Construction of Circles: Take out the compasses from your geometrical box and open its arms. Now, take a point \(O\) on the page of your notebook and place the steel-end of the compasses at \(O,\) and let the pencil-end touch the page at the point, say \(P.\) Keeping the steel-end of the compasses fixed at \(O,\) rotate the other arm around, the pencil-end always touching the page. Continue the rotation till the pencil-end returns to the starting point \(P.\) The figure you will obtain on the page of your notebook is a circle. A circle is the most familiar figure in our surroundings. Also, in our day-to-day life, we come across several objects or figures that remind us of a circle.

An object suggesting a circle is said to be circular. The wheel, the rupee coin, the five rupee coin, the full moon etc., are all examples of circular shaped objects. In the following sections, we shall describe a circle in geometric terms and introduce other terms associated with a circle or its parts.

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Circle

Definition: A circle is the collection of all the points in a plane whose distance from a fixed point is always the same.

Different Components of a Circle

Centre: The centre of the circle is a point in the circle from which all the distances to the points on the circle are equal.

Radius: The circle’s radius is a line segment that joins the centre and any point on the circumference.

Diameter: The circle’s diameter is the line segment starting from any point on the circle’s circumference, passing through the centre and ending at the point on the circumference at the opposite side of a circle. The length of a diameter is twice the length of the radius in a circle.

Circumference: The fixed point is the centre of a circle, and the circle’s boundary is known as the circle’s circumference.

Chords: A line segment that joins any two points of the circle is known as the chord of the circle.

Construction of Circles Steps

We will discuss few basic constructions of the circle:

Construction of a circle with the radius \({\rm{ = 3}}\,{\rm{cm}}\)

Steps of construction:

You have to mark the point \(O\)

Then, with \(O\) as centre and radius \({\rm{ = 3}}\,{\rm{cm}},\) draw the circle first by using the compasses. \(O\) is the centre of the circle with radius \({\rm{ = 3}}\,{\rm{cm}}.\)

To construction a circle as the given line segment \(AB\) as its diameter.
Steps of construction:

1. Draw the perpendicular bisector of \(AB\) to intersect it at \(O.\)

Take \(OA=OB\) as radius, and with \(O\) as the centre draw the circle with the compasses.

To construct the circle to pass through the endpoints \(A, B\) of a line segment \(AB{\rm{ = 3}}\,{\rm{cm}}\) and having a radius\({\rm{ = 2}}{\rm{.4}}\,{\rm{cm}}\).

Steps of the construction:

1. Draw \(XY\) the perpendicular bisector of \(AB.\)
2. With \(B\) as the centre (or \(A\)) and radius \({\rm{ = 2}}{\rm{.4}}\,{\rm{cm}},\) draw an arc to cut \(XY\) at \(O.\)
3. With \(O\) as centre and radius \({\rm{ = 2}}{\rm{.4}}\,{\rm{cm}},\) draw the circle.

Construction of Concentric Circles

The circles that are drawn with the different measures of the radii and with the same centre are known as concentric circles. The centre is called the common centre.

In the given diagram, \({C_1}\) and \({C_2}\) are the two circles having the same centre \(O\) with the different radii \({r_1}\) and \({r_2}\) respectively.

Circles \({C_1}\) and \({C_2}\) are known as concentric circles.

The area bounded between the two circles is called the circular ring.

Width of the circular ring\( = OB – OA = {r_2} – {r_1}\left( {{r_2} > {r_1}} \right)\)

The steps of the construction of the concentric circles are given below:

1. First, draw the rough diagram and then mark the given measurements.
2. Then take any point \(O\) and then mark it as the centre.
3. Here, with \(O\) as the centre and draw the circle of the radius \(OA = \,3\,{\rm{cm}}{\rm{.}}\)
4. Finally, with \(O\) as the centre and then draw the circle of the radius \(OB = \,5\,{\rm{cm}}{\rm{.}}\)

Hence, the concentric circles \({C_1}\) and \({C_2}\) are drawn as shown below:

Construction of Isometric Circles

The steps of the construction of the isometric circles using the four centre method are given below:

1. First, you have to locate the centre of an ellipse.
2. Then you have to construct the isometric square.
3. After this, you need to construct the perpendicular bisector from each of the tangent points.
4. Now, locate the four centres.
5. Finally, you have to draw the arcs with these centres and the tangents to the isometric square.

Construction of Magic Circle

It is the arrangement of the natural numbers on the circles where the sum of each circle and the sum of the numbers on the diameter are similar. One of his magic circles was constructed from the number \(33\) natural numbers from the numbers \(1\) to \(33\) arranged on the four concentric circles, with the number \(9\) at the centre.

Solved Examples: Constructions of Circles

Q.1. Identify the length of the chord of a circle with radius \({\rm{7}}\,{\rm{cm}}{\rm{.}}\) Also, the perpendicular distance from the chord to the circle is \({\rm{4}}\,{\rm{cm}}{\rm{.}}\) Use chord length formula.
Ans: The given parameters are as follows:
Radius \({\rm{r = 7}}\,{\rm{cm}}\)
The perpendicular distance from the centre to a chord, \({\rm{d = 4}}\,{\rm{cm}}\)
Now, use the formula for the chord length as given below:
\({C_{len}} = 2 \times \sqrt {\left( {{r^2} – {d^2}} \right)} \)
\({C_{len}} = 2 \times \sqrt {\left( {{7^2} – {4^2}} \right)} \)
\( = 2 \times \sqrt {(49 – 16)} \)
\( = 2 \times 5.744\)
\( = 11.48\;{\rm{cm}}\)
Hence, the length of the chord will be \(11.48\;{\rm{cm}}.\)

Q.2. The circle radius is \({\rm{14}}\,{\rm{cm}}{\rm{,}}\) and the perpendicular distance from the chord to the centre is \({\rm{8}}\,{\rm{cm}}{\rm{.}}\) Identify the length of a chord?
Ans: Given radius, \({\rm{r = 14}}\,{\rm{cm}}\)
The perpendicular distance \({\rm{d = 8}}\,{\rm{cm}}\)
Now, the formula. Length of the chord\( = 2\sqrt {\left( {{r^2} – {a^2}} \right)} \)
Substitute.
Length of the chord\(\left. { = 2\sqrt {\left( {{{14}^2} – {8^2}} \right.} } \right)\)
\( = 2\sqrt {(198 – 64)} \)
\( = 2\sqrt {(132)} \)
\( = 2 \times 11.5\)
\( = 23\;{\rm{cm}}\)
Hence, the length of a chord is \( = 23\;{\rm{cm}}.\)

Q.3. The perpendicular distance from the centre of the circle to a chord is \({\rm{8}}\,{\rm{m}}{\rm{.}}\) Calculate the chord length if a circle’s diameter is \({\rm{34}}\,{\rm{m}}{\rm{.}}\)
Ans: Distance is \(d = 8\,{\rm{m}}{\rm{.}}\)
The diameter is \(D = 34\,{\rm{m,}}\) and radius is \(r = \frac{D}{2} = \frac{{34}}{2} = 17\;{\rm{m}}\)
The length of a chord\( = 2\sqrt {\left( {{r^2} – {d^2}} \right)} \)
The length of chord \(2\sqrt {\left( {{{17}^2} – {8^2}} \right)} \)
\( = 2\sqrt {(289 – 64)} \)
\( = 2\sqrt {(225)} \)
\( = 2 \times 15\)
\( = 30\;{\rm{m}}\)
Thus, the length of the chord is \({\rm{30}}\,{\rm{m}}{\rm{.}}\)

Q.4. If the coordinates of its centre are \(( – g, – f) = (5, – 3)\) and \(c = 9,\) then find the radius of the circle.
Ans: We know that radius of the circle is \(\sqrt {{g^2} + {f^2} – c} \)
Now, \(\sqrt {{{(5)}^2} + {{( – 3)}^2} – 9} = \sqrt {25 + 9 – 9} \)
\( = \sqrt {25} = 5\,{\rm{units}}\)
Hence, the radius of the circle is \(5\,{\rm{units}}{\rm{.}}\)

Q.5. Find the equation of the circle given \(f = 3,g = – 5\) and \(c = 11.\)
Ans: Given, \(f = 3,g = \, – 5\) and \(c = 11.\)
Substitute the given values in the general form of the circle’s equation \({x^2} + {y^2} + 2gx + 2fy + c = 0.\)
That is, \({x^2} + {y^2} + 2 \times ( – 5) \times x + 2 \times 3 \times y + 11 = 0\)
\( \Rightarrow {x^2} + {y^2} – 10x + 6y + 11 = 0\)
Hence, the required circle’s equation is \({x^2} + {y^2} – 10x + 6y + 11 = 0.\)

Summary

In this article, we have learnt about the definition of the circle along with its parts, then talked about the steps of the construction of circles. Then we discussed the construction of concentric circles followed by the concentric circle and the isometric circle. We also glanced at the construction of the magic circles. Finally, we have provided the solved examples along with a few FAQs on the construction of circles

The learning outcome of this article is that we learned how to construct the circles using different tools and methods.

Frequently Asked Questions (FAQs) – Construction of circles

Q.1. What are the different methods of constructing a circle?
Ans: The different methods of construction of the circle are given below:
1. When only the centre and length of the radius is given.
2. When the length of the diameter is given.
3. When the length of radius and chord is given.

Q.2. How do you construct the radius of a circle?
Ans: The construction of the radius of we require the ruler and the compasses.
1. Place the compass’s pointer at the starting point of the ruler and then extend the other end of the pencil, measuring the required length of the radius from the initial point.
2. Mark the point on the piece of the paper. This point is supposed to be the centre of the circle that you are about to construct.
3. Then place the pointer of the compass at the point.
4. Finally, turn the compass slowly through 360° to draw the circle.
5. Join the centre of the circle with any point on the circumference. This is the radius of the circle.

Q.3. What is the structure of a circle?
Ans: The circle is a plane figure bounded by one curved line, such that all straight lines drawn from a certain point within it to the bounding line are equal. The bounding line is called its circumference, and the point is the centre.

Q.4. What are the different parts of a circle?
Ans: The different parts of the circle based on the shape are written below:
1. Centre
2. Diameter
3. Radius
4. Circumference
5. Arc
6. Chord
7. Segment

Q.5. Why are circles used in architecture?
Ans: The architectural plan, when used symbolically, communicate its meaning through its shape. From prehistoric times and in many cultures, the circle, with its suggestion of the planets and other manifestations of nature, gained a symbolic, mystical significance and was used in the plans of houses, tombs, and religious structures.

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