Lesson Area of n-sided polygon circumscribed about a circle

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<H2>Area of n-sided polygon circumscribed about a circle</H2>

In this lesson you will learn how to calculate the area of an &nbsp;n-sided polygon circumscribed about a circle.

<H3>Theorem 1</H3>The area of an &nbsp;n-sided polygon circumscribed about a circle equals half the product of the perimeter of the polygon and the radius of the circle.

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 <TD>
Proof

Let us consider the circle in a plane with the radius &nbsp;{{{R}}}&nbsp; and with the center&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

at the point &nbsp;O&nbsp; (Figure 1a). &nbsp;Let &nbsp;A1A2A3...A(n-1)An&nbsp; be an &nbsp;n-sided polygon

in the plane circumscribed about the circle, &nbsp;where the points &nbsp;A1, &nbsp;A2,

A3, . . . , &nbsp;A(n-1), &nbsp;An&nbsp; are the vertices of the polygon.

Figure 1a&nbsp; shows the &nbsp;5-sided polygon circumscribed about the circle.

Note that the polygon in the &nbspTheorem&nbsp is not necessary a regular polygon.

For the proof, let us connect the center of the circle with the vertices by the

straight segments &nbsp;OA1, &nbsp;OA2, &nbsp;OA3, . . . , &nbsp;OA(n-1), &nbsp;and &nbsp;OAn &nbsp;(blue lines in

the &nbsp;Figure 1b). &nbsp;Then the interior of the polygon is divided in &nbsp;n&nbsp; triangles

{{{DELTA}}}OA1A2, &nbsp;{{{DELTA}}}OA2A3, . . . , &nbsp;{{{DELTA}}}OA(n-1)An, &nbsp;{{{DELTA}}}OAnA1.

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 <TD>
{{{drawing( 240, 240, -1.5, 1.5, -1.15, 1.85, 
 circle( 0.0, 0.0, 1.0), 
 circle( 0.0, 0.0, 0.02), 
 locate(-0.03, -0.02, O),

line  ( 0.0, 0.0, 0.48, 0.866),
            locate( 0.25,  0.4, R),

line  (-0.52, -1.0,  0.52, -1.0),
            line  (-1.15,  0.0, -0.52, -1.0),
            line  ( 1.25,  0.1,  0.55, -1.0),
            line  (-1.15,  0.0, -0.13,  1.7),
            line  ( 1.25,  0.1, -0.13,  1.7),

locate(-0.60, -1.0,   A1),
            locate( 0.50, -1.0,   A2),
            locate( 1.28,  0.19,  A3),
            locate(-0.17,  1.865, A4),
            locate(-1.37,  0.09,  A5)
)}}}

&nbsp;&nbsp;Figure 1a. &nbsp;To the &nbsp;Theorem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>
{{{drawing( 240, 240, -1.5, 1.5, -1.15, 1.85, 
 circle( 0.0, 0.0, 1.0),

locate(-0.60, -1.0,   A1),
            locate( 0.50, -1.0,   A2),
            locate( 1.28,  0.19,  A3),
            locate(-0.17,  1.865, A4),
            locate(-1.37,  0.09,  A5),

blue(line  ( 0.0,  0.0, -0.52, -1.0)),
       blue(line  ( 0.0,  0.0,  0.52, -1.0)),
       blue(line  ( 0.0,  0.0, -1.15,  0.0)),
       blue(line  ( 0.0,  0.0, -0.13,  1.7)),
       blue(line  ( 0.0,  0.0,  1.25,  0.1)),

circle( 0.0,   0.0,  0.03), 
            locate( 0.03,  0.09, O),

green(line  ( 0.0,  0.0,  0.0,   -1.0)),
      green(line  ( 0.0,  0.0, -0.866, -0.5)),
      green(line  ( 0.0,  0.0,  0.866, -0.5)),
      green(line  ( 0.0,  0.0,  0.76,   0.65)),
      green(line  ( 0.0,  0.0, -0.85,   0.53))
)}}}

Figure 1b. To the proof of the Theorem 1

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</TABLE>

Therefore, &nbsp;the area of the polygon &nbsp;A1A2A3...A(n-1)An&nbsp; is the sum of the areas of the triangles &nbsp;{{{DELTA}}}OA1A2, {{{DELTA}}}OA2A3, . . . , &nbsp;{{{DELTA}}}OA(n-1)An, &nbsp;{{{DELTA}}}OAnA1:

{{{S}}} = {{{S[OA1A2]}}} + {{{S[OA2A3]}}} + . . . + {{{S[OA(n-1)An]}}} + {{{S[OAnA1]}}}.

The area of each of the triangles &nbsp;{{{DELTA}}}OA1A2, {{{DELTA}}}OA2A3, . . . , &nbsp;{{{DELTA}}}OA(n-1)An&nbsp; is half the product of the measure of the corresponding side of the polygon &nbsp;(the base of the triangle)&nbsp; and the radius &nbsp;{{{R}}}, &nbsp;because the radius drawn to the tangent point is perpendicular to the tangent segment &nbsp;(see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Circles/A-tangent-line-to-a-circle-is-perpendicular-to-the-radius-drawn-to-the-tangent-point.lesson>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</A>&nbsp; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site). So, you have

{{{S[OA1A2]}}} = |A1A2|*{{{R/2}}} + |A2A3|*{{{R/2}}} + . . . + |A(n-1)An|*{{{R/2}}} + |AnA1|*{{{R/2}}}= ( |A1A2| + |A2A3| + . . . + |A(n-1)An| + |AnA1| ) * {{{R/2}}} = {{{1/2}}}.{{{P*R}}},

where &nbsp;{{{P}}}&nbsp; is the perimeter of the polygon. &nbsp;It is exactly what the &nbsp;Theorem&nbsp; states.

For triangles, &nbsp;this Theorem was proved in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-area-of-a-triangle-via-the-radius-of-the-inscribed-circle.lesson>Proof of the formula for the area of a triangle via the radius of the inscribed circle</A>&nbsp; under the current topic in this site.

For quadrilaterals, &nbsp;it was proved in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-sircumscribed-quadrilateral.lesson>Area of a quadrilateral circumscribed about a circle</A>.

<H3>Example 1</H3>Find the area of a polygon circumscribed about a circle, &nbsp;if the radius of the circle is of &nbsp;5 cm and the perimeter of the polygon is of &nbsp;50 cm.

Solution

Apply the &nbsp;Theorem 1. &nbsp;According to this &nbsp;Theorem, &nbsp;the area of the polygon is half the product of the perimeter and the radius of the circle.

So, &nbsp;the area of the polygon is &nbsp;&nbsp;{{{1/2}}}.50*5 = 125 {{{cm^2}}}.

Answer. &nbsp;The area of the polygon is &nbsp;125 {{{cm^2}}}.

My other lessons on the area of polygons in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-regular-n-sided-polygon-via-the-radius-of-the-circumscribed-circle.lesson>Area of a regular n-sided polygon via the radius of the circumscribed circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-regular-n-sided-polygon-via-the-radius-of-the-inscribed-circle.lesson>Area of a regular n-sided polygon via the radius of the inscribed circle</A>

under the topic &nbsp;Area and surface area&nbsp; of the section &nbsp;Geometry, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-regular-polygons.lesson>Solved problems on area of regular polygons</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.