Circles

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Circles — Quick Facts

Key Definitions:

• Radius (r): Distance from centre to any point on the circle
• Diameter (d): $d = 2r$ (passes through centre)
• Chord: Line segment joining two points on the circle
• Arc: Part of the circle’s circumference
• Sector: Region bounded by two radii and an arc
• Segment: Region bounded by a chord and an arc

Essential Formulas:

• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$
• Arc length: $s = r\theta$ (radians) or $s = \frac{\theta}{360} \times 2\pi r$ (degrees)
• Sector area: $A = \frac{1}{2}r^2\theta$ (radians) or $\frac{\theta}{360} \times \pi r^2$ (degrees)
• Segment area: $A_{\text{sector}} - A_{\text{triangle}}$

⚡ CAT Exam Tip: In most CAT circle problems, use $\pi = \frac{22}{7}$ unless told otherwise. Convert angles to radians when using $r\theta$ formulas.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Circles — Study Guide

Equation of a Circle:

Centre at origin: $x^2 + y^2 = r^2$

Centre at $(h, k)$: $(x - h)^2 + (y - k)^2 = r^2$

General form: $x^2 + y^2 + 2gx + 2fy + c = 0$ Centre = $(-g, -f)$, radius $r = \sqrt{g^2 + f^2 - c}$

Example: Find centre and radius of $x^2 + y^2 - 4x + 6y + 9 = 0$

$(x^2 - 4x) + (y^2 + 6y) = -9$ $(x - 2)^2 + (y + 3)^2 = -9 + 4 + 9 = 4$

Centre = $(2, -3)$, radius = $2$

Perpendicular from Centre to Chord:

If a chord is at distance $d$ from the centre and has length $c$: $$c = 2\sqrt{r^2 - d^2}$$

Example: Find chord length if $r = 5$ cm and chord is 4 cm from centre.

$c = 2\sqrt{25 - 16} = 2\sqrt{9} = 6$ cm

Inscribed Angles:

An angle subtended by an arc at the circumference equals half the central angle subtending the same arc.

$$\angle \text{inscribed} = \frac{1}{2} \times \angle \text{at centre}$$

Angles in the Same Segment: Equal

⚡ Common Student Mistake: Confusing when to use degrees vs radians. Arc length $s = r\theta$ uses radians. Sector area $A = \frac{1}{2}r^2\theta$ uses radians. Convert: $\pi$ radians $= 180°$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Circles — Comprehensive Notes

Intersecting Chords Theorem:

If two chords AB and CD intersect at P inside the circle: $$PA \times PB = PC \times PD$$

Example: Chords AB and CD intersect at P. If PA = 3 cm, PB = 8 cm, and PC = 4 cm, find PD.

$3 \times 8 = 4 \times PD \Rightarrow PD = 6$ cm

Tangent-Secant Theorem:

If a tangent touches at T and a secant passes through to intersect at P and Q: $$(PT)^2 = PA \times PQ$$

Cyclic Quadrilateral:

If a quadrilateral is inscribed in a circle (cyclic), opposite angles sum to $180°$: $$\angle A + \angle C = 180°$$ $$\angle B + \angle D = 180°$$

Exterior Angle of a Cyclic Quadrilateral: The exterior angle equals the interior opposite angle.

Area of Segment:

Area of minor segment = Area of sector − Area of triangle

Using central angle $\theta$ in radians: $$A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$$

Tangents from an External Point:

From an external point P, two tangents can be drawn to a circle. They are equal in length: $$PT_1 = PT_2$$

The line from P to the centre bisects the angle between the two tangents.

Common CAT Circle Problems:

Problem Type 1: Find area of region bounded by two concentric circles (annulus) $$A = \pi(R^2 - r^2)$$

Problem Type 2: Two circles touching externally. Distance between centres = $r_1 + r_2$

Problem Type 3: Two circles touching internally. Distance between centres = $|r_1 - r_2|$

Problem Type 4: Maximum number of intersection points between two circles = 2

Problem Type 5: Angle formed by two chords intersecting inside the circle: $$\angle = \frac{1}{2}(\text{sum of arcs intercepted})$$

JAMB Pattern Analysis (CAT 2015-2024):

• 2015: Equation of circle from centre and radius
• 2017: Arc length and sector area
• 2019: Tangent properties
• 2021: Intersecting chords
• 2023: Cyclic quadrilateral angle sums
• 2024: Area of segment using $\frac{1}{2}r^2(\theta - \sin\theta)$

⚡ Exam Strategy: Draw a clear diagram, label all known values. Use the right triangle formed by radius, perpendicular to chord, and half-chord. This is the key to most chord problems.