In geometry, there are terms such as length, height, slant height, and surface area used to represent various shapes. One of the most important characteristics that identify a circle is the diameter. A diameter is a straight line going through the center of a circle and touching two points on its edge. It is also referred to as the longest chord of the circle. Likewise, the diameter of a sphere can be described in two different ways, but in both instances, it is a fundamental measurement. The diameter is vital because it not only describes the circle but also calculates the distance across the circle from one end to the other, going through the center. The understanding of the term diameter is essential in solving geometry problems and hence is an important concept to master for achieving success in U.S. competitive exams such as SAT and ACT exam.

Diameter of a Circle

The circumference of a circle, measured from edge to other, includes its entire length from center to center. This length right here is the entire thing. The letter “d” stands for a circle’s diameter.

A circle’s radius is equal to twice the diameter of the circle. There are multiple diameters within a circle. The diameter of a circle is the distance along which two locations located on its periphery are connected by a line that runs through the circle’s center.

Each circle contains four essential characteristics. They are as follows:

A radius is a line segment that extends from the center to any point on the circumference. Its sign in formulae is ‘r’.

A diameter is a line that begins at one point on the circle’s perimeter, travels through its centre, and then ends on a point on the circle diametrically opposite to the initial point. Its symbol is ‘d’.

The circumference of a circle is its length from edge to edge. C is its symbol.

Area: the entire circumference of the circle. Its designation is A.

Steps: Create a circle’s diameter.

To create a “chord,” draw a line through the circle.

Create the circle’s diameter by creating the chord’s perpendicular bisector.

To find the circle’s center, create the perpendicular bisector of that diameter.

Relation between Radius and Diameter

A circle’s radius is the distance a point on the circle’s circumference travels from its center.

The circle’s diameter is a straight line that passes through its center and has its endpoints on the circle.

Radius is therefore equal to half of the diameter.

Diameter = 2(Radius)

\( Diameter = 2\times r \)

Diameter of a Circle Formula

The diameter of a circle is calculated using the diameter formula. The diameter formula can be used to determine the separation between the circle’s two edges.

Diameter of a Circle Using Circumference

The diameter formula can be simply derived from the circumference.

Calculating a circle’s circumference uses the following formula:

\( C = \pi d \)

\( d = \frac{C }{\pi } \)

\( Diameter = \frac {Circumference }{\pi } \)

For example:

A circle’s circumference is 38 meters, therefore determine its diameter.

Solution:

Circumference = 38 meters

\( C = \pi d \)

38 = \( \pi d \)

\( \Rightarrow \frac{22}{7}\times d\) = 38

\( \Rightarrow d = \frac{38\times 7}{22} \)= 12

Diameter of a Circle Using Radius

The radius of a circle is the distance along a line drawn from its center to one of its ends, whereas its diameter is equal to double that distance.

Using this concept, the diameter formula is

\( d = r\times 2 \)

Where,

d = diameter

r = radius

For example:

Joy drew a circle with a 7 unit radius. What is the circle’s diameter?

Solution:

Circle radius = 7 unit

Circle’s diameter = \( 2\times radius \)

\( = 2\times 7 \)

Circle’s diameter = 49

Circle diameter is 49 units

Diameter Formula Using Area of Circle

Using the formula for the area of a circle, we can calculate the diameter of a circle, which is,

\( A = \pi \left ( r \right )^{2} \)

Here, r = radius

A = area of circle

Substituting the value of radius as \(\frac{d}{2}\) then we obtain

\(\frac{A}{\pi } = \left (\frac{d}{2}\right)^{2} \)

\(\Rightarrow\frac{d}{2} = \sqrt{\left ( \frac{A}{\pi } \right )}\)

\(\Rightarrow d = 2\times \sqrt{\left ( \frac{A}{\pi } \right )} \)

For example:

If a circle’s area is \( 380 cm^{2} \) what is its diameter?

Solution:

\( A = \pi r^{2} \)

\( 380 = \left ( 3.14 \right )r^{2} \)

\( 121 = r^{2} \)

11 = r

\( d = 2\times r \)

\( d = 2\times 11 \)

d = 22

The diameter of circle is 22 cm

Diameter Solved Examples

Problem: 1 A circle has a surface area of 379.94 \( cm^{2} \). Determine the diameter of circle.

Solution:

Area of circle = 379.94 \(cm^{2}\).

As we know that

\(A = nr^{2}\)

\(379.94 = \left[ 3.14 \right ]r^{2}\)

\(\frac{379.94}{3.14} = r^{2}\)

\(121= r^{2}\)

Square roots from both sides are then taken.

r = 11

Diameter (d) = 2r

\(d = 2\times 11 \)

Diameter of circle is 22 cm

Problem: 2 If the diameter of a circle is 38 units, can you calculate its radius?

Solution:

Diameter = 38 units

The radius divided by two gives the diameter. This indicates that the radius is equal to the diameter divided by two.

\( Diameter = 2\times Radius \)

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

\( r = \frac{38}{2} \)

= 19

The radius of circle is 19 units

Conclusion

It is crucial to comprehend the definition of a circle's diameter in order to solve geometry problems efficiently. Whether you are solving for the diameter based on the radius, circumference, or area, knowing these formulas by heart will make it easier to solve complex problems and increase your confidence during exams. Keep in mind that the diameter is the longest chord that goes through the center of the circle and touches two opposite points on the circle. Using these principles, you can easily solve related questions and enhance your problem-solving speed. If you're looking to excel on your math exams, practice is the key! Sites like Testbook provide in-depth study materials, weekly practice tests, and expert-guided live classes. Download the Testbook App right now and take the first step to reaching your academic potential.