Interaction between Circle and Polygon: Inscribed, Circumscribed, Formulas

Interaction between Circle and Polyon: Certain geometric shapes can be created by combining circles with other geometric figures, such as polygons. There are two basic ways to link a circle and a polygon together. There is an inscription and a circumscription, respectively. Each vertex of a polygon that is inscribed in a circle crosses the circle. It is perpendicular to the circle for each side of a polygon that is encircled by one.

A circle that circumscribes a polygon is said to be a circumcircle around a polygon. A circle that inscribes a polygon is said to be an incircle into the polygon. In this article, let’s learn everything about Interaction between Circle and Polygon in detail.

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Circles and Polygons | Inscription and Circumscription

There are two types of markings: inscriptions and circumscriptions. When a polygon is inscribed in a circle, it signifies that each vertice crosses the circle. When a polygon is circumscribed around a circle, each of the polygon’s sides is perpendicular to the circle.

Here, the hexagon \(ABCDEF\) is inscribed in the circle with centre \(G\) and the quadrilateral \(ABCD\) is circumscribed around the circle with centre \(E\).

Circumscription and inscription are notions that can be expanded to three (or more) dimensions. Cones can be circumscribed around pyramids when their vertices coincide, and their bases coincide.

Relationship Between Circle and Polygon

Polygons (straight-sided geometric shapes) whose corners are on an exterior circle or whose sides are touched at one point each by an interior circle is circumscribed and inscribed, respectively (i.e., whose sides are all tangent to a circle).
Consider drawing a circle around a triangle, so that the circle touches all three vertices of the triangle. Therefore, the triangle is said to be inscribed within the circle, while the circle is said to be circumscribed around it.

Study Properties of Polygon

Concentric Circles

Circles with the same centre are known as concentric circles. One circle is inside another does not imply concentrically; they must share the same centre point. Any number of circles can be concentric if they all have the same centre.

Circle Inscribed in a Polygon Formula

The perimeter of a regular \(n-\)sided polygon inscribed in a circle equals \(n\) times the polygon’s side length, which can be calculated as:

\({P_n} = n \times 2r\sin \left( {\frac{{360}}{{2n}}} \right)\)

Circumcircle and Incircle of a Regular Hexagon Formula

• I. Circumference of circumcircle \(= 2 \pi a\) units
• II. Area of circumcircle \(= \pi a^2\) sq. units
• III. Circumference of incircle \( = 2\pi \times \frac{{\sqrt 3 a}}{2} = \sqrt 3 \pi a\) units
• IV. Area of incircle \( = \pi {\left( {\frac{{\sqrt 3 a}}{2}} \right)^2} = \pi \frac{{3{a^2}}}{4} = \frac{{3\pi {a^2}}}{4}\) sq. units,
• where \(a = \) side of the hexagon.

Polygon Inscribed in a Circle

A cyclic polygon is inscribed in a circle, and the circle is its circumscribed circle or circumcircle. The radius of the inscribed circle or sphere, if one exists, is the inradius or filling radius of a particular outer figure.

Inscribed Circles of Triangles

The largest circle contained within a triangle is called an inscribed circle. At one point, the inscribed circle will touch each of the triangle’s three sides. The incentre of the triangle, or the place where the angle bisectors of the triangle meet, is the centre of the circle inscribed in a triangle.

The circumscribed circle of a triangle is the circle that passes through all three of the triangle’s vertices. The circumcentre of the triangle, where the perpendicular bisectors of the sides meet, is the centre of the circumscribed circle.

To make the circumscribed circle, follow these steps:

• 1. Construct the circumcentre.
• 2. Construct a circle that crosses through one of the vertices and is centred at the circumcentre. The same circle should connect all three vertices.

How to Construct Angle Bisectors?

Draw a triangle with your pencil. Construct two of the angles’ angle bisectors. Why is the incentre of the circle the point of intersection of the two angle bisectors?

Construct the angle bisector of one of the angles using your compass and straightedge.

Repeat with another angle.

The incentre is the point where the angle bisectors cross. Because all three angle bisectors intersect at the same place, they do not need to be constructed. The third angle bisector adds nothing to the picture.

Constructing Perpendicular Lines

Construct a line perpendicular to one of the triangle’s sides that passes through the triangle’s incentre.

Make a line perpendicular to one of the triangle’s sides that passes through the incentre with your compass and straightedge.

Constructing Inscribed Circles

Construct a circle with the incentre centred at the point of intersection of the triangle’s side and the perpendicular line from the above problem.

Polygon Circumscribed About a Circle

The circumscribed circle, also known as the circumcircle of a polygon, is a circle that travels across all the polygon’s vertices. The circumcentre and circumradius are the names given to the circle’s centre and radius, respectively. A circumscribed circle is not present in every polygon.

Example: Construct a circumcircle for an equilateral triangle \(6\;\rm{cm}\) long.

Solution: Steps of construction:

1. Construct an equilateral triangle with a given dimension and name the triangle as \(PQR\).

2. Draw perpendicular bisector of the line segment \(PQ\).

3. Draw perpendicular bisector of the line segment \(QR\).

4. The circumcentre is the point of intersection. The label that intersection point as \(O\).
5. With \(O\) as centre and \(OR\) as radius, draw a circle.

6. The circumcircle of this triangle is the circle that goes through \(P\), \(Q\), and \(R\).

Solved Examples on Interaction Between Circle and Polygon

Q.1. Construction of a segment of a circle on a given line segment containing an angle \(\theta\).
Ans: Construction:
Step 1: Draw a line segment \(\overline {AB} \)
Step 2: At \(A\), take \(\angle BAE = \theta \). Draw \(AE\).
Step 3: Draw, \(AF \bot AE\).
Step 4: Draw the perpendicular bisector of \(AB\) meeting \(AF\) at \(O\).
Step 5: With \(O\) as centre and \(OA\) as radius, draw a circle \(ABC\).
Step 6: Take any point \(C\) on the circle; by the alternate segment’s theorem, the major arc \(ACB\) is the required segment of the circle containing the angle \(\theta\).

Q.2. Construct the circumcircle of the right triangle with the hypotenuse of a right-angled triangle is \(12\;\rm{cm}\) long, whereas the other side is \(5\;\rm{cm}\) long.
Ans: Steps of construction:
1. Construct a right-angled triangle with a given dimension and name the triangle as \(PQR\).

2. Draw perpendicular bisector of the line segment \(PQ\).

3. Draw perpendicular bisector of the line segment \(QR\).

4. The circumcentre is the point of intersection. The label that intersection point as \(O\).
5. With \(O\) as centre and \(OR\) as radius, draw a circle.

6. The circumcircle of this triangle is the circle that goes through \(P\), \(Q\), and \(R\).

Q.3. Construct the circumcircle of a triangle with a length of \(6.7\;\rm{cm}\) and two angles of \(75^\circ\) and \(55^\circ\) adjacent to one side.
Ans: Steps of construction:
1. Construct a triangle with a given dimension and name the triangle as \(PQR\).

2. Draw perpendicular bisector of the line segment \(PQ\).

3. Draw perpendicular bisector of the line segment \(QR\).

4. The circumcentre is the point of intersection. The label that intersection point as \(O\).
5. With \(O\) as centre and \(OR\) as radius, draw a circle.

6. The circumcircle of this triangle is the circle that goes through \(P\), \(Q\), and \(R\).

Q.4. Construct an incircle of the right-angle triangle \(PQR\) with sides \(8\;\rm{cm}\), \(6\;\rm{cm}\) and \(10\;\rm{cm}\).
Ans: Steps of construction:
1. Draw a line segment \(QR = 8\,{\rm{cm}}\).
2. With width \(6\;\rm{cm}\), draw an arc from the point \(Q\).
3. Take an arc of \(10\;\rm{cm}\) with centre \(R\), construct an arc.
4. Name the intersection point of these arcs as \(P\).
5. So, triangle \(PQR\) is the required right-angled triangle.
6. Now, bisect \(\angle P\) and \(\angle R\). Name the intersection point of these bisectors as \(O\).
7. Taking \(O\) as a centre, draw an incircle. Here, \(OS\) is the radius.

Q.5. Construct a circumscribing circle of a given regular hexagon with each side of the given regular hexagon is \(4\;\rm{cm}\).
Ans: Steps of construction
1. Using the given data, construct the regular hexagon \(ABCDEF\) with each side equal to \(4\;\rm{cm}\).
2. Draw the perpendicular bisectors of sides \(AB\) and \(AF\) which intersect each other at point \(O\).
3. With \(O\) as centre and \(OA\) as radius, draw a circle that will pass through all the vertices of the regular hexagon \(ABCDEF\).

Summary

In this article, we learnt about circles and polygons inscription and circumscription, the relationship between circle and polygon, concentric circles, a circle inscribed in a polygon formula, polygon inscribed in a circle, polygon circumscribed about a circle, solved examples on the interaction between circle and polygon and FAQs on the interaction between circle and polygon.

We understand how to make an inscribed and circumscribed polygon of a circle after reading this article.

FAQs on Interaction Between Circle and Polygon

Q.1. What is the relationship between a circle and a regular polygon?
Ans: Any regular polygon can be used to circumscribe a circle. A circle \(C\) passes through each vertex of the regular polygon, ensuring that all the polygon’s sides are included within the circle with boundary \(C\).
A circle can inscribe any regular polygon. A circle \(C\) touches each side of the regular polygon, and the circle is contained within the closed region bounded by the polygon.

Q.2. What does it mean for a circle to be inscribed in a polygon?
Ans: The incircle of any polygon is called its incircle, and the polygon is then referred to as a tangential polygon. A cyclic polygon is inscribed in a circle, and the circle is its circumscribed circle or circumcircle.

Q.3. What polygons can be circumscribed by a circle?
Ans: A circumscribed circle is not present in every polygon. Because its vertices are concyclic, a polygon with one is termed a cyclic polygon, or occasionally a concyclic polygon. There are cyclic triangles, regular simple polygons, rectangles, isosceles trapezoids, and right kites.

Q.4. Can any polygon be inscribed in a circle?
Ans: For any \( n \geqslant 3\), every circle has an inscribed regular polygon with \(n\) sides, and every regular polygon may be inscribed in some circle (referred to as its circumcircle).

Q.5. Why is a circle not a polygon?
Ans: A polygon is not a circle. A polygon is a closed figure on a plane made up of a finite number of end-to-end line segments. Because a circle is curved, it can’t be made from line segments and hence doesn’t meet the criteria for being a polygon.

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