Circles

A circle is a 2-dimensional closed shape that has a curved side whose ends meet to form a round shape. The word ‘Circle’ is derived from the Latin word 'circulus' which means a small ring. Let us learn more about the circle definition, the circle formulas, and the various parts of a circle with a few circle practice problems on this page.

What is Circle?

A circle is a two-dimensional figure formed by a set of points that are at a fixed distance (radius) from a fixed point (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius. Observe the following figure to see the basic parts of a circle, the center, the radius, and the diameter of a circle.

Parts of a Circle

There are many parts of a circle that we should know to understand its properties. A few important parts of a circle are given below.

Circumference: It is also referred to as the perimeter of a circle and can be defined as the length of the boundary of the circle.

Radius of Circle: Radius is the distance from the center of a circle to any point on its boundary. A circle has an infinite number of radii.

Diameter: A diameter is a straight line passing through the center that connects two points on the boundary of the circle. We should note that there can be multiple diameters in the circle, but they should:

• Pass through the center.
• Be straight lines.
• Touch the boundary of the circle at two distinct points which lie opposite to each other.

Chord of a Circle: A chord is any line segment touching the circle at two different points on its boundary. The longest chord in a circle is its diameter which passes through the center and divides it into two equal parts.

Tangent: A tangent is a line that touches the circle at a unique point and lies outside the circle.

Secant: A line that intersects two points on an arc/circumference of a circle is called the secant.

Arc of a Circle: An arc of a circle is referred to as a curve which is a part or portion of its circumference.

Segment in a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments - minor segment and major segment.

Sector of a Circle: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors - minor sector, and major sector.

Observe the figure given below which shows all the important parts of a circle.

Properties of Circle

Let us move ahead and learn about some interesting properties of circles that make them different from other geometric shapes. Here is a list of properties of a circle:

• A circle is a closed 2D shape that has one curved face.
• Two circles can be called congruent if they have the same radius.
• Equal chords are always equidistant from the center of the circle.
• The perpendicular bisector of a chord passes through the center of the circle.
• When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points.
• Tangents drawn at the endpoints of the diameter are parallel to each other.

Circle Formulas

There are many formulas related to a circle. Let us see the list of a few basic circle formulas pertaining to any circle.

• Area of a Circle Formula: The area of a circle refers to the amount of space covered by the circle. It totally depends on the length of its radius → Area of a circle = πr 2 , where 'r' is the radius and π = 3.14
• Circumference of a Circle Formula: The circumference is the total length of the boundary of a circle → Circumference of a circle = 2πr, where 'r' is the radius and π = 3.14
• Arc Length Formula: An arc is a section (part) of the circumference. Length of an arc = θ × r. Here, θ is in radians, and 'r' is the radius.
• Area of a Sector Formula: If a sector makes an angle θ (measured in radians) at the center, then the area of the sector of a circle = (θ × r 2 ) ÷ 2. Here, θ is in radians.
• Length of Chord Formula: It can be calculated if the angle made by the chord at the center and the value of the radius is known. Length of chord = 2 r sin(θ/2). Here, θ is in radians.
• Area of Segment Formula: The segment of a circle is the region formed by the chord and the corresponding arc covered by the segment. The area of a segment = r 2 (θ − sinθ) ÷ 2. Here, θ is in radians.

☛ Related Topics

Check these interesting articles related to circles in math.

• Constructing Circles
• Circumference of Circle Calculator
• Equation of Circle Calculator
• Symmetry of Any Circle
• Chords and Diameters

Examples on Circles

Example 1: If the radius of a circular pool is 20 units, what is the length of the diameter of the pool? Solution: Given: Radius = 20 units ⇒ Diameter of pool (circle) = 2 × r. Therefore, the length of the diameter of the circular pool = 2 × 20 = 40 units.

Example 2: John went swimming in a circular swimming pool. After swimming, he ran one round along the boundary of the pool. If the radius of the pool is 35 feet, can you find the distance that John ran around the pool? Solution: To find the distance that John ran, we need to know the circumference of the circle (pool). For this, we need to know the value of π and r, where r is the radius of the pool. Given: r = 35 feet, and π = 22/7. Using the formula, Circumference (C) = 2πr ⇒ C = 2 × 22/7 × 35 = 220 feet. Therefore, John ran 220 feet.

Example 3: A tabletop is in the shape of a circle. If the radius of the tabletop is 21 inches, find the area of the tabletop. Solution: Given: r = 21 inches, π = 22/7. Using the formula, Area = πr 2 ⇒ A = 22/7 × 21 × 21 = 1386 square inches. Therefore, the area of the tabletop is 1386 square inches.

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