Basic Geometrical Ideas

Introduction to Circles: Properties & Area of a Circle

Circles in maths refer to a closed 2D shape made up of points that are equidistant from a central point. A common circle we see in everyday life is the wheel, which is perhaps the most significant invention transferred to us from the Mesopotamian civilisation! Historically, there are references to circles and the theorems on circles dating back to 650 BC. the word circle is derived from the Greek word ‘’kirkos’ that means ‘ring’. Here, we delve into the basics of circles, right from their definition to how to draw circle, parts of circle, circle formulas, and properties of circles.

Table of Contents:

Defining a Circle

Circle Shaped Objects

How to Draw Circle?

Parts of Circle

Circle Formulas

Area and Circumference of a Circle

Properties of Circles

Solved Examples

Frequently Asked Questions on Circles

Defining a Circle

Circles in maths refers to a set of points in a plane that are at the same distance from a particular point at the centre. Few real-world examples of circles include coins, dining plates, pizza, and others. The distance from the centre and the points is called radius. We’ve covered how to draw circle in the next section. In coming sections, we will cover circle formulas and properties of circles.

The line that divides a circle into half is called its diameter. If we fold a circle into half, the parts that make up either side of the circle will be exactly the same in terms of area and shape. So, the diameter is the line of symmetry and the circle is said to be a symmetrical figure. The line that divides a figure into equal symmetrical parts is called the line of symmetry. In a circle, we can draw an infinite number of lines of symmetry. This means, a circle can be divided into similar or to put it more accurately, ‘congruent’ parts.

Apart from having infinite lines of symmetry, a circle also has rotational symmetry. Rotational symmetry is the property in which a figure if rotated around its centre appears exactly the same as it appeared before it was rotated. If we take a circle and rotate it, it will appear the same as it was before rotating. So, a circle has rotational symmetry.

Circle Shaped Objects

Circle is an important shape in geometry. Out of all the geometric shapes and figures, circles in maths are a fascinating form present around us. Be it the sun, moon and even a bottle cap, circular objects are quite common. Other examples of circular objects include coins, wall clocks, wheels, dartboards, rings, cakes, bangles, buttons and many more. In the next section, we’ll learn how to draw circle.

 

How to Draw Circle?

Drawing a perfect circle isn’t easy by hand. If you want to draw circles in maths, a compass (a geometric tool) is preferred. Here are the steps on how to draw circle:

  1. First, take the compass and a ruler.
  2. Depending on the radius (we’ll learn about this in the parts of circle section) of the circle you wish to draw, measure the distance between the tip of the compass point and the attached pencil using the ruler.
  3. Next, mark a centre point on the paper.
  4. Place the tip of the compass needle on the point and press it.
  5. Keeping the needle tip in place, rotate the pencil around the point to create a circle.

Here’s a video to learn more about circles in maths: https://youtu.be/tEleNEkB6ds

Parts of Circle

Now that we’ve seen how to draw circle, let’s move on to parts of circle.

Diameter

The diameter can be termed as a line which is drawn across a circle passing through the center.

Radius

The distance from the middle or center of a circle towards any point on it is a radius. Interestingly, when you place two radii back-to-back, the resultant would hold the same length as one diameter. Therefore, we can call one diameter twice as long as the concerned radius.

Circle Area

In circles in maths, the area can be stated as π times the square of the radius. It is written as: A = π r2. Taking into consideration the diameter: A = (π/4) × D2 In the previous section, we saw how to draw circle, the border of the circle is called circumference of a circle.

Chord

A line segment that joins two points present on a curve is called as the chord. In geometry , the usefulness of a chord (parts of circle) is focused on describing a line segment connecting two endpoints which rest on a circle.

Tangent & Arc

A line that slightly touches the circle on its travel to a different direction is Tangent. On the other hand, a part of the circumference is an Arc.

Sector & Segment

A sector is a part of a circle surrounded by two radii of it together with their intercepted arc. The segment is that region that is enclosed by a chord together with the arc subtended by the chord.

Common Sectors

In geometry, Quadrant and Semicircle are known as two special versions of a sector. Other parts of circle include         quadrant and semicircle.

A circle’s quarter is termed as Quadrant. Half a circle is known as a Semicircle.

So far, we have covered parts of circle. Let’s move on to formulae and equations associated with circles.

Area and Circumference of a Circle

Area of a circle

The area of circles in maths refers to the entire space enclosed by the circle. The circumference of a circle refers to the perimeter or outline of the circle. There are several ways to calculate the area of circles in maths. One method to calculate these circle formulas is by using the rectangle method:

First, draw a circle. Refer to the previous section on how to draw circle. Next, divide the circle into 8 equal parts and then arrange them in a rectangular form.

Now, if we can calculate the area of the rectangle, it will be equal to the area of the square.

Area of a rectangle is = length × breadth

The breadth of the rectangle = radius of the circle (r)

Compare the length of the rectangle and the circumference of the circle shows that length of the rectangle is ½ the circumference of the circle. Based on this, we can write the formula of the area of circles in maths as:

Area of circle = Area of rectangle = ½ (2πr) × r = πr2, where r, is the radius of the circle

Value of π is 22/7 or 3.14.

In the case of concentric circles, the region between two concentric circles is called the area of the ring. Concentric circles are those circles that have the same centre but different radii.

Circumference

Circumference of a circle is the length of the border of the circle. In section, how to draw circle, we’ve seen that the border of the circle that we drew is the circumference. It is also sometimes called ‘perimeter’ but the word perimeter is more apt for figures that are made of straight lines, like rectanges, squares and polygons. Circle formulas for the circumference of a circle is C = πd = 2πr, where π = 3.1415.

Learn more about parts of circle (circumference, tangent, and so on) and circle formulas.

Circle Formulas

  • Area of a circle: Area of a circle refers to the space enclosed by the circumference of the circle. For circles in maths, area can be calculated using radius or diameter (parts of circle). A = π r2 or A = (π/4) × D2
  • Circumference of circle: The circumference of a circle is the perimeter or the whole length of the arc. Circle formulas for circumference = 2πr
  • Arc length: An arc is a section of the circumference. Arc length = θ × r, where θ is in radians.
  • Area of a sector: A sector makes an angle θ at the center. Area of a sector = (θ × r2)÷ 2.
  • Length of chord: A chord is line segment touching the circle at two different points on its border. Length of chord = 2 r sin(θ/2).
  • Area of Segment: A segment is a region formed by a chord and the corresponding arc covered by that segment. Circle formulas for area of a segment = r2(θ − sinθ) ÷ 2.

Equation of Circle

We have seen circle formulas in the previous section. To derive the equation of circles in maths, let us draw a circle of radius ‘r’ (we have covered definition of radius in parts of circle section), and put it on the coordinate plane. We have seen how to draw circle in a previous section. Take the centre at O(h,k) and then a point A on the circumference and assume its coordinates to be (x,y) and join OA. Now, we will assume a point B inside the given circle such that OB is ⊥ to AB which gives us a right angled triangle AOB with right angle at B. This makes OA, hypotenuse.

From Pythagoras Theorem, we can say that

OB2 + AB2 = OA2

Here, OA = r; OB = difference in the x-coordinate = x – h; AB = difference in the y-coordinate = y – k, which means, (x−h)2+(y−k)2=r2, and this is the equation of circles in maths. Now if you’re thinking that this is just at one point, then stop yourself right there. You can take a similar point on any part of the circumference and still you’ll get the same result.

So, this equation is more like a collection of all such points on the circumference and is, hence, the equation of circles in maths. The centre of this circle is at (h, k) and if you move it to the origin then the equation will become x2+y2=r2

Equation of circle in parametric form – Parametric Equation of circle with centre (h,k)  and radius R is given by x=h+Rcosθ & y=k+Rsinθ, where θ is the parameter.

In the next section, we’ll cover the different properties of circles.

Learn more about circles in maths and parts of circle, including circle formulas.

Properties of Circles

Focusing on geometry, there are numerous properties of circles in maths. Further, the relation of it to straight lines, polygons, and angles can also be proved. All of these facts together are properties of the circle. Now that we know how to draw circle, let us try to learn the primary properties of circles in order to enhance our understanding.

  • Circles holding equal radii are known to be congruent.
  • Circles with different radii are seen as similar.
  • In a circle, the central angle that intercepts an arc is known to be double to any inscribed angle which intercepts the same arc.
  • Chords are parts of circle that are equidistant from the center are known to be of the same length.
  • A radius perpendicular to a particular chord does bisect the chord.
  • The tangent is always at right angles to the radius considering the point of contact.
  • Two tangents which are drawn on a circle from an exterior point are equal in length.
  • The circumference of two diverse circles in maths is proportional to the corresponding radii.
  • The angle subtended at the circle’s center by its circumference is known to be equivalent to four right angles.
  • Arcs associated with the same circle are termed proportional to their corresponding angles.
  • Another significant one of the properties of circles is that equal chords hold equal circumferences.
  • Equal circles hold equal circumferences.
  • Radii linked to the same or equal circles are known to be equal.
  • The longest chord is the diameter.

Solved Examples for Circles in Maths

Question 1: Calculate the area of a circle having radius 1.2 m.

Answer: The area of a circle here can be calculated using the circle formulas, Area = πr2

Therefore, A = π × 1.22 = 3.14159… × (1.2 × 1.2) = 4.52 (to 2 decimals).

Question 2: If the number of units in the circumference of a circle is same as the number of units in the area of a circle, then the radius of the circle will be:

Answer: Circumference of circle = 2πr and Area of circle = πr2

Given, the circumference of the circle = area of the circle.

For circles in maths, 2πr=πr2

∴ r= 2

Question 3: If two circles have their radii in the ratio of 3:2, then their areas are in the ration of?

Answer: D =  9/4, area of a circle = πr2

a1 / a2 =  r12 / r22 = (3)2 / (2)2 = 9 / 4

 

Question 4: If the radius of a circle is 5 cm and the measure of the arc is 110˚, what is the lengthof the arc?

Answer: Based on circle formulas, Arc length = 2πr × m/360° = 2π × 5 × 110°/360° = 9.6 cm

 

Question 5: If we have a circle of radius 20 cm with its centre at the origin, the circle can be described by the pair of equations?

Answer: We know that for parametric form of equation of circle, x=h+Rcosθ & y=k+Rsinθ

Here, since the centre is at (0, 0), so, h = k = 0 and it is already mentioned that radius is 20 cm. Thus, x=20cosθ &  y=20sinθ are the required pair of equations

Frequently Asked Questions on Circles

Question 1: How to find the area of a circle?

Answer: The formula for finding the area of circles in maths is πr2. Moreover, you can find the area of a circle by multiplying the value of pi that is 3.14 or 22/7 by the square of the radius. However, in place of radius. diameter of a circle is mentioned in the question then half it as the radius is half of the diameter.

Question 2: Is the circumference of a circle squared?

Answer: Usually, we can define pi as the ratio of the circumference of a circle to its diameter, hence the circumference of a circle is pi times the diameter, or 2 pi times the radius. So, this provides a geometric proof that the area of a circle really is πr2.

Question 3: Give a simple definition of circles in maths?

Answer: It refers to a round 2D shape in which all the points on the edge of the circle are at the same distance from the center. We have already covered how to draw circle in a previous section. Moreover, the diameter of a circle is equal to twice its radius. In addition, the circumference of a circle is the line that goes around the center of the circle. Parts of circle include diameter, radius, chord, tangent, etc.

Question 4: Define the radius of a circle?

Answer: It refers to the distance between the centers to any point on its circumference. The simplest way to find radius is to half the diameter.

Question 5: What are the properties of circles?

Answer: When we look at the essential properties of a circle, we see that it has many. A circle is congruent when it had equal radii. Moreover, looking at the diameter of the circle, we see that it is the longest chord of a circle. Further, equal chords and equal circles have the equal circumference.

Question 6: What is 2π?

Answer: A full circle comprises of 2π radians (approximately 6.28). You will often come across this in circle formulas. Moreover, always remember that an arc of a circle is what defines a radian. Further, the length of the arc is equal to the radius of a circle.

Question 7: What is 90 degrees in a circle?

Answer: First, let us make clear that a circle divides into 360 equal degrees. Thus, this means that a right angle is 90°. So, the angle of an equilateral triangle comprises 60 degrees. Then again, scientists, engineers, and mathematicians usually measure angles in radians.

Question 8: Is circle a polygon?

Answer: Polygon refers to a closed plane figure that has three or more sides that are all straight. Besides a circle is not a polygon because it does not have straight side, instead it has round sides.

We have covered various aspects of circles here, including how to draw circle, parts of circle, circle formulas and properties of circles.

 

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