Radius
- What is a Radius of a Circle?
- Radius Formulas of a Circle
- Radius from Diameter
- Radius from Circumference
- Radius from Area
- Equation of Circle Using Radius
- Practice Problems
- Frequently Asked Questions
What is a Radius of a Circle?
To comprehend what is a radius, we must first grasp the fundamental concept of a circle. A radius is essentially a line segment that connects the center of a circle to any point on its circumference. It defines the distance from the center to any point on the circle's edge. Imagine a wheel where the radius would be the distance from the center of the wheel to its outer edge. In mathematical terms, we denote the radius as \( r \).
Radius Formulas of a Circle
Understanding the various radius formulas associated with a circle can provide valuable insights into its properties and dimensions. These formulas are essential tools in geometry and mathematics, allowing us to calculate different aspects of a circle such as its radius, diameter, circumference, and area. Let's delve into some fundamental formulas that enable us to work with circles effectively.
Radius from Diameter
The diameter of a circle is the longest distance across the circle, passing through its center. It is a line segment that connects two points on the circle's circumference and passes through the center, effectively dividing the circle into two equal halves. The diameter is twice the length of the radius of a circle.
Example: What is the radius of a circle with a diameter of `5` centimeters?
Solution:
To find the radius of a circle when the diameter is known, we use the formula:
\( \text{Radius} = \frac{\text{Diameter}}{2} \)
Given that the diameter of the circle is 5 centimeters, we can substitute this value into the formula:
\( \text{Radius} = \frac{5}{2} \)
\( \text{Radius} = 2.5 \text{ centimeters} \)
Therefore, the radius of the circle is `2.5` centimeters.
Radius from Circumference
The circumference of a circle is the total distance around the outer edge of the circle. If you know the circumference of a circle, you can find the radius using the formula \( r = \frac{C}{2\pi} \), where \( r \) represents the radius and \( C \) represents the circumference. This formula divides the circumference by \( 2\pi \) to obtain the radius.
Example: If the circumference of a circle is `31.4` units, what is the radius of the circle?
Solution:
To determine the radius of a circle from its circumference, we employ the formula \( r = \frac{C}{2\pi} \), where \( r \) denotes the radius and \( C \) signifies the circumference.
Given that the circumference of the circle is `31.4` units, we can substitute this value into the formula:
\( r = \frac{31.4}{2\pi} \)
Using the approximation \( \pi \approx 3.14 \):
\( r = \frac{31.4}{2 \times 3.14} \)
\( r = \frac{31.4}{6.28} \)
\( r \approx 5 \)
Hence, the radius of the circle is approximately `5` units.
Radius from Area
The area of a circle involves the space enclosed within its boundary on a flat surface. Radius of a circle can also be determined if you know its area. The formula to find the radius from the area is \( r = \sqrt{\frac{A}{\pi}} \), where \( r \) represents the radius and \( A \) represents the area of the circle. This formula involves taking the square root of the quotient of the area divided by \( \pi \).
Example: If the area of a circle is \( 78.5 \) square units, what is the radius of the circle?
Solution:
To determine the radius of a circle from its area, we utilize the formula \( r = \sqrt{\frac{A}{\pi}} \), where \( r \) denotes the radius and \( A \) signifies the area of the circle.
Given that the area of the circle is \( 78.5 \) square units, we can substitute this value into the formula:
\( r = \sqrt{\frac{78.5}{\pi}} \)
Using the approximation \( \pi \approx 3.14 \):
\( r = \sqrt{\frac{78.5}{3.14}} \)
\( r = \sqrt{25} \)
\( r = 5 \)
Hence, the radius of the circle is \( 5 \) units.
Equation of Circle Using Radius
The equation for the radius of a circle on the cartesian plane is represented as:
\( (x - h)^2 + (y - k)^2 = r^2 \)
In this equation:
- \( h \) and \( k \) represent the coordinates of the center of the circle.
- \( r \) represents the radius of the circle.
- \( (x, y) \) are the coordinates of any point on the circumference of the circle.
This equation indicates that any point \( (x, y) \) on the circle's circumference satisfies the condition that the square of the difference between its horizontal coordinate \( x \) and the center's horizontal coordinate \( h \), plus the square of the difference between its vertical coordinate \( y \) and the center's vertical coordinate \( k \), equals the square of the radius \( r \).
Example `1`: Given a circle with a radius of `6` units and its center at the point `(3, 4)` on the cartesian plane, what is the equation of the circle.
Solution:
The equation for a circle on the cartesian plane with its center at the point `(h, k)` and a radius of `'r'` units is given by:
\( (x - h)^2 + (y - k)^2 = r^2 \)
Given that the center of the circle is at points `(3, 4)` and the radius is `6` units, we substitute these values into the equation:
\( (x - 3)^2 + (y - 4)^2 = 6^2 \)
\( (x - 3)^2 + (y - 4)^2 = 36 \)
Therefore, the equation of the circle with a radius of `6` units and center at `(3, 4)` is \( (x - 3)^2 + (y - 4)^2 = 36 \).
Example `2`: The center of a circle is at `(3, 4)` on the cartesian plane. Point `(7,4)` is a point on the circle. What is the radius of the circle?
Solution:
The equation for a circle on the cartesian plane with its center at the point `(h, k)` and a radius of `'r'` units is given by:
\( (x - h)^2 + (y - k)^2 = r^2 \)
Given that the center of the circle is at points `(3, 4)`, we substitute these values into the equation:
\( (x - 3)^2 + (y - 4)^2 = r^2 \)
Point `(7,4)` is a point on the circle. We further substitute `x = 7` and `y = 4` into the equation
\( (7 - 3)^2 + (4 - 4)^2 = r^2 \)
\( 4^2 + 0^2 = r^2 \)
\( r^2 = 16 \)
\( r = 4 \)
Therefore, the radius of the circle with a center at `(3, 4)` and a point at `(7,4)` is `4` units.
Practice Problems
Q`1`. What is the radius of a circle with a diameter of `12` units?
a) `3` units
b) `6` units
c) `9` units
d) `12` units
Answer: b
Q`2`. If the circumference of a circle is `31.4` units, what is its radius? (Consider `π = 3.14`)
a) `5` units
b) `7` units
c) `10` units
d) `15` units
Answer: a
Q`3`. The area of a circle is \(50 \pi\) square units. What is its radius?
a) `5` units
b) `7` units
c) `10` units
d) `15` units
Answer: a
Q`4`.The equation of a circle is \( (x - 3)^2 + (y + 4)^2 = 25 \). What are the coordinates of its center?
a) `(3, 4)`
b) `(-3, -4)`
c) `(3, -4)`
d) `(-3, -4)`
Answer: c
Q`5`. If a circle's equation is \(x^2 + y^2 = 16\), what is its radius?
a) `2` units
b) `4` units
c) `8` units
d) `16` units
Answer: b
Q`6`. The radius of a circle is `10` units. What is its circumference? Write your answer in terms of `π`.
a) `40π` units
b) `25π` units
c) `30π` units
d) `20π` units
Answer: d
Frequently Asked Questions
Q`1`. What is the definition of a circle's radius?
Answer: The radius of a circle is the distance from its center to any point on its circumference.
Q`2`. What is the formula for calculating the circumference of a circle using its radius?
Answer: The formula for the circumference of a circle using its radius is \( C = 2πr \), where \( r \) is the radius of the circle.
Q`3`. How can I determine the area of a circle if I know its radius?
Answer: You can determine the area of a circle if you know its radius by using the formula \( A = πr^2 \), where \( r \) is the radius of the circle.
Q`4`. What is the relationship between the radius and the diameter of a circle?
Answer: The diameter of a circle is twice the length of its radius. In other words, diameter \( d = 2 \cdot\) \( r \).
Q`5`. How is the radius used in geometry and mathematics?
Answer: In geometry and mathematics, the radius is a fundamental parameter used to calculate various properties of circles, such as circumference, area, and arc length. It is also used in formulas for calculating the volume and surface area of spheres, which are three-dimensional counterparts of circles.
Q`6`. Can the radius of a circle be negative?
Answer: No, the radius of a circle cannot be negative. It represents a distance, which is always non-negative. However, it can be zero if the circle degenerates into a single point.
Q`7`. What are some real-life examples of radius?
Answer: The concept of the radius is encountered in various real-life scenarios, such as measuring the radius of a bicycle wheel, a cooking pot, or the Earth. In architecture and engineering, the radius is used to describe the curvature of arches, bridges, and other structures. In sports, the radius is used to describe the curvature of sports arenas, tracks, and fields.