Circle equations, angles, arcs and radians
Connect circle geometry and coordinates, including tangents, radian measure and unit-circle values.
What you will tackle
- Use circumference, area, arcs, sectors and circle-angle relationships
- Use tangent–radius perpendicularity and central/inscribed angle relationships
- Convert degrees and radians and interpret unit-circle sine/cosine
- Construct, transform and complete-square circle equations
- Use distance to test point membership and recover circle dimensions
An arc has angular and length descriptions
A full turn is 360° = 2π radians. For radius r and angle θ in radians, arc length is rθ and sector area is r²θ/2. An inscribed angle subtending an arc is half the central angle subtending the same arc. A tangent is perpendicular to the radius at the point of contact.
Coordinates encode the same geometry
The circle (x − h)² + (y − k)² = r² has centre (h,k). The left side is squared distance from the centre, so equality tests membership. Completing squares converts a general equation to this form. On the unit circle, the point at angle θ has coordinates (cos θ,sin θ); signs depend on the quadrant, and tan θ = sin θ/cos θ when cos θ ≠ 0.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
For radius 6 and central angle 120°, find the radian angle, arc length and sector area. Give the corresponding inscribed angle and the tangent–radius angle.
Show a starting hint
Convert to radians before using rθ.
Compare with the explained answer
- 120° = 2π/3 radians.
- Arc length = 6·2π/3 = 4π; sector area = 36·(2π/3)/2 = 12π.
- The inscribed angle is 60° for the same arc; tangent and radius meet at 90°.
Answer / response criteria: 2π/3; 4π; 12π; 60°; 90°.
Guided 2
Use the unit circle to find sine and cosine at 120°, and convert it to radians. Find tan 120° using the ratio definition.
Compare with the explained answer
- 120° = 2π/3 and has reference angle 60° in quadrant II.
- The unit-circle coordinates are (−1/2,√3/2).
- Tangent is (√3/2)/(−1/2) = −√3.
Answer / response criteria: cos120° = −1/2; sin120° = √3/2; tan120° = −√3; 2π/3 radians.
Independent practice
Independent 1
Find the centre and radius of x² + y² − 4x + 6y − 12 = 0, and test (7,−3). Give (cos θ,sin θ) at θ = 150°.
Compare with the explained answer
- Complete squares: (x − 2)² + (y + 3)² = 25.
- The centre is (2,−3), radius 5; (7,−3) has squared distance 25 and lies on it.
- 150° is in quadrant II with reference angle 30°, giving (−√3/2,1/2).
Answer / response criteria: Centre (2,−3), radius 5; point on circle; (−√3/2,1/2).
Independent 2
Find (cos θ,sin θ) at 225° and convert to radians. Explain the signs and find tan θ.
Compare with the explained answer
- 225° = 5π/4, in quadrant III with reference angle 45°.
- Both coordinates are negative: (−√2/2,−√2/2).
- Their ratio is positive, so tangent equals 1.
Answer / response criteria: (−√2/2,−√2/2); 5π/4; tan225° = 1.
Application practice
Application 1
A circular path has centre (−1,4) and passes through (2,8). Write its equation, then translate the whole path 3 units right and 2 down. Find a quarter-turn arc length.
Compare with the explained answer
- The radius is √(3² + 4²) = 5.
- The original equation is (x + 1)² + (y − 4)² = 25; the new centre is (2,2), giving (x − 2)² + (y − 2)² = 25.
- A quarter turn is π/2 radians, so its arc length is 5π/2.
Answer / response criteria: Original (x + 1)² + (y − 4)² = 25; translated (x − 2)² + (y − 2)² = 25; arc 5π/2.
Application 2
A point on a wheel of radius 4 has position (4cos θ,4sin θ) relative to its centre. Starting on the positive x-axis, it turns 150° counterclockwise. Find its coordinates and distance travelled along the circular path.
Compare with the explained answer
- At 150°, cosine is −√3/2 and sine is 1/2.
- The coordinates are (−2√3,2).
- The angle is 5π/6 radians, so arc length is 4·5π/6 = 10π/3.
Answer / response criteria: Coordinates (−2√3,2); arc distance 10π/3.
What can you explain now?
A point on a wheel of radius 4 has position (4cos θ,4sin θ) relative to its centre. Starting on the positive x-axis, it turns 150° counterclockwise. Find its coordinates and distance travelled along the circular path.
Compare with the explanation
Coordinates (−2√3,2); arc distance 10π/3.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide