One of the most common shapes we all know is a circle. What do you imagine when you hear the word circle? You probably imagine something like the round shape. For example, the shape of the ball or a pizza. You are absolutely right. We all know what a circle looks like but there is more than that to a circle than just its shape. In this tutorial, we are going to learn some amazing things about the geometry of a circle like what is its radius, diameter, area, arc, equations, etc. Just like you love pizza or playing with your ball, you are going to love this topic as well.
So, let’s get going!
What is a Circle
If we talk about the definition of circle, a circle is a 2-dimensional, closed plane curve in which every point on the curve is at an equal distance (equidistant) from its center.
Radius of circle
The distance from the center of the circle to the circle is known as the radius. Let us denote radius by r.
Double that distance and you will get the diameter of the circle. Thus, the distance of the line segment which has two endpoints on the circle and is passing through the center of the circle is known as diameter. Let us denote diameter by d.
d = 2r, diameter is 2 times the radius.
Circumference of a circle
After knowing the radius and diameter, the next important term in the geometry of a circle is circumference. The circumference or perimeter is the distance around the circle. Consider you have a circular rubber band, if you cut it and straighten it then the length of that straightened circle will be the circumference of the circle. We refer to the length of the shapes like square or triangle as the perimeter. So, it is just a fancier word for the perimeter. Let us denote the circumference by C.
The formula for the perimeter of the circle is given as:
C = πd
In terms of radius, perimeter or circumference is given as
C = 2πr
The central angle in a circle is the angle between two radii (plural of radius) of the circle with vertex at the circle’s center.
The central angle of the entire circle is 360o.
Let us now talk about some of the important topics related to the circle.
Tangent to the circle
A tangent is a line that touches the circle at only one point. It is always outside the circle. It never enters into the circle. The common point of the tangent and the circle is the point at which they both intersect each other and the point is known as the point of tangency. The radius and the tangent are perpendicular to each other at that point. The tangent to the circle is used in many problems like finding the equation or slope of the tangent line.
Do you know how many tangent a circle can have?
A circle can have an infinite number of tangents, as there are infinite points in the periphery of a circle and at each point, we can draw a tangent. But, note that no tangent can be drawn through the circle, as told earlier because a tangent always passes from only one point in any curve.
Secant of a circle
A secant in a circle is a line that touches two points of the circle and passes through it. Thus, there will be two points of intersection between the circle and the secant. If A and B are the two points of intersection and if A = B i.e. A and B coincides, then the secant becomes a tangent.
A chord is a line segment that has two endpoints on the circle. It is completely inside the circle. If it passes through the center of the circle, it becomes the diameter.
- A chord is different from the secant is the sense that the endpoint of the chord lies on the circle whereas the endpoint of the secant lies outside of the circle.
- Diameter is the longest chord in a circle.
How to find area of a circle
The surface area of the circle is calculated by the following formula
Area = A = πr2
Area of a circle with diameter
The surface area of a circle can be written in the form of diameter, by substituting the ‘r’ by ‘d/2’.
A = π(d/2)2
A = (π/4)*d2
Can we find the volume of the circle?
No, we can not find the volume of a circle because a circle is a 2 dimensional in nature but we can find the volume of only 3-dimensional objects. The 3D object which is analogous to the circle is a sphere, which is formed by rotating the circle about its vertical or horizontal axis passing through the origin.
Let us talk about Arc now
Arc of a circle is defined as the portion of the circumference of the circle.
Arc can be measured in two ways. These are:
- By measuring the central angle
- By measuring the length
Measure of angle
The measure of the central angle is taken as the arc. In other words, If A and B are the endpoints of an arc, then the angle formed between the radii to these points with vertex at the center is the arc.
There are two different types of arcs based on the measure of the angle.
- Major arc
- Minor arc
If the measure of an angle is greater than 0o and less than 180o, it is referred to as minor arc.
If the angle measure is greater than 180o and less than 360o, it is known as the major arc.
If the central angle is 180o, it is known as a semi-circle. As the name suggests, the circle is cut into half.
Measure of length:
As the name suggests, this measurement of the arc takes into account the distance or the length of the arc just like we measure the circumference.
But how to find the length of an arc?
Let us say ‘s’ represents the length of the arc and ‘θ’ represents the central angle. Then, arc length can be found by multiplying the radius with the angle subtended by the arc.
Arc length = s = rθ
Remember θ is measured in radians here. The conversions from degrees to radians can be done by the formula,
2π = 360o
For example, if the angle in degrees is 120o, then,
(120o / 360o) × 2π = 2π / 3 radians.
Sector and its area
What is sector of a circle?
Sector is the portion of the circle which is enclosed by the two radii and one arc. Sector can be considered as slice of a circular cake if it is cut up to the center of the circle.
Area of sector of a circle – formula
The area of the whole circle = A = πr2
Let us arrange this formula as follows:
A = (θ / 2π) × πr2
This represents the area of any portion circle. If θ = 2π (remember θ is measured in radians), area becomes πr2, which is the area of the whole circle. If θ = π, the area becomes πr2/2, which is the area of the semi-circle i.e. the half the area of the complete circle.
Simplifying the formula, we get,
Area of sector of circle = (θ / 2) × r2
= (θ / 2) × r × r
We know that s = rθ,
So, modified formula for area of sector of a circle = 1/2sr
Segment and the Area of the Segment
What is segment of a circle?
A segment is a region enclosed by the chord.
How to find area of segment of a circle?
If we take the endpoints of the chord and draw two radii from the center of the circle to that points. Then,
Area of the sector = Area of the segment + Area of the remaining portion
The remaining portion is actually a triangle.
Asector = Asegment + Atriangle
Asegment = Asector – Atriangle
This formula reduces to
Asegment = [(θ – sin(θ)) / 2] × r2 where θ is in radians.
Let’s solve some questions on circle to understand the topic in more detail.
If the measure of the angle is 180o. What will be the length of the arc of the circle if the radius is 4 cm?
Angle = θ = 180o = π radian
radius = 4 cm
Measure of the length of the arc is given as:
s = rθ
s = 4 × π
s = 4 × 3.14159
s = 12.56 cm
The length of the arc = s =12.56 cm
Find the area of the sector if the central angle is 30o and radius is 9cm.
radius = r = 9 cm
θ = 30o
We know that,
2π = 360o
(30o / 360o) × 2π = π / 6 radians.
Area of the sector is given as:
Area of sector= 1/2sr
s = rθ
Asector = 1/2 × r2 × θ
= 1/2 × 92 × (π/6)
Area of the sector is 21.20cm
Equation of the circle
One of the most important thing while understanding any geometry is the equation. After learning this topic you will have complete knowledge in the geometry of the circle.
Standard Form of the Equation
Consider a circle in the Cartesian coordinate system. Let the center of the circle be at the point (a, b) and let r be its radius. If (x, y) is any point on the circle, then, the equation of the circle in the standard form is given as:
(x – a)2 + (y – b)2 = r2
If the circle is located at the origin i.e. (0,0), the equation becomes,
(x – 0)2 + (y – 0)2 = r2
x2 + y2 = r2
General Form of the Equation of the Circle
The general equation of the circle is:
x2 + y2 + 2gx + 2fy + c = 0
The center of the circle is (-g, -f) and r = √(g2 + f2 – c)
We can go from standard form to general form by expanding the perfect square and we can go from general form to standard form by the method of completing the square.
Its time to solve some questions on equation of circle.
Write the equation x2 + y2 – 4x – 8y + 8 = 0 in the standard form.
The above equation can be converted into the standard form by using completing square method.
x2 + y2 – 4x – 8y + 8 = 0
x2 – 4x + y2 – 8y + 8 = 0
x2 – 2(2)x + y2 – 2(4)y + 8 = 0
Adding 4 and 16 on both sides
x2 – 2(2)x + y2 – 2(4)y + 8 + 4 + 16 = 4 +16
x2 – 2(2)x + 4 + y2 – 2(4)y + 16 = 4 + 16 – 8
We know that a2 + 2ab + b2 = (a+b)2
(x-2)2 + (y-4)2 = (√12)2
The standard form of the circle is (x-2)2 + (y-4)2 = (√12)2
What will be the coordinates of the center and the radius if the equation of the circle is
(x + 3)2 + (y – 6)2 = 25
The standard form of the equation is given as
(x – a)2 + (y – b)2 = r2
The coordinates of the center are (a, b) and the radius is r.
Comparing the given equation with the standard form.
(x – (-3))2 + (y – 6)2 = 52
The coordinates of the center are (-3, 6) and the radius is 5.