Circle definition

A circle is a planar closed curve made of a set of all the points that are at a given distance from a given point - the center. Alternately, you can define the circle as the locus of points at the same distance from a given fixed point. The name circle derives from Greek, meaning hoop or a ring.

The circle, technically speaking, is only the boundary, not the area inside the shape. The whole figure is called a disc. A disc may be closed if it contains the circle that constitutes its boundary and open if it doesn't. In everyday use, the circle is sometimes understood as the disc – the plane surface bounded by such a curve.

Special lines of the circle

Circle terminology consists of the definitions of:

Circumference – the distance around the circle;
Radius – a line segment joining the center of the circle with any point on the circle;
Diameter – a line segment whose endpoints lie on the circle and which passes through the center; and
Chord – a line segment whose endpoints lie on the circle.

There are many other terms associated with the circle – such as the arc, secant, or tangent – but for our basic explanations, you need to understand only the ones described above.

Circle properties

A circle is a simple, distinctive shape with many unusual properties:

Has the largest area for a given length of perimeter;
Highly symmetric – reflection symmetry occurs for every line through the center, rotational symmetry around the center for every angle; and
May be constructed through any three points on the plane (not all on the same line). Such a circle is a unique circle.

Moreover, all circles are similar, and unique circles in every triangle may be inscribed and circumscribed. A circle has dozens of other interesting properties, which you can discover on your own.

Example

Solving Circle Calculations

A circle is a geometric shape defined by all points equidistant from a center point. Various calculations related to a circle include finding its circumference, area, diameter, and radius.

The general formulas for circle calculations are:

Circumference: \( C = 2\pi r \) or \( C = \pi d \)
Area: \( A = \pi r^2 \)
Diameter: \( d = 2r \)
Radius: \( r = \frac{d}{2} \)

Finding the Circumference of a Circle

The circumference of a circle is the total distance around it. It can be calculated using:

\[ C = 2\pi r \]

Example:

If a circle has a radius of 7 cm, the circumference is:

Step 1: Use the formula \( C = 2\pi r \).
Step 2: Substitute \( r = 7 \): \( C = 2\pi(7) \).
Step 3: Approximate using \( \pi \approx 3.1416 \): \( C \approx 2(3.1416)(7) = 43.98 \) cm.

Finding the Area of a Circle

The area of a circle is the total space enclosed by the circle. It is given by:

\[ A = \pi r^2 \]

Example:

If a circle has a radius of 5 cm, the area is:

Step 1: Use the formula \( A = \pi r^2 \).
Step 2: Substitute \( r = 5 \): \( A = \pi (5^2) \).
Step 3: Calculate \( 5^2 = 25 \), so \( A = 25\pi \).
Step 4: Approximate using \( \pi \approx 3.1416 \): \( A \approx 25 \times 3.1416 = 78.54 \) cm².

Finding the Radius Given the Area

If you know the area and want to find the radius, use:

\[ r = \sqrt{\frac{A}{\pi}} \]

Example:

If the area of a circle is 50 cm², the radius is:

Step 1: Use the formula \( r = \sqrt{\frac{A}{\pi}} \).
Step 2: Substitute \( A = 50 \): \( r = \sqrt{\frac{50}{\pi}} \).
Step 3: Approximate using \( \pi \approx 3.1416 \): \( r \approx \sqrt{\frac{50}{3.1416}} \).
Step 4: Calculate \( r \approx \sqrt{15.92} \approx 3.99 \) cm.

Real-life Applications of Circle Calculations

Circle calculations have many practical applications, such as:

Determining the size of wheels, plates, and other circular objects.
Calculating the amount of material needed for circular designs in construction or manufacturing.
Estimating distances traveled by rotating objects like tires and gears.
Understanding planetary orbits in astronomy.

Common Circle Formulas

Circumference: \( C = 2\pi r \) or \( C = \pi d \)

Area: \( A = \pi r^2 \)

Radius: \( r = \frac{d}{2} \)

Diameter: \( d = 2r \)

Circle Calculator Examples Table
Calculation Type	Description	Formula	Example
Calculate Circumference	Finds the distance around the circle.	\( C = 2\pi r \) or \( C = \pi d \)	If \( r = 7 \), then \( C = 2\pi(7) = 14\pi \approx 43.98 \).
Calculate Area	Finds the space inside the circle.	\( A = \pi r^2 \)	If \( r = 7 \), then \( A = \pi(7^2) = 49\pi \approx 153.94 \).
Calculate Diameter	Finds the longest distance across the circle.	\( d = 2r \)	If \( r = 7 \), then \( d = 2(7) = 14 \).
Calculate Radius	Finds the radius when given the diameter.	\( r = \frac{d}{2} \)	If \( d = 14 \), then \( r = \frac{14}{2} = 7 \).