What is a Circle?
The definition of a circle is based on one important aspect. Every circle is a closed two dimensional shape, and all the points on the boundary, or circumference of the circle are equidistant from the center of the circle.
This definition can also be stated as the locus of points that are at the same distance from the center of the circle.
Having defined a circle, let’s define another part of the circle called the radius of the circle.
What is the Radius of a Circle?
Suppose we draw a circle, and then we draw a line from the center to the circumference of the circle. This line is known as the radius of the circle.
The radius is denoted by either ‘R’ or ‘r’. We have to note that the measure of the circumference and even the measurement of the area of a circle is calculated based on the radius of the circle.
When we are calculating the circumference of a circle, the formula for the circumference is,
Circumference, C = 2πr, in which ‘r’ is the measure of the radius of the circle.
When it comes to calculating the area of a circle, the formula for this calculation is,
Area, \(A=\pi r^2\) in which, as seen in the previous formula, ‘r’ is the radius of the circle.
Radius in Other Shapes
Radius as a measure is also found in multiple two dimensional or flat shapes, as well as in solid three dimensional shapes. When we look at two dimensional shapes the radius appears as a measure in semi-circles.
In three dimensional shapes, spheres, cones with a circular base and cylinders with circular bases have the radius as a measure. The radius in solid shapes is also used to calculate the surface area as well as the volume of these three dimensional shapes.
Definition of the Diameter
Another important part of a circle is the diameter. We can define the diameter of a circle as any line that joins two points on the circumference of the circle and this line also passes through the center of the circle.
We must also note that the diameter is twice the measure of the radius. This implies that ‘D’ which is used to represent the diameter is equal to ‘2r’.
Or, D = 2r or conversely, r = \(\frac{D}{2}\).
Hence, as shown in the image we can easily notice that the diameter is twice the length of the radius.
Note:
In addition, there is another line that connects two points on the circumference and which does not pass through the center of the circle. This line is known as a chord.
As shown in the image, a chord touches two points on the circle but it does not pass through the center of the circle.
Formulas Related to the Radius
There are a few formulas that are linked to the radius. Let us look at the formulas that require the value of the radius.
- Area of a circle – \(\pi r^2\)
- Circumference of a circle – 2\(\pi\)r or \(\pi\)D, in which, D replaces 2r.
Rapid Recall
Solved Examples
Example 1: There is a circular lake with an area of 56 square miles. Calculate the circumference of the lake. Take the value of π to be 3.14.
Solution:
The details provided,
Area of circular lake = 56 square miles
Hence, \(Area=\pi r^2=56\) square miles Write area formula of a circle
So, \(r^2=\frac{56}{3.14}\) Substituting 3.14 as the value of \(\pi\)
\(\Rightarrow r^2=17.83\)
This gives us,
r = \(\sqrt{17.83}\) ≈ 4.22 miles
Now that we have the radius of the circular lake, we can find the circumference.
Circumference = 2\(\pi\)r Write the formula for the circumference of a circle.
Substituting the values,
Circumference = 2 x 3.14 x 4.22 = 26.5 miles
Hence the circumference of the circular lake is 26.5 miles.
Example 2: The distance traveled by a race car around a circular track is 82 miles. Find the radius of the race track.
Solution:
The distance covered by the race car around the track is 82 miles.
This distance is the circumference of the race track.
Hence, 2\(\pi\)r = 82 miles Writing the formula for the circumference of a circle
Hence, \(r=\frac{82}{2\pi}\)
\(\Rightarrow r=\frac{82}{2~\times~3.14}\) Substituting 3.14 as the value of \(\pi\)
\(\Rightarrow r=\frac{82}{6.28}\)
\(\Rightarrow\) r = 13.05 miles
Therefore the radius of the race track is 13.05 miles.
Example 3: The wave produced when a rock is dropped on the surface of a pond travels 70 feet in all directions. Find the total area covered by the wave. Take the value of \(\pi\) to be \(\frac{22}{7}\).
Solution:
The wave on the surface covers a distance of 70 feet in all directions.
Hence the radius of the area covered by the wave will be 70 feet.
Using this value as the radius, we can find the measure of the area covered by the wave.
So, \(Area=\pi r^2\) Write the formula for the area of a circle
\(\Rightarrow Area=\frac{22}{7}~\times~70~\times~70\) Substituting the values of \(\pi\) and r
\(\Rightarrow Area=15,400\) square feet
Therefore the total area covered by the wave is 15,400 square feet
A circle is a closed two dimensional shape. The unique aspect of a circle is that any point on the boundary of the circle is equidistant from the center of the circle.
Many circles with a common center are called concentric circles.
The measure of the boundary of a circle is called the circumference of a circle. The formula to calculate the circumference of a circle is twice the radius times the value of pi.
When a sphere is sliced the surface of the cross section is a circle.