Radius, Diameter, and Circumference

Welcome to the world of geometry! In this discussion, we will explore three fundamental concepts that form the backbone of circles: radius, diameter, and circumference. These terms are not only vital in geometry but also find applications in various fields.

Table of Contents:

• What is Circle?
• Radius
• Diameter
• Circumference
• FAQs

Circle

A circle is a geometric shape defined as a closed curve where all points on the curve are equidistant from a single fixed point called the centre. The distance from the centre to any point on the circle is called the radius.

The length of the curve that makes up the circle is called the circumference. The diameter of a circle is the distance across the process passing through the centre and is equal to twice the radius.

Circles are found in many areas of mathematics and science, including geometry, trigonometry, and physics. They are also commonly used in everyday objects like wheels, clocks, and CDs.

Properties of a Circle

Here are some properties of a circle:

• Center: A circle is defined by a central point called the center. All points on the circle are equidistant from the center.

• Radius: The radius is the distance from the center of the circle to the circumference of the circle.

• Diameter: The diameter is the distance across the circle passing through the center. It is equal to twice the radius.

• Circumference: The circumference is the length of the curve that makes up the circle. It is equal to 2π times the radius or π times the diameter.

• Area: The area of a circle is the amount of space inside the circle. It is equal to π times the square of the radius or one-fourth times π times the square of the diameter.

• Chord: A chord is a line segment that connects two points on the circle.

• Tangent: A tangent is a line that touches the circle at only one point.

• Arc: An arc is a portion of the circumference of a circle.

• Sector: A sector is a portion of the circle that is bounded by two radii and an arc.

• Segment: A segment is a portion of the circle that is bounded by a chord and an arc.

These are some of the main properties of a circle, and they are important in various areas of mathematics and science.

Radius

The radius of a circle is a line segment that connects the center of the circle to any point on the circle’s circumference. It is the distance between the center of the circle and any point on the circle’s edge.

The radius is an important property of a circle, as it is used to calculate other important properties, such as the diameter, circumference, and area of the circle. The length of the radius is denoted by the letter “r”.

The diameter of a circle is equal to twice the radius, and the circumference is equal to 2π times the radius, where π (pi) is a mathematical constant approximately equal to 22/7 or 3.14159.

The area of a circle is equal to π times the square of the radius, or πr^2. Therefore, the radius plays a crucial role in determining the size, shape, and other properties of a circle.

Formula for Radius

The formula for radius (r) of a circle is given by:

r = C / 2π

where C is the circumference of the circle.

Alternatively, the formula for radius can be derived from the formula for the area of a circle (A), which is given by:

A = πr^2

By rearranging the above formula, we get:

r = sqrt(A / π)

where sqrt() denotes the square root.

Therefore, the radius of a circle can be calculated using either the circumference or the area of the circle. These formulas are important in solving problems related to circles in various fields of mathematics and science.

Solved Examples for Radius

Example 1: Find the radius of a circle whose circumference is 30 cm.

Solution: We can use the formula for radius in terms of circumference:

r = C / 2π

Substituting the given value of circumference, we get:

r = 30 cm / 2π

r ≈ 4.77 cm (rounded to two decimal places)

Therefore, the radius of the circle is approximately 4.77 cm.

Example 2: The area of a circle is 78.5 sq cm. Find the radius of the circle.

Solution: We can use the formula for radius in terms of area:

r = sqrt(A / π)

Substituting the given value of area, we get:

r = sqrt(78.5 / π)

r ≈ 5 cm (rounded to one decimal place)

Therefore, the radius of the circle is approximately 5 cm.

Example 3: The diameter of a circle is 10 cm. Find its radius.

Solution: We know that the diameter of a circle is twice the radius, so we can use the formula for radius in terms of diameter:

r = d / 2

Substituting the given value of diameter, we get:

r = 10 cm / 2

r = 5 cm

Therefore, the radius of the circle is 5 cm.

These examples illustrate how to use the formula for radius in different situations, and how it can be derived from other formulas related to circles.

Diameter

The diameter of a circle is a line segment that passes through the center of the circle and connects two points on the circle’s circumference. It is the longest chord of the circle and it divides the circle into two equal halves.

The diameter of a circle is denoted by the letter “d”. It is twice the length of the radius, and can be expressed mathematically as:

d = 2r

where r is the radius of the circle.

The diameter is an important property of a circle, as it is used to calculate other important properties, such as the circumference and the area of the circle. The length of the diameter is a fundamental measure of the size of the circle.

The diameter is also important in many real-world applications. For example, the diameter of a wheel is important in determining its size and compatibility with other parts of a vehicle. In addition, the diameter of a pipe is important in determining its flow capacity and suitability for various applications.

Formulae of a Diameter

The formula for the diameter (d) of a circle is given by:

d = 2r

where r is the radius of the circle.

Alternatively, the formula for the diameter can be derived from the formula for the circumference (C) of the circle, which is given by:

C = 2πr

By rearranging the above formula, we get:

d = C / π

Therefore, the diameter of a circle can be calculated using either the radius or the circumference of the circle. These formulas are important in solving problems related to circles in various fields of mathematics and science.

Solved examples for Diameter

Example 1: Find the diameter of a circle whose radius is 6 cm.

Solution: We can use the formula for diameter in terms of radius:

d = 2r

Substituting the given value of radius, we get:

d = 2 x 6 cm

d = 12 cm

Therefore, the diameter of the circle is 12 cm.

Example 2: The circumference of a circle is 31.4 cm. Find its diameter.

Solution: We can use the formula for diameter in terms of circumference:

d = C / π

Substituting the given value of circumference, we get:

d = 31.4 cm / π

d ≈ 10 cm (rounded to one decimal place)

Therefore, the diameter of the circle is approximately 10 cm.

Example 3: The radius of a circle is 8 m. Find its diameter in centimeters.

Solution: We can use the formula for diameter in terms of radius:

d = 2r

First, we need to convert the radius from meters to centimeters, since the diameter is required in centimeters. One meter is equal to 100 centimeters, so:

r = 8 m x 100 cm/m

r = 800 cm

Substituting the converted value of radius, we get:

d = 2 x 800 cm

d = 1600 cm

Therefore, the diameter of the circle is 1600 cm.

These examples illustrate how to use the formula for diameter in different situations, and how it can be derived from other formulas related to circles.

Diameter from Radius

To find the diameter of a circle when given the radius, you can use the formula:

d = 2r

where “d” represents the diameter and “r” represents the radius.

This formula simply states that the diameter is equal to twice the radius of the circle. Therefore, if you know the radius of a circle, you can simply multiply it by 2 to find the diameter.

For example, if the radius of a circle is 5 cm, then the diameter can be calculated as:

d = 2 x 5 cm = 10 cm

So, the diameter of the circle is 10 cm.

This formula is useful when you only have the radius of the circle and need to find its diameter. It is important to remember that the diameter is always twice the length of the radius, regardless of the size or shape of the circle.

Circumference

The circumference of a circle is the distance around its outer edge. It is also defined as the perimeter of the circle.

The circumference of a circle is denoted by the letter “C”. It can be calculated using the formula:

C = 2πr

where “r” is the radius of the circle and “π” (pi) is a mathematical constant approximately equal to 3.14159.

This formula states that the circumference of a circle is equal to twice the radius multiplied by pi. Pi is an irrational number, which means that its decimal representation never ends or repeats.

The circumference of a circle is an important property of the circle, as it is used to calculate other important properties, such as the area of the circle. It is also used in various real-world applications, such as in determining the length of wire needed to go around a circular object, or the distance traveled by a wheel in one revolution.

It is important to note that the circumference and diameter of a circle are related by the formula:

C = πd

where “d” is the diameter of the circle. This formula states that the circumference of a circle is equal to pi multiplied by the diameter.

Circumference from Diameter & Radius

The circumference of a circle can be calculated using either the diameter or the radius of the circle.

If you know the diameter of the circle (d), you can use the formula:

C = πd

where “C” is the circumference of the circle and “π” (pi) is a mathematical constant approximately equal to 3.14159.

Alternatively, if you know the radius of the circle (r), you can use the formula:

C = 2πr

where “C” is the circumference of the circle and “π” (pi) is a mathematical constant approximately equal to 3.14159.

These formulas show that the circumference of a circle is proportional to its diameter and radius, and is directly related to the value of pi. In other words, as the diameter or radius of a circle increases, so does its circumference.

For example, suppose the diameter of a circle is 10 cm. We can use the formula C = πd to find its circumference:

C = π x 10 cm

C ≈ 31.4 cm

Therefore, the circumference of the circle is approximately 31.4 cm.

Alternatively, suppose the radius of a circle is 4 m. We can use the formula C = 2πr to find its circumference:

C = 2π x 4 m

C ≈ 25.13 m

Therefore, the circumference of the circle is approximately 25.13 m.

Solved Examples of Circumfrence

Find the circumference of a circle with a diameter of 8 cm.

Solution:

C = πd

C = π x 8 cm

C ≈ 25.12 cm

Therefore, the circumference of the circle is approximately 25.12 cm.

A circular garden has a radius of 6 m. What is the circumference of the garden?

Solution:

C = 2πr

C = 2π x 6 m

C ≈ 37.7 m

Therefore, the circumference of the garden is approximately 37.7 m.

A wheel has a diameter of 50 cm. What is the circumference of the wheel?

Solution:

C = πd

C = π x 50 cm

C ≈ 157.08 cm

Therefore, the circumference of the wheel is approximately 157.08 cm.

A circular pizza has a circumference of 30 inches. What is the diameter of the pizza?

Solution:

C = πd

30 inches = πd

d = 30 inches / π

d ≈ 9.55 inches

Therefore, the diameter of the pizza is approximately 9.55 inches.

The circumference of a circle is 12π cm. What is the radius of the circle?

Solution:

C = 2πr

12π = 2πr

r = 6 cm

Therefore, the radius of the circle is 6 cm

Radius, Diameter, and Circumference FAQS

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