# Circle

Definition: A **circle** is a simple shape, consisting of those points in a plane that are a given distance from a given point - the centre.

**Origin**: the center of a circle

**Radius**: the distance from the center of a circle to any point on it.

**Diameter**: the longest distance from one end of a circle to the other. The diameter = 2 × radius (d = 2r).

**Circumference**: the distance around the circle.

Circumference $= \pi \times diameter$.

Circumference $= \pi \times d = 2 \times \pi \times r$

**$\pi$ - pi**: a number equal to 3.141592... or $\approx \frac{22}{7}$, that is $\frac{\text{the circumference}}{\text{the diameter}}$ of any circle.

**Arc**: a curved line that is part of the circumference of a circle.

The arc of a circle is measured in degrees or radians - for example: 90° or $\frac{\pi}{2}$ - a quarter of the circle,

180° or $\pi$ - a half of the circle.

The arc is smaller than 360°(or $2\pi$) because that is the whole circle.

**Chord**: a line segment within a circle that touches 2 points on the circle.

**Sector**: is like a slice of pie (a circle wedge).

**Tangent**: a line perpendicular to the radius that touches ONLY one point on the circle.

### Formulas

The formula for finding the **circumference** of a circle is $\pi \cdot \text{diameter} = 2 \cdot \pi \cdot \text{radius}$

The formula for finding the area of a circle is $\pi \cdot \text{radius}^2$

The standard notation for a radius is r, for a diameter - d, for a circumference - P and for area A.

$A = \pi \cdot r^2$

## Angles

### Central angle

If the length of an arc is $\theta$ degrees or radians then the measurement of the central angle is also $\theta$(degrees or radians).

If you know the length of an arc(in inches, yards, foots, centimeters, meters...) you can find the measurement of its corresponding central angle($\theta$) by the formula:

$l$ is the length of the arc.

### Inscribed angle

An inscribed angle is formed by two chords and vertex on the circle.

$\angle APB$ is an inscribed angle.

Example:

The arc $\widehat{AB}$ measures 84°.
The measurement of $\angle APB = \frac{84}{2} = 42^\circ$

### Angles between two secants

**Case 1:** two secants intersect **inside** a circle.

In the figure, arc AB is 60° and the arc CD is 50°.

So angle 1 and 2 measure ½(60° + 50°) = 55°

**Case 2:** two secants intersect **outside** a circle.

The measurement of the angle formed is equal to half the difference of the arcs.

$\angle ABC =\frac{1}{2}(x - y)$

For example: if the bigger arc measures 80° and the smaller one 30° then $\angle ABC = \frac{1}{2}(80 - 30) = \frac{1}{2} \cdot 50 = 25^\circ$

### Formula for intersecting chords

When two chords intersect inside a circle then:

### Area of sector

$\theta$ is the measurement of the central angle.

If the angle θ is in degrees, then the area = $\frac{\theta}{360} \pi r^2$

If the angle θ is in radians, then the area = $\frac{\theta}{2} r^2$

### Area of a Circular Ring

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

### Area of a Circular Trapezoid

Area = $\pi(R^2 - r^2)\frac{\theta}{360}$