Geometry: Inscribed and Circumscribed Polygons
This is a free lesson from our course in Geometry
 
   
Introduction: In this lesson you’ll explore concepts and relationships about the Inscribed and circumscribed polygons. You may recall from earlier geometry learning that polygon is a closed plane figure bounded by straight line segments as sides. A regular polygon may be a polygon which is equiangular (all angles are equal in measure) or equilateral (all sides have the same length). Also a regular polygon may be convex or star. A regular n-sided polygon has rotational symmetry of order when all the vertices of a regular polygon lie on a common circle, i.e., every regular polygon has a circumscribed circle. Along with the property of equal-length sides, this means that every regular polygon also has an inscribed circle. The examples of simplest regular polygons are the equilateral triangle, the square, the regular pentagon etc. You can also take help from Encyclopedia or Wikipedia for higher level details. Another important topic covered here is inscribed angle.  (More text below video...)
<h2> Geometry - Inscribed and Circumscribed Polygons</h2> <p> polygon, circle, area, inscribed, preperties, find, video, segment, tangent, circumscribed, inscribed polygon, solution, example, explaination, circumscribed polygon, geometry help, TA-CO method, practice questions, quizzes</p> <p> An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the circle. A circumscribed regular polygon is a polygon whose segments are tangent to a circle.</p>
Other useful lessons:
Measuring Circles
Equation of a circle in standard form
Circles- arcs, chords, tangents, sector, segment, secant
(Continued from above) Inscribed angle: an inscribed angle in Geometry is formed when two secant lines of a circle intersect on the circle. For easy understanding- ‘inscribed angle may be defined by two chords of the circle when each side of a polygon is tangent to a circle, and the polygon is circumscribed about the circle (Fig: 1). Inscribed angle
 
Inscribed and Circumscribed Polygons:
In this lesson, you’ll learn the concepts of Inscribed and Circumscribed polygons and their properties which will prove a big help to do geometry home work, as well as to solve real-world problems. You will explore all this with the help of some examples and practice questions with solution. Not that this might seem complicated here in text, but once you have instructor explain it to you in the voice and handwriting and using video, it would look easy and much simpler to understand.
An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the circle and each of its sides is a chord. A circumscribed regular polygon is a polygon whose segments are tangent to a circle. Thus these two differ in the sense- "Inscribed is a polygon inside a circle with all points on a given point in the circle and Circumscribed is a circle inside a polygon with any given point touching just one point on the polygon". (Fig: 2, 3).
Inscribed and Circumscribed Polygons
Further in this section you’ll look at the related summary on- Inscribed polygon, Circumscribed polygon, Circumcircle about a polygon, Incircle into a polygon, Radius of an incircle into a triangle, Radius of a circumcircle about a triangle, Regular polygon and its center, Apothem, Relations between sides and radii of a regular polygon.
Also the circumference, passing through vertices of a polygon (Fig: 2), is called a circumcircle and the circumference, for which sides of the polygon are tangents (Fig: 3), is called an incircle. E.g. in case of a triangle as polygon, radius r  of an incircle is expressed in terms of sides a, b, c of the triangle as:
r = , where s = (a + b + c) / 2
AND
radius R of a circumcircle is expressed as,

Look at the following examples to have deeper understanding to help geometry home work:
Example 1: if you are asked to find out if it is possible to cut out a square with a side 30 cm from a circle with a radius of 20 cm. To solve it, the steps are as follows:
Recall from above learning that the biggest square that is included in a circle, is an inscribed square. Now take help of the formula:
The relation between sides and radii of regular polygon is:  a = 2 * radius =  1.141 r
Since radius here is 20 cm, side equals to 1.141 * 20 = 28 cm approx.
Therefore, it is possible to cut a square with side 30 cm from a circle of 20 cm radius or 40 cm diameter, as the final answer.
Example 2: here you may learn how to find out the area of inscribed and circumscribed polygons using the TA-CO method method, in which the cut out (CO) is subtracted from the total area (TA) to get the required area.
E.g. in case of a circumscribed regular polygon- if a square floor of side 6m is covered by a circular shape rug, then the area of the floor uncovered by the rug is 62 - 8 (3)2 = (36 - 9)m2.
Example 3: identify two polygons inscribed in the circle.
Step 1: In case of polygon inscribed in a circle, each of its sides is a chord.
Step 2: For SQP, SQR, SRP, QRP and Quadrilateral SPQR, each side is a chord of the circle.
Step 3: For SRO, SPO, QPO, QRO, some of the sides are not the chords of the circle.
Step 4: Therefore, SQP, SQR, SRP, QRP and Quadrilateral SRQP are the inscribed polygons.
Hence the two polygons inscribed in the circle are: SQR & SQP, as the final answer.
Inscribed Angle:
An inscribed angle of a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle (Fig: 5).
Inscribed Angle
Theorems:
The measure of an inscribed angle of a circle is equal to one-half the measure of its intercepted arc.
Proof   If one of the sides of the inscribed angle contains a diameter of the circle.
Consider first an inscribed angle (Fig: 6), ABC, with diameter of circle O.
mOAB = mOBA = x
=> mAOC = x + x = 2x
Also, mAOC = m = 2x 
=>  mABC = x = ½ m
Corollary
• an angle inscribed in a semicircle is a right angle.
• if two inscribed angles of a circle intercept the same arc, then they are congruent.
• if a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.
Example : find the measure of arc in Fig: 7.
By the theorem stated above, A and C are supplementary. Therefore, C equals 95.
From the theorem, measure of an arc is double that of its inscribed angle. Therefore, arc 190, as the final answer.
Remember: 
• It is possible to inscribe a circle in a quadrangle, if sums of its opposite sides are the same. E.g. in case of a rhombus (square), the center of an inscribed circle is located at the intersection point of diagonals.
• It is possible to circumscribe a circle around a quadrangle, if sum of its opposite angles is equal to 180. E.g. in case of a rectangle (square), the center of circumscribed circle is located at the intersection point of diagonals. But in case of trapezoid, it is possible to circumscribe a circle only if the trapezoid is isosceles one.
The video above will explain more in detail with the help of several examples; about the Inscribed and Circumscribed Polygons, which will give you deeper understanding and also support Geometry homework help.
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