Basic Properties of Circles

1Summary 重點摘要

Learn central angle, inscribed angle, radius diameter basic properties.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know the parts of a circleA circle is the set of all points at the same distance from a fixed point called the centre; that distance is the radius. A diameter is a chord passing through the centre with both ends on the circle, and its length is twice the radius. You should also know the chord (a segment joining two points on the circle), the arc (a part of the circumference) and the tangent (a line touching the circle at exactly one point). Marking the centre O and a radius early makes every later theorem easier to apply and helps you avoid missing conditions.
2. Step 2: Circumference and area formulasThe circumference is C = 2πr and the area is A = πr², both expressed in the radius r. If the diameter d is given, divide it by 2 to get the radius, or use C = πd and A = πd²/4 directly. Two errors are common: substituting the diameter as if it were the radius, and forgetting to square r in the area formula. Keep π as a symbol or use the calculator's π value instead of approximating with 3.14 too early.
3. Step 3: Angle at centre is twice angle at circumferenceThe central angle theorem states that the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circumference. For example, if arc AB subtends 100° at the centre O, then the angle it subtends at any point on the circumference is 50°. Always confirm that both angles really stand on the same arc, which is the easiest place to go wrong; mark the arc and both angles on the diagram, check carefully, and only then write the statement 'angle at centre = twice angle at circumference'.
4. Step 4: Angle in a semicircleThe angle-in-a-semicircle result follows directly from the central angle theorem: a diameter subtends 180° at the centre, so the angle formed by the two ends of the diameter and any point on the circle, that is, the angle in the semicircle, must be half of 180°, namely 90°. Whenever a triangle is formed by a diameter and a point on the circle, treat it immediately as a right triangle; this property is often combined with Pythagoras' theorem to find the third side.
5. Step 5: Equal chords subtend equal arcsThe equal-chord theorem says that in the same circle, or in congruent circles, two chords are equal if and only if the arcs they subtend are equal, and the perpendicular distances from the centre to the chords are equal as well. Depending on what is given and what is required, you may quote 'equal chords subtend equal arcs' or the converse. Remember that arcs come in major and minor forms; the theorem normally compares the corresponding minor arcs, and in proofs you should first state that the chords lie in the same circle before quoting it.
6. Step 6: The perpendicular from the centre bisects a chordThe line drawn from the centre perpendicular to a chord always bisects the chord, and it also bisects the arc cut off by the chord. This property is used to find chord lengths or perpendicular distances: with radius r, distance d from the centre and half-chord a, Pythagoras' theorem gives r² = d² + a². The usual mistake is forgetting that the perpendicular bisects only half the chord and substituting the full chord length; divide the chord length by 2 first, then put it into the equation.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Diameter = 2 × radius
• Circumference = 2πr, area = πr²
• Central angle = 2 × inscribed angle same arc
• Inscribed angle subtended by diameter = 90°
• Center perpendicular to chord always bisects chord