Grade 10 · Grade 10: Circle Geometry and Trigonometry · Circle Geometry, Trigonometry and Solids
Circle Theorems, Arcs and Sectors
Prove circle relationships and measure arc length, sector area and circle equations.
Learning objectives
- Prove and apply theorems about inscribed and central angles, chords, radii and tangents.
- Derive and use arc length and sector area formulas.
- Use the equation of a circle and find intersections with lines.
AERO Mathematics alignment
AERO.M10.GM.1
Prove and apply theorems about circles, including relationships among inscribed angles, central angles, radii, chords and tangents.
AERO.M10.GM.2
Derive and use formulas for arc length and sector area, including the use of radian-style proportional reasoning.
AERO.M10.GM.4
Derive and use the equation of a circle, and solve problems involving intersections of lines and circles algebraically.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Two students measure an angle at the centre of a circle and an angle at the circumference standing on the same arc.
Think about it
What relationship do you expect between the two angles?
Hint: Measure a few examples before you generalise.
EXPLAIN
Circle theorems and measurement
| Theorem | Statement |
|---|---|
| Angle at the centre | The central angle is twice the inscribed angle on the same arc |
| Angles in the same segment | Inscribed angles on the same arc are equal |
| Angle in a semicircle | An angle inscribed in a semicircle is 90° |
| Cyclic quadrilateral | Opposite angles add to 180° |
| Tangent and radius | A tangent meets the radius at 90° at the point of contact |
| Equal tangents | Two tangents from an external point are equal in length |
Arcs and sectors
A central angle of θ degrees cuts off a fraction θ/360 of the circle. So arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². A 60° sector of radius 9 cm has arc length 3π ≈ 9.42 cm.
The equation of a circle
A circle of radius r centred at (a, b) has equation (x − a)² + (y − b)² = r², which is simply the distance formula rearranged. Substituting a line equation into it gives a quadratic whose discriminant tells you whether the line cuts, touches or misses the circle.
Where this is used
GPS trilateration, collision detection in games and gear design all depend on circle equations and tangent properties.
INVESTIGATION
Try it yourself
Draw a circle and test the angle-at-the-centre theorem with three different inscribed angles on the same arc. Record your measurements, then write a short proof using the isosceles triangles formed by the radii.
Watch
Circle Theorems: Angle at the Centre
Circle geometry explainer · 10:00
Walks through the central-angle theorem with a proof and exam-style questions.
Before you watch: How does the angle at the centre compare with the angle at the circumference?
- An inscribed angle is 40°. Find the central angle on the same arc.
- Find the arc length for a 60° sector of radius 9 cm.
Interactive simulation · PhET
Trig Tour
Track the angle around the unit circle and connect arc length to the central angle.
While you explore
- How is arc length related to the central angle?
- What fraction of the circle is a 45° sector?
Key vocabulary
- Sector
- A slice of a circle between two radii.
- Inscribed angle
- An angle with its point on the circle.
Practice questions
0/1 correct
Level 1 · Criterion A
An inscribed angle measures 35°. What is the central angle standing on the same arc?
MYP criterion tasks
Level 2 · Criterion C
Find the arc length and sector area for a 60° sector of radius 12 cm.
Reflect & track
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