Welcome to Circles and Circle Theorems!
Circles are everywhere around us — from bicycle wheels and clock faces to Ferris wheels and planetary orbits. In this chapter, we are going to explore the geometry of circles, learn how to calculate arc lengths and sector areas, and uncover the secret rules (known as circle theorems) that govern the angles inside them.
Don't worry if geometry feels challenging at first! Circle theorems are just like visual puzzles. Once you learn the distinct patterns to look for, you will be able to spot the clues and crack any exam question with confidence.
1. Circle Anatomy: The Essential Vocabulary
Before jumping into the theorems, let's review the basic parts of a circle. Knowing these names is vital because exam questions use them to describe shapes and lines.
• Centre: The fixed point right in the middle, usually labelled \(O\).
• Radius (\(r\)): A straight line from the centre to any point on the outside edge. (Plural: radii. All radii in the same circle are equal in length!)
• Diameter (\(d\)): A straight line going from one side of the circle to the other, passing straight through the centre. Notice that \(d = 2r\).
• Circumference (\(C\)): The perimeter or the total distance all the way around the outside edge.
• Chord: A straight line connecting any two points on the circumference without necessarily passing through the centre.
• Tangent: A straight line outside the circle that touches the circumference at exactly one point.
• Arc: A curved section of the circumference. A small piece is a minor arc; the larger remaining piece is a major arc.
• Sector: A region bounded by two radii and an arc. Think of it like a slice of pizza!
• Segment: A region bounded by a chord and an arc. Think of it like slicing off a piece of crust.
Quick Memory Tip: A sector is sliced like a pizza (from the centre by radii), while a segment is cut off by a straight blade (a chord).
2. Circle Measurements: Arcs, Sectors, and Perimeters
You already know the basic formulas for a full circle:
• Circumference: \(C = 2\pi r = \pi d\)
• Area: \(A = \pi r^2\)
Working with Fractions of a Circle
When dealing with a sector of a circle with an angle of \(\theta\) (theta) at the centre:
• Fraction of circle: \(\frac{\theta}{360^\circ}\)
• Arc Length: \(\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r\)
• Sector Area: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\)
• Perimeter of a Sector: \(\text{Perimeter} = \text{Arc Length} + 2r\)
Common Mistake to Avoid: When asked for the perimeter of a sector, students often calculate just the curved arc length and forget to add the two straight radii to close the shape!
Step-by-Step Example
Question: A sector has a radius of \(6\text{ cm}\) and a central angle of \(60^\circ\). Find (a) the arc length, (b) the total perimeter, and (c) the area. Give answers to \(1\) decimal place.
Step 1: Find the fraction of the circle: \(\frac{60^\circ}{360^\circ} = \frac{1}{6}\).
Step 2: Arc length \(= \frac{1}{6} \times 2 \times \pi \times 6 = 2\pi \approx 6.3\text{ cm}\).
Step 3: Perimeter \(= 6.28... + 6 + 6 = 18.3\text{ cm}\).
Step 4: Sector Area \(= \frac{1}{6} \times \pi \times 6^2 = 6\pi \approx 18.8\text{ cm}^2\).
3. The Circle Theorems
Circle theorems are established geometric facts. In CCEA GCSE exams, you must state the exact mathematical reason alongside your numerical answer to secure full method marks.
Theorem 1: Angle in a Semicircle
The Rule: The angle subtended by a diameter at the circumference is always a right angle (\(90^\circ\)).
Visual Clue: Look for a triangle where one side is the circle's diameter.
Exam Reason to Write: "Angle in a semicircle is a right angle (\(90^\circ\))."
Example: Triangle \(ABC\) is drawn inside a circle where \(AB\) is the diameter. If angle \(CAB = 35^\circ\), what is angle \(ABC\)?
Since \(AB\) is the diameter, angle \(ACB = 90^\circ\).
Therefore, angle \(ABC = 180^\circ - 90^\circ - 35^\circ = 55^\circ\).
Theorem 2: Angle at the Centre
The Rule: The angle subtended by an arc at the centre is twice the angle subtended by it at the circumference.
Formula: \(\text{Angle at centre} = 2 \times \text{Angle at circumference}\)
Visual Clue: Look for an "arrowhead" shape pointing to the circumference, with its tail at the centre.
Exam Reason to Write: "Angle at the centre is twice the angle at the circumference."
Example: If the angle at the circumference is \(42^\circ\), the angle at the centre is \(2 \times 42^\circ = 84^\circ\).
Theorem 3: Angles in the Same Segment
The Rule: Angles at the circumference subtended by the same arc (or chord) are equal.
Visual Clue: Look for a "bow-tie" or "butterfly" shape inside the circle where both angles touch the circumference from the same baseline.
Exam Reason to Write: "Angles in the same segment are equal."
Theorem 4: Opposite Angles in a Cyclic Quadrilateral
The Rule: The opposite angles of a cyclic quadrilateral (a 4-sided polygon where all 4 vertices touch the circle) add up to \(180^\circ\).
Formula: If the opposite angles are \(A\) and \(C\), then \(A + C = 180^\circ\).
Visual Clue: A four-cornered shape where every single corner touches the circumference edge.
Exam Reason to Write: "Opposite angles in a cyclic quadrilateral add up to \(180^\circ\)."
Watch Out: If one corner sits on the centre point \(O\) instead of the outer edge, it is not a cyclic quadrilateral!
Theorem 5: Tangent Meets Radius
The Rule: The angle between a tangent and a radius drawn to the point of contact is \(90^\circ\).
Visual Clue: A tangent line skimming the circle with a radius line meeting it.
Exam Reason to Write: "The angle between a tangent and a radius is \(90^\circ\)."
Theorem 6: Tangents from an External Point
The Rule: Tangents to a circle from the same external point are equal in length.
Visual Clue: An "ice-cream cone" shape with two straight tangent lines meeting at an outside point \(P\).
Key Properties: This creates two identical (congruent) right-angled triangles when connected to the centre, forming a kite.
Exam Reason to Write: "Tangents from an external point are equal in length."
Theorem 7: Alternate Segment Theorem
The Rule: The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment.
Visual Clue: Look for a triangle touching a tangent line at one of its vertices.
Exam Reason to Write: "Alternate segment theorem."
How to spot it: Put your finger on the angle between the tangent and the triangle's side (the chord). The equal angle is the corner of the triangle directly opposite that chord!
Theorem 8: Perpendicular from the Centre to a Chord
The Rule: A line drawn from the centre of a circle that is perpendicular to a chord bisects (cuts in half) the chord.
Exam Reason to Write: "Perpendicular from the centre bisects the chord."
Top Tip: This often creates a right-angled triangle, allowing you to use Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) where the hypotenuse is the radius \(r\).
4. The Secret Weapon: Hidden Isosceles Triangles
Many circle theorem problems require one extra step that isn't a named theorem: spotting isosceles triangles formed by radii.
• Whenever two sides of a triangle are radii (lines from the centre \(O\) to the circumference), those two sides are equal in length.
• Because two sides are equal, the base angles must also be equal!
Exam Reason to Write: "Radii are equal, forming an isosceles triangle."
5. Step-by-Step Strategy for Angle Problems
When faced with a complex circle diagram in your exam, follow these four steps:
1. Highlight the centre: Look for centre \(O\). Are there radii? Mark them as equal. Look for diameters and isosceles triangles.
2. Look for tangents: If you see a tangent line, check for a \(90^\circ\) angle with a radius or the alternate segment theorem.
3. Count the vertices: If you see a 4-sided shape, check if all 4 corners touch the circumference (cyclic quadrilateral).
4. Write your reasons: For every angle you calculate, state its size and write down the theorem used.
Quick Review Summary
• Semicircle angle \(= 90^\circ\)
• Centre angle \(= 2 \times\) Circumference angle
• Angles in same segment are equal (Bow-tie)
• Opposite angles in cyclic quadrilateral \(= 180^\circ\)
• Radius meets Tangent at \(90^\circ\)
• Two tangents from a single point are equal in length
• Tangent-chord angle \(=\) Angle in alternate segment
• Radius to chord at \(90^\circ\) bisects chord