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.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide