Construction of Regular Polygons: A polygon is a two-dimensional closed figure that is formed by using three or more line segments. Regular polygons are polygons with all equal measures; they have equal sides and angles.

We have different regular polygons such as equilateral triangle, square, pentagon, hexagon, etc. Polygons are constructed by using the ruler, compass, scale, etc. This article will describe how to construct regular polygons like equilateral triangles, squares, pentagons, and hexagons.

Regular Polygons

A polygon is a two-dimensional closed figure that is formed by using three or more line segments. A polygon should have at least a minimum of three line segments. Polygons are classified into regular, irregular, concave, and convex polygons.

Regular polygons are polygons with all equal measures; they have equal sides and angles. All regular polygons are convex polygons, I.e., the sum of the interior angles are less than \({180^ \circ }\). Some of the regular polygons are:

1. Equilateral triangle: Polygon with three equal sides
2. Square: A polygon with four equal sides
3. Hexagon: A polygon with six equal sides

A regular polygon has various parts such as vertices, sides, interior, exterior angles, diagonals. 

Let us explore this by using a regular hexagon:

1. Vertices: \(A, B, C, D, E, F\) (The intersection point of two sides)
2. Sides: \(AB, BC, CD, DE, EF, AF\) (Line segments joining the vertices)
3. Diagonals: \(BE\) (Line segments joining two vertices)
4. Interior angles: Shown with the blue colour.
5. Exterior angles: Shown with the red colour.

Properties of a Polygon

Regular polygons are polygons with all equal measures; they have equal sides and angles. Some of the properties of the polygons are listed below:

1. Polygons have a minimum of three sides.
2. Interior angle can be found by using \({180^ \circ } \times \frac{{(n – 2)}}{n} = n\) number of sides.
3. The exterior angle can be found by using \(\frac{{{{360}^ \circ }}}{n},n -\) number of sides.
4. The number of diagonals can be found by using \(\frac{{n(n – 3)}}{2},n – \) number of sides.

Construction of Polygons

Polygons are constructed by using the ruler, compass, scale, etc. The ruler is used to draw line segments, and the compass is used to measure and draw the arcs.

To construct regular polygons, we required the only length of one side.

Construction of Regular Polygons – Equilateral Triangle

The equilateral triangle is the regular three-sided polygon with three equal sides and angles. Let us discuss the construction of the equilateral triangle \(ABC\) with the help of suitable steps as discussed below:

1. Draw a line segment \(AB\) of any length with the help of a ruler and pencil.

2. Measure the length of the line segment \(AB\) with the help of a compass.

3. Draw an arc above the line segment \(AB\) by taking the same length and placing the tip of the compass at any one point of the line segment, say \(A.\)

4. Draw another arc above the line segment \(AB\) by taking the same length and placing the tip of the compass at the other end of the line segment, say \(B.\)

5. Now, mark the intersection point of two arcs as \(C.\)

6. Join the points \(A\) and \(C, B\) and \(C.\)

7. Thus, \(\Delta ABC\) is the regular polygon formed with three equal sides.

Thus, an equilateral triangle is a polygon with three equal sides and three equal angles, each being \({{{60}^ \circ }}\)

Construction of Regular Polygon – Square

A regular polygon with four equal sides is a square. The square is the regular polygon with four equal sides and four equal angles, each being \({90^ \circ }\). Let us discuss the construction of the square \(PQRS\) by using the suitable steps as mentioned below:

1. Draw a line segment \(PQ\) of any length.

2. Extend the line segment and draw two arcs on either side of the point \(Q,\) and named the points as \(U\) and \(V.\)

3. Draw arcs from the points \(U\) and \(V\) by using a compass with the same length.

4. Mark the intersection point as \(W.\)

5. Draw an arc of radius \(PQ\) above the point \(P\) by placing the tip of the compass at the point \(P.\)

6. Cut the line \(QW\) by placing the tip of the compass at the point \(Q\) with radius \(PQ.\)

7. Mark the intersection point as \(R;\) this gives the vertex of the square.

8. Draw an arc that intersects the previous arc drawn above the point \(P\) by placing the tip of the compass at the point \(R\) and radius \(PQ.\)

9. Mark the intersection point as \(S.\)

10. Join the points \(P\) and \(S, S\) and \(R.\)

11. Thus, \(PQRS\) gives the regular polygon with four equal sides.

Another way of constructing the square, inscribed in the circle is as follows:

1. Draw a circle of any radius.

2. Draw any diameter for the given circle.

3. Draw another diameter perpendicular to the given diameter.

4. Now mark the endpoints of the diameters as \(P, Q, R, S.\)

5. Join \(PQ, QR, RS, SP.\)

6. Thus, \(PQRS\) gives the required square.

Construction of Regular Polygon – Hexagon

Hexagon is a regular polygon with all six equal sides. All the six angles of the hexagon are equal, and each being equal to \({{{60}^ \circ }}\). Let us discuss the construction of hexagon with the help of suitable steps as mentioned below:

1. First, draw a circle of any radius.

2. Mark any point on the circle by placing the tip of the compass at the centre of a circle.

3. Draw another arc on the circle by placing the tip of the compass at the previous point, taking a radius equal to the radius of the circle.

4. Again draw another arc on the circle by placing the tip of the compass at the previous point, taking a radius equal to the radius of the circle.

5. Ans, repeat the steps mentioned above and plot the total six points \(A, B, C, D, E, F\) on the circle.

6. Join \(AB, BC, CD, DE, EF, AF.\)

7. Thus, \(ABCDEF\) gives the required hexagon.

Solved Examples – Construction of Regular Polygons

Q.1. Construct a regular polygon \(XYZ\) each side being \(6\) units.
Ans: We know that a regular polygon with three equal sides is called an equilateral triangle. Construction of equilateral triangle \(XYZ\) as mentioned below:
1. Step-1: Draw a line of length \(YZ=6\) units by using scale and pencil.
2. Step-2: Place the metal tip of the compass at point \(Y,\) and draw an arc of length \(6\) units by using a compass.
3. Step-3: Place the metal tip of the compass at point \(Z,\) and draw an arc of length \(6\) units by using a compass.
4. Step-4: Mark the intersection as \(X\) and join \(XY, XZ. XYZ\) is the required polygon.

Q.2. Construct an equilateral triangle with the length of sides \((8 \mathrm{~cm}).\)
Ans: The steps of constructing the equilateral triangle are mentioned below:
1. Step-1: Draw a line segment of a given length \((8 \mathrm{~cm})\) by using scale and pencil.

2. Step-2: Place the metal tip of the compass at one end of the line segment and draw an arc of the given length \(8 \mathrm{~cm}\) by using a compass.

3. Step-3: Place the metal tip of the compass at the other end of the line and draw an arc of the same length by using a compass.

4. Step-4: Mark the intersection and join with other points of the line. The figure gives the required equilateral triangle with a side \(8 \mathrm{~cm}\)

Q.3. Construct the regular polygon with four equals sides inscribed in the circle of radius \(5 \mathrm{~cm} .\)
Ans: We know that a regular polygon with four equal sides is known as the square. Steps of construction of regular polygon of four sides (square) inscribed in the circle are listed below:
1. Draw a circle with a radius of \(5 \mathrm{~cm} .\)

2. We know that diameter is twice the radius, so the circle’s diameter is \(2 \times 5 \mathrm{~cm}=10 \mathrm{~cm} .\)

3. Draw any diameter of the circle that passes through the centre of the given circle.

4. Now, draw another diameter that passes through the centre and is perpendicular to the previous diameter.

5. Mark the endpoints of the diameters on the circle \(P, Q, R, S.\)

6. Join \(PQ, QR, RS, SP.\) We know that the diagonals of the square are equal and perpendicular to each other.

7. Thus, \(PQRS\) is the required polygon.

Q.4. Construct a square of length \(3\,{\rm{cm}}.\)
Ans: Steps of construction of square are listed below:
1. Draw a line segment \(A B=3 \mathrm{~cm}\).
2. From point \(A,\) construct an angle of \({90^ \circ }\) by using the compass, draw a perpendicular line \(AX.\)
3. Cut the line \(AX,\) of length \(3 \mathrm{~cm}\) placing the compass’s tip at point \(A.\)
4. Mark the intersection point as \(D.\)
5. From point \(D,\) draw an arc of length \(3 \mathrm{~cm}\)
6. Now, draw another arc from point \(B\) of the same length cutting the previous arc at \(C.\)
7. Join \(CD.\)
8. Thus, \(ABCD\) is the required square.

Q.5. Construct a hexagon inscribed in a circle of radius \(5\,{\rm{cm}}\)
Ans:
1. First, draw a circle of radius \(5 \mathrm{~cm} .\)

2. Mark any point on the circle by placing the tip of the compass at the centre of a circle.

3. Draw another arc on the circle by placing the tip of the compass at the previous point, taking the same radius \((5 \mathrm{~cm} .)\)

4. Again draw another arc on the circle by placing the tip of the compass at the previous point, taking the same radius \((5 \mathrm{~cm} .)\)

5. And, repeat the steps mentioned above and plot the total six points \(A, B, C, D, E, F\) on the circle.

6. Join \(AB, BC, CD, DE, EF, AF.\)

7. Thus, \(ABCDEF\) gives the required hexagon.

Summary

In this article, we have studied the definitions of polygons and regular polygons. We also discussed the definitions of equilateral triangle, square, and hexagon. This article also gives the properties of the polygons. 

In this article, we have discussed the construction of the equilateral triangle, square, and hexagon with the help of suitable steps. This article also gives the solved examples that help us to construct the polygons easily.

FAQs

Q.1. What is a regular polygon?
Ans: Regular polygons are polygons with all equal measures; they have equal sides and angles.

Q.2. How do you construct an equilateral triangle?
Ans: Draw a line segment. Draw arcs from the endpoints of the line segment, join the intersection point with the endpoints.

Q.3. How do you construct a regular polygon having four sides?
Ans: Draw a circle, Draw two diameters perpendicular to each other, join the endpoints to get the regular polygon of four sides.

Q.4. What are tools used for construction?
Ans: The tools used for construction are a ruler, pencil and the compass, etc.

Q.5. What is a regular polygon that has six equal sides?
Ans: A regular polygon having six equal sides is a hexagon.

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