Geometry Unit 9: Circles — Free Review Games.
This unit covers central and inscribed angles, arc length, tangent lines and secants and chords — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Pick a mode. Play.
Answer questions as fast as you can. 2 minutes on the clock. Build streaks for bonus points!
Don't want to play?
Review the questions traditionally. Click to expand.
Questions loading...
Focus on understanding.
Focus on understanding core concepts before memorizing details. Use the game modes to test yourself repeatedly — spaced repetition is proven to boost long-term retention.
This unit covers central and inscribed angles, arc length, tangent lines and secants and chords — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Central And Inscribed Angles
A central angle equals the arc it intercepts. An inscribed angle equals half the intercepted arc. Inscribed angles that intercept the same arc are congruent.
Key Points
- Central angle = intercepted arc (1:1 ratio)
- Inscribed angle = ½ × intercepted arc
- An inscribed angle that intercepts a semicircle is always 90°
- Two inscribed angles intercepting the same arc are equal
An inscribed angle intercepts an arc of 84°. Find the inscribed angle measure.
Apply the inscribed angle theorem: inscribed angle = ½ × intercepted arc. So the angle = ½ × 84° = 42°. This is the most common exam setup — given the arc, halve it to find the inscribed angle.
2 Arc Length
Arc length is a portion of the circle's circumference, determined by the central angle. The formula is Arc Length = (central angle / 360°) × 2πr. You must know both the radius and the central angle.
Key Points
- Arc Length = (θ/360) × 2πr, where θ is the central angle in degrees
- Arc length is a distance (units: cm, in, etc.), not a degree measure
- A larger central angle produces a longer arc on the same circle
- Do not confuse arc length with arc measure — arc measure is in degrees, arc length is in linear units
A circle has radius 9 cm. Find the arc length intercepted by a central angle of 80°.
Plug into the formula: Arc Length = (80/360) × 2π(9) = (2/9) × 18π = 4π ≈ 12.57 cm. Simplify the fraction first to avoid arithmetic errors. Exams may leave the answer in terms of π.
3 Tangent Lines
A tangent line touches a circle at exactly one point and is always perpendicular to the radius drawn to that point. Two tangent segments drawn from the same external point are congruent.
Key Points
- Tangent ⊥ radius at the point of tangency — this creates a 90° angle
- Two tangents from an external point are equal in length
- Tangent-chord angle = ½ × intercepted arc
- In right triangle problems, use the Pythagorean theorem with the radius and tangent segment
From external point P, a tangent segment to circle O has length 12. The radius is 5. Find the distance from P to the center O.
The radius to the point of tangency is perpendicular to the tangent, forming a right angle. Apply the Pythagorean theorem: PO² = 12² + 5² = 144 + 25 = 169, so PO = 13. This is a classic 5-12-13 right triangle setup common on exams.
4 Secants And Chords
When two chords intersect inside a circle, the products of their segments are equal. When two secants are drawn from an external point, there is a specific angle and segment relationship to apply. Angle measures depend on whether the intersection is inside, on, or outside the circle.
Key Points
- Two chords intersecting inside: (segment 1a)(segment 1b) = (segment 2a)(segment 2b)
- Angle formed by two chords inside = ½(sum of intercepted arcs)
- Angle formed by two secants from outside = ½(difference of intercepted arcs)
- Two secants from external point: (whole segment 1)(external part 1) = (whole segment 2)(external part 2)
Two chords AB and CD intersect inside a circle at point E. AE = 6, EB = 4, CE = 3. Find ED.
Use the intersecting chords theorem: AE × EB = CE × ED. Substitute: 6 × 4 = 3 × ED, so 24 = 3 × ED, giving ED = 8. Always identify the two pairs of segments before multiplying — pairing them incorrectly is the most common exam mistake.
Questions, answered.
What is Circles?
Circles is Unit 9 of Geometry, covering central and inscribed angles, arc length, tangent lines and secants and chords.
How to study for Geometry Unit 9?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.