Chapters
In this article, we will discuss a circle and all its elements in detail. Let us start with the definition of the circle.
A circle is defined as:
A closed curved line whose all points are located at the same distance from a fixed point called the center
A circle with the radius "r" is shown below. All the points in the circle are located at a fixed distance from its center c.
In mathematics and geometry, a circle is a figure which is measured in terms of its radius. A circle refers to a special type of ellipse that has zero eccentricity. The two foci of this two-dimensional figure are coincident. We can also say that a circle is a locus of the points drawn at an equal distance from the center. A plane is divided into interior and exterior regions through the circle. Some examples of circle-shaped objects include bangles, rings, coins, wheels, CDs/Discs, and buttons.
Properties of the Circle
The properties of the circle are as follows:
- The circle's outer line is equidistant from the center of the circle.
- A circle is divided into two equal parts by its diameter.
- Two circles are congruent if they have the same radii.
- The diameter of the circle is double its radius and is referred to as the largest chord of the circle.
Center Circle
The center is the point that is equidistant from all points in the circle. It is in fact the midpoint of the circle. In figure 1.1, the midpoint of the circle is "c" which is known as the center of the circle.
Radius of the Circle
The radius is the distance from the center to the edge of the circle. It refers to the line segment that joins the center of the circle to any point on the circle itself. We denote the radius of the circle by "R" or "r".
Chord
A chord is a line segment that joins two points in the circle. Both endpoints of the chord lie on the circle.
Diameter
The diameter is a chord that passes through the center of the circle. It is actually a line segment that passes through the center of the circle and both its endpoints lie on the circle. It is twice the radius, which means that d = 2r. If we are given the diameter of the circle, we can find the radius easily through the formula .
The diameter of the circle is shown in the figure below:
Arc
An arc is each of the segments of the circumference that a chord divides. It basically denotes the connected curve of the circle. It is usually associated with the lower arc.
Semicircle
A semicircle is each of the two equal parts in which a diameter divides a circle.
Area of a Circle
An area of the circle refers to the amount of space occupied by the circle. The formula for the area of the circle is:
, where
Example 1
Calculate the area of the circle whose diameter is 8cm.
Solution
We know that to calculate the area of the circle, the radius should be known. Radius is half of the diameter. If the diameter of the circle is 8cm, then its radius must be 4 cm. The formula for the area of the circle is:
Substitute r = 4 in the above formula to get the area:
Example 2
Calculate the shaded area of the figure below if the radius of the larger circle is 6 cm and the radius of the smaller circles is 2 cm.
Solution
To calculate the area of the shaded region, first, we will find the area of the bigger circle and the smaller circles. In the next step, we will subtract these areas. So, let us begin with finding the area of the bigger circle.
Radius of the bigger circle = 6cm
Area of the bigger circle =
Area of a smaller circle =
Area of four smaller circles =
Area of the shaded region = Area of the bigger circle - Area of the smaller circles
=
=
Example 3
In a circular park with a radius of 250 m, there are 7 lamps whose bases are circles with a radius of 1 m. The entire area of the park has grass except for the bases for the lamps. Calculate the total lawn area.
Solution
Radius of the circular park = 250 m
Area of the circular park =
=
Radius of one lamp = 1m
Area of one lamp =
=
=
Area of seven lamps =
Area of the park that has grass = Area of the park - Area covered by lamps
=
Example 4
Calculate the shaded area, knowing that the side of the outer square is 6 cm and the radius of the circle is 3 cm.
Solution
To find the area of the shaded region, first, we will find the area of the square and circle, and then we will take the difference.
Length of one side of a square = 6 cm
Area of a square =
Radius of a circle = 3 cm
Area of the circle =
Area of the shaded region = Area of the square - Area of the circle
=
Circumference of a Circle
The circumference of the circle can be defined as follows:
Distance around the circle is known as the circumference of the circle
Circumference of the circle is like the perimeter of the polygons. The perimeter of the polygons represents the distance around them like the circumference of the circle. The formula for calculating the circumference of the circle is:
, where r = 3.14
Example
The radius of the circle is 7 cm. Calculate its circumference.
Solution
Simply, substitute r = 7 cm in the below formula to get the circumference:
Equation of the Circle
Each line passing through the circle forms the line of reflection symmetry. This line also has rotational symmetry around the center at each and every angle. The equation of a circle in a plane is given below:
Here, (x,y) is the coordinate points
(h,k) represent coordinates of the center of the circle
"r" represents the radius of the circle
Example
Solution
The coordinates of the center of the circle are given, So, we can say that:
Radius of the circle = 7 units
Equation of the circle =
Substitute values of 'r', 'h', and 'k' in the above equation:
Expand this equation like this:
I’m just curious if the area between the polygon and the circumscribed circle has a name.
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