Ximpledu
English

Circle (Geometry)

See how to solve circle problems
(arc/segments/angles/area).
25 examples and their solutions.

Circumference of a Circle

Formula


C = 2πr

C = πd

Example

Circumference?

Solution

Example

Circumference?

Solution

Measure of an Arc

Definition


mAB = θ

Example

mAB = ?

Solution

Length of an Arc

Formula

Example

Length of AB = ?

Solution

Chord of a Circle

Definition

A chord is a line segment
whose endpoints are on the circle.

Property

A line segment that passes through the center
perpendicularly bisects the chord.

Example

AB = ?

Solution

Inscribed Angle

Definition

Formula


θ = 12mAB
θ: Inscribed angle
mAB: Intercepted arc

Example

Solution

Example

mABC = ?

Solution

Example

Solution

Inscribed Right Triangle

Property

The hypotenuse of an inscribed right triangle
passes through the center of the circle.

Example

Solution

Example

Solution

Inscribed Quadrilateral

Property



m∠1 + m∠2 = 180
m∠1' + m∠2' = 180
The opposite angles are supplementary angles.

Example

Solution

Tangent to a Circle

Definition

A tangent to a circle is a line
that touches the circle.
From a point exterior of the circle,
you can draw two tangents.

Property

The tangent and the radius
are perpendicular
at the intersecting point of the circle.

Example

PB = ?

Solution

Property

Example

Perimeter of △ABC?

Solution

Angle Formed by a Tangent and a Chord

Formula


θ = 12mAB

Example

mAB = 110, x = ?

Solution

Example

Solution

Angle Formed by Two Chords

Formula


θ = 12(mAC + mBD)

Example

Solution

Angle Formed by Two Secants

Formula


θ = 12(mBD - mAC)

Example

Solution

Angle Formed by a Tangent and a Secant

Formula


θ = 12(mBC - mAC)

Example

Solution

Angle Formed by Two Tangents

Formula


θ = 12(mABC - mAC)

Example

Solution

Segments Formed by Two Chords

Formula


APPB = CPPD

Example

Solution

Segments Formed by Two Secants

Formula


PAPB = PCPD

Example

Solution

Segments Formed by a Tangent and a Secant

Formula


PA2 = PBPC

Example

Solution

Area of a Circle

Formula


A = πr2

Example

Area?

Solution

Example

Area?

Solution

Area of the Sector of a Circle

Formula

Example

Area?

Solution