 64) = 2√105 ≈ 2×10.25 = 20.5 cm. - What is the area of a sector with a central angle of 60° in a circle of radius 9 cm?
Solution: Sector area = (θ/360)·πr² = (60/360)·π·9² = (1/6)·π·81 ≈ 13.5π ≈ 42.4 cm². - Explain why a radius drawn to the point of contact of a tangent is perpendicular to the tangent.
Solution: By definition, a tangent touches the circle at exactly one point. If the radius were not perpendicular, the line would intersect the circle at two points, contradicting the definition of a tangent.