Circles: Practice Questions & Notes

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The constant distance is called the radius.

Unlike polygons, a circle has no angles and no sides, but it carries some of the most powerful geometric properties you'll encounter.

Key Terminology

Term	Definition
Radius (r)	Distance from the center to any point on the circle
Diameter (d)	A chord that passes through the center; longest chord $d = 2r$
Circumference (C)	Perimeter of the circle $C = 2\pi r$
Area	Space enclosed $A = \pi r^2$
Chord	A line segment with endpoints on the circle
Arc	A part of the circle's circumference
Sector	Portion of a circle enclosed by two radii and the arc
Segment	Region between a chord and the corresponding arc
Central Angle	Angle subtended at the center by an arc
Minor/Major Arc	Smaller/larger of the two arcs subtended by a chord

Important Formulas

1. Circumference

C = 2\pi r = \pi d

2. Area

A = \pi r^2

3. Length of an Arc (central angle $\theta^\circ$ )

\text{Length} = \frac{\theta}{360} \times 2\pi r

4. Area of a Sector

\text{Area} = \frac{\theta}{360} \times \pi r^2

5. Area of a Segment

\text{Segment Area} = \text{Sector Area} - \text{Area of Triangle (formed by radii)}

Geometric Properties and Theorems

1. Angle at the Centre vs. Angle at the Circle

The angle subtended at the center is twice the angle subtended at the circumference:

\angle AOB = 2 \cdot \angle ACB

2. Angle in a Semicircle

Any angle inscribed in a semicircle is a right angle (90°).

3. Cyclic Quadrilateral

All vertices lie on a circle
Sum of opposite angles = 180°

4. Perpendicular from Center to a Chord

Always bisects the chord.

5. Equal Chords

Subtend equal arcs and are equidistant from the center.

6. Tangents

A tangent is a line that touches the circle at exactly one point.
Always perpendicular to the radius at the point of contact.
From an external point, lengths of tangents to the circle are equal.

Special Cases and Advanced Formulas

1. Area of Ring (Annulus)

A ring formed between two concentric circles with radii $R$ and $r$ :

\text{Area} = \pi(R^2 - r^2)

2. Number of Tangents from a Point

From a point outside the circle: 2 tangents
From a point on the circle: 1 tangent
From a point inside: No tangent

Visual Explanations

Circle Components Diagram:

$O$ = Center
$AB$ = Diameter
$r$ = Radius
Circumference: Full boundary
Area: Entire region enclosed

Conceptual Tips and Common Mistakes

Mistake	Tip
Using radius instead of diameter (or vice versa)	Always confirm what's given — $d = 2r$
Confusing arc length with sector area	Arc = length (1D), Sector = area (2D)
Forgetting unit consistency	Keep all values in same units: cm, m, etc.
Misusing the tangent property	Tangents from an external point are equal and perpendicular to radius

Shortcuts and Tricks

Semicircle implies a 90° angle at the circumference.
Use Pythagoras when tangents or radii create right triangles.
Memorize that the longest chord = diameter.
For rapid MCQ solving:
- Angle subtended by diameter = 90°
- Angle subtended by chord at center = $2 \times$ angle at arc

Examples

Example 1:

A circle has radius 7 cm. Find circumference and area.

C = 2\pi r = 2 \times \pi \times 7 = 44 \text{ cm} \\ A = \pi r^2 = \pi \times 49 \approx 154 \text{ cm}^2

Example 2:

Find the length of an arc that subtends 60° in a circle of radius 6 cm.

\text{Arc Length} = \frac{60}{360} \times 2\pi \times 6 = \frac{1}{6} \times 12\pi = 2\pi \approx 6.28 \text{ cm}

Example 3:

Two tangents are drawn from a point 10 cm away from the center to a circle of radius 6 cm. Find the length of each tangent.

Use Pythagoras in the right triangle:

l = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm}