What is a Circle? Definition, Formulas, Examples

What is a Circle?

A circle is a closed two-dimensional figure in which the set of all points in the plane are equidistant from a given point called the “center”. Every line passing through the circle forms a line of reflection symmetry. Also, it has rotational symmetry around the center for every angle.

Circle Definition

Circle in Detail

A circle is a two-dimensional geometric figure characterized by all points equidistant from a central point. Here are some key features.

Radius (r)

The distance from the center of the circle to any point on its circumference.

Diameter (d)

Twice the radius; the distance across the circle through its center.

Circumference (C)

The perimeter or the distance around the circle.

$C = 2\pi r$ or $C = \pi d$.

Area (A)

The region enclosed by the circle.

$A = \pi r^2$.

Some Examples Related to Circles along with their Answers

Example 1: Circumference Calculation

Question: If a circle has a radius of 7 cm, find its circumference.

Answer-  C = 2π × 7 = 14π cm (or approximately 43.98 cm).

Example 2: Diameter and Radius Relationship

Question: If the diameter of a circle is 10 units, what is its radius?

Answer- The radius is half of the diameter,

$r = \frac{10}{2} = 5$ units.

Example 3: Area Calculation

Question- Find the area of a circle with a radius of 3 meters.

Answer-

$A = \pi \times (3)^2 = 9\pi$ square meters (or approximately 28.27 square meters).

Example 4: Circle Sector Angle

Question- In a circle with a radius of 8 cm, an angle at the center is 45 degrees. Find the length of the arc it subtends.

Answer-

$\text{Arc length} = \frac{45}{360} \times 2\pi \times 8$ cm.

Example 5: Tangent Line Length

Question: If a tangent is drawn to a circle with a radius of 6 cm from an external point, find the length of the tangent if it makes an angle of 60 degrees with the radius.

Answer- Use the tangent line formula.

$\text{Tangent length} = \sqrt{r^2 + r^2 – 2 \cdot r \cdot r \cdot \cos(\theta)}$, where $\theta$ is the angle in radians.

Example 6: Inscribed Angle

Question: In a circle, an inscribed angle measures 90 degrees. What is the corresponding central angle?

Answer- The central angle is twice the inscribed angle, so Central angle = 2 × 90 = 180 Central angle = 2 × 90 = 180 degrees.

Example 7: Circles in Nature

Question: The diameter of a circular tree cross-section is 60 cm. What is its circumference?

Answer- C = π × 60 cm.

Example 8: Circular Pizza

Question: A pizza has a diameter of 16 inches. What is its area?

Answer-

$A = \pi \times \left(\frac{16}{2}\right)^2$ square inches.

Example 9: Wheel of a Bicycle

Question: If the diameter of a bicycle wheel is 26 inches, what is its circumference?

Answer- C = π × 26 inches.

Example 10: Circular Coin

Question: The diameter of a coin is 2 cm. Find its radius.

Answer-

$r = \frac{2}{2} = 1$ cm.

Example 11: Circular Mirror

Question: The diameter of a circular mirror is 18 inches. What is its radius?

Answer-

$r = \frac{18}{2} = 9$ inches.

Example 12: Circular Garden

Question: A circular garden has a radius of 5 meters. Calculate its area.

Answer-

$A = \pi \times (5)^2 = 25\pi$ square meters.

Example 13: Circular Swimming Pool

Question: The circumference of a circular swimming pool is 50 feet. Find its radius.

Answer-

$r = \frac{50}{2\pi}$ feet.

Example 14: Concentric Circles

Question: Draw two concentric circles. Label the radius of the larger circle as R and the radius of the smaller circle as r. What is the difference in their areas?

Answer-

$\text{Area difference} = \pi \times (R^2 – r^2)$.

Read Also

Quadratic Equation Definition Formulas and History

Some Examples Related to Circles

Calculating Circumference– If a circle has a radius of 5 units, its circumference is 2π × 5

Diameter to Radius Relationship- The diameter is always twice the radius.

Pizza as a Circle- Pizza slices are often shaped like sectors of a circle.

Wheel of a Bicycle- Bicycle wheels are circular.

Circular Tabletops- Many tables have circular tops.

Circular Coins- Coins are often circular.

Circular CDs and DVDs- Optical discs are circular.

Circles in Nature- The cross-section of a tree trunk is approximately circular.

Circular Stamps- Postage stamps are often circular.

Circular Traffic Signs- Many road signs are circular.

Circular Clocks- Clock faces are typically circular.

Ferris Wheels- The shape of Ferris wheels involves circles.

Target Boards- Targets used in shooting sports are circular.

Circular Moon- The moon appears circular.

Circular Buttons- Buttons on clothing can be circular.

Circular Plates- Plates are often circular in shape.

Circular Mirrors- Mirrors are often circular.

Circular Hula Hoops- Hula hoops are circular toys.

Circular Water Ripples- When you drop an object into water, it creates circular ripples.

Circular Watch Faces- Watch faces are typically circular.

Tires on Cars- Car tires are circular.

Circular Lenses- Eyeglass lenses are often circular.

Circular Coffee Cups- The tops of coffee cups are circular.

Circular Doilies- Decorative doilies are often circular.

Circular Garden Pots- Many plant pots have a circular shape.

Circular Cookies- Cookies are often circular.

Circular Stickers- Stickers can be circular in shape.

Circular Steering Wheels- Steering wheels are typically circular.

Circular Buttons on Websites- Online buttons are often circular.

Circular Frisbees- Frisbees are circular flying discs.

Circular Rings- Rings can be considered as circles.

Circular Platters- Serving platters are often circular.

Circular Coins- Many coins are circular.

Circular Earrings- Earrings can have a circular design.

Circular Street Manholes- Manhole covers are often circular.

Circular Cookies- Cookies are often circular.

Circular Ice Cream Cones- Ice cream cones have a circular base.

Circular Pies- Pies are often circular in shape.

Circular Stairs- The shape of staircases can involve circles.

Circular Buttons on Electronic Devices- Buttons on devices like smartphones can be circular.