Radius of the Circle: Definition, Formula with Solved Examples

Lines and circles are one of the first things you learn to draw when you start with mathematics in elementary classes. However, these simple figures have more to offer than what meets the eye. They have several elements and properties, some of which we will go through in this article before we finally learn how to find the radius of a circle.

What is a Circle?

A circle can be defined in many ways. 

• It is the collection of all the points in a plane, which lie at a fixed distance from a set point in the plane. The fixed point here is the center, called “O.”
• It is a closed two-dimensional figure with an area, i.e., the region in a 2D plane bordered by it. It also has a perimeter, which is also called the circumference, i.e., the distance around the circle. 
• It is a figure in which all the points in the plane are “equidistant” from the center, “O.”

Applications of Circles in Real Life

• Focal lengths of camera lenses are calculated by using the radius of curvature of the lens. 
• Odometers are instruments used to calculate the distance traveled in automobiles. This is done by counting the number of rotations and the circumference of the wheel, which is defined by its radius. 
• The diameter of round pans calculates pizza and cake sizes.

Some important elements of a circle are:

• Circumference: It is the boundary of the circle.
• Center – It is the midpoint of a circle.
• Diameter – This is the line that passes through the center of the circle, touching the two points on the circumference. It is represented as “D” or “d.” Diameters should be straight lines and touch the circle’s boundary at two distinct points which are opposite to each other.
• Arc – It is a curved part of the circumference of the circle. The biggest arc is called the “major arc,” while the smaller arc is called the “minor arc.”
• Sector – It is a section or portion of a circle determined by two radii and includes an arc of the circle
• Chord – It is a straight line that joins any two points on the circumference of a circle.
• Tangent – It is a line that connects the circumference of a circle at a point.
• Secant – It is any line that intersects the circle at two distinct points.
• Annulus – This is the region determined by two concentric circles, which resemble a ring-shaped object.
• Radius: Denoted by “R” or “r,” the radius is the line from the center of the circle to the circumference. 

Properties of a Circle

Circles have properties that determine their quality and functions. Some of them are given below:

• Circles are two-dimensional and not polygons. 
• Circles are purported to be congruent if they have the same radius, i.e., equal radii
• The longest chord in a circle is the diameter.
• Equal chords of a circle demarcate equal angles at the center of a circle
• Any radius drawn perpendicular to a chord in a circle will bisect the chord
• A circle can circumscribe any shape – rectangles, triangles, trapeziums, kite squares, etc.
• Circles can be inscribed within a square, kit, and triangle
• Chords that are at an equal distance from the center have the same length
• The distance that exists between the center of the circle to the diameter (the longest chord) is zero
• When the length of the chord increases, the perpendicular distance from the center of the circle decreases 
• Tangents are parallel to each other if they are drawn at the end of the longest chord or diameter

Circle Formulas

Certain formulas are used in geometry to solve solutions involving circles. Some of these formulas are:

• Area of a circle:

A = πr² sq unit

• Circumference of a circle:

2πr units OR πd.

Where, Diameter = 2 x r

Therefore, d = 2r Where “r” = radius of a circle.

What is a Radius?

• A radius can be defined as the line from the center “O” of the circle to the circumference of the circle. It is a line segment represented by the letter “R” or “r.” 
• A circle’s radius length remains the same from the middle point to any point on the boundary. A radius is half the length of the diameter of a circle or sphere. So, the radius of the circle or sphere can be expressed as d/2, where “d” represents the diameter. 
• The term “radii” is the plural of radius, used when talking about the radius of two or more circles. 
• A circle can have multiple radii within itself because the circumference of a circle has infinite points. Thus, circles can have an infinite amount of radii, and all of these radii have the same length of distance from the center of the circle. 

Formulas for Finding the Radius of the Circle

The radius formula is derived by splitting the diameter of the circle in half. When a point on the circumference of a circle is connected to the exact center, the line segment that appears is called the radius of the ring or circle.

The radius of a circle can be derived using three basic radius formulas, i.e., when the diameter, the circumference, or the area is provided. Given below are the radius formulas for a circle. 

Radius Formula from Diameter:

As mentioned earlier, the diameter is twice the radius of a circle. Therefore, the formula of a radius can be derived by dividing the diameter by 2. Mathematically, the diameter is represented by the letter “D” and is written as:

Diameter = 2 × radius. 

When the diameter of a circle is provided in a problem, the radius formula is written as:

radius = diameter/2 units 

= D/2 units.

Radius Formula from Circumference:

The circumference of a circle is represented by “C.” It can be expressed as C = 2πr units, where C = circumference, r = radius of the circle, and π = 3.14159. The ratio of circumference to 2π is the radius. The radius formula derived from the circumference of a circle is written as:

R= C/2π units

Radius Formula with Area:

The relationship between the radius and area is represented by the formula:

Area of the circle = πr² square units. 

Where r represents the radius and π is the constant, 3.14159. The radius formula derived from the area of a circle is written as:

Radius = √(Area/π) units

Example: If the diameter is given as 24 units, then the radius is 24/2 = 12 units. If the circumference of a circle is provided as 44 units, then its radius can be calculated as 44/2π. This implies, (44×7)/(2×22) = 7 units. And, if the area of a circle is given as 616 square units, then the radius is ⎷(616×7)/22 = ⎷28×7 = ⎷196 = 14 units.

Examples:

Q1. If a basketball has a diameter of 24cm, find the radius of the ball.

Sol: Given, diameter = 24cm

We know, d= 2r

Therefore, r=d/2

=24/2cm

=12cm

therefore the radius of the ball =12cm

Q2. If the area of the orange is 13cmsq, find the radius of the orange.

Sol: Given, area = 13cmsq,

We know, that area=2πR²

Therefore, R= ⎷(A/2π)

R= ⎷(13/2*3.14)cmsq

= 2.03 cm or 2cm approx.

Therefore, the radius of the orange is 2cm.

Q3. If the diameter of the cherry is 9 cm, find its radius.

Sol: Given, Diameter (d) = 9 cm

We know that diameter = 2r,

Radius=d/2

Therefore, radius = 9/2 = 4.5 cm

Now, Area = πr² (where r=radius)

Therefore, the Area of the circle = 3.14 x 4.5 x 4.5 = 63.585 cm sq.

Thus, we find that the radius of the cherry is 4.5 cm.

WORKSHEET

Here is a worksheet for you to work on your skills to find the radius of a circle.

Q1. Solve the following word problems:

1. Find the radius of the circle with a diameter of 89 cm. 
2. If the area of a circle is 152 cm sq, find the radius of the circle.
3. Find the radius of the circle with a circumference of 375 cm.

Q2. Find the radius of the circle based on the diameter provided:

1. D= 354 cm
2. D = 79 cm
3. D = 1001 cm

Q3. Find the radius of the circle based on the area provided:

1. A = 200 cm sq
2. A = 2346 cm sq
3. A = 623 cm sq

Q4. Find the radius of the circle based on the perimeter provided:

1. P = 67 cm
2. P = 107 cm
3. P = 482 cm

Q5. Find the circumference of the circle whose radius is:

(a) 57 cm 

(b) 23.1 cm

(c) 90.09 cm

This article has provided an in-depth explanation and demonstration of how to find the radius of a circle. Now that you are ready with your knowledge of the elements, properties, and formulas of a circle and its parts, solving geometry questions will now be a piece of cake, or rather, a sector of a circle.