Lesson Area of a regular n-sided polygon via the radius of the circumscribed circle
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<H2>Area of a regular n-sided polygon via the radius of the circumscribed circle</H2> In this lesson you will learn how to calculate the area of a regular n-sided polygon via the radius of the circumscribed circle. <H3>Theorem</H3><TABLE> <TR> <TD>The area {{{S}}} of a regular n-sided polygon equals {{{S}}} = {{{n}}}.{{{R^2}}}.{{{cos(pi/n)}}}.{{{sin(pi/n)}}} = {{{1/2}}}.{{{n}}}.{{{R^2}}}.{{{sin(2pi/n)}}}, (1) where {{{R}}} is the radius of the circle circumscribed about the polygon. <B>Proof</B> (<B>Figure 1a</B> and <B>Figure 1b</B> show a regular pentagon and a regular hexagon along with their circumscribed circles as the examples). For the given regular n-sided polygon, let us draw the circumscribed circle. You know from the lesson <A HREF=http://www.algebra.com/algebra/homework/Polygons/Regular-polygons.lesson>Regular polygons</A> that it is always possible to do so. The center of this circle coincides with the center </TD> <TD> {{{drawing( 240, 240, -1.2, 1.2, -1.2, 1.2, circle( 0.0, 0.0, 1.0), circle( 0.0, 0.0, 0.02), locate(-0.03, -0.02, O), line ( 0.8090, -0.5878, 0.8090, 0.5878), line ( 0.8090, 0.5878, -0.3090, 0.9511), line (-0.3090, 0.9511, -1.0000, 0.0000), line (-1.0000, 0.0000, -0.3090, -0.9511), line (-0.3090, -0.9511, 0.8090, -0.5878), line ( 0.8090, -0.5811, 0.8090, 0.5811), blue(line ( 0.00, 0.00, 0.8090, 0.00)), green(line ( 0.00, 0.00, 0.8090, 0.5811)), green(line ( 0.00, 0.00, 0.8090, -0.5811)), locate( 0.34, 0.44, R), arc ( 0.0, 0.0, 0.35, 0.35, 324, 36), arc ( 0.0, 0.0, 0.41, 0.41, 324, 36), locate( 0.86, 0.49, a/2), locate( 0.86, -0.09, a/2), locate( 0.86, 0.05, D), arc ( 0.0, 0.0, 0.48, 0.48, 324, 360), locate( 0.27, 0.95, a), locate( 0.865, -0.51, A1), locate( 0.865, 0.69, A2), locate(-0.340, 1.13, A3), locate(-1.190, 0.05, A4), locate(-0.340, -0.98, A5) )}}} <B>Figure 1a</B>. To the <B>Theorem 1</B> </TD> <TD> {{{drawing( 240, 240, -1.2, 1.2, -1.2, 1.2, circle( 0.0, 0.0, 1.0), circle( 0.0, 0.0, 0.02), locate(-0.03, -0.02, O), line ( 0.86, -0.50, 0.86, 0.50), line ( 0.86, 0.50, 0.00, 1.00), line ( 0.00, 1.00, -0.86, 0.50), line (-0.86, 0.50, -0.86, -0.50), line (-0.86, -0.50, 0.00, -1.00), line ( 0.00, -1.00, 0.86, -0.50), blue(line ( 0.00, 0.00, 0.86, 0.00)), green(line ( 0.00, 0.00, 0.86, 0.50)), green(line ( 0.00, 0.00, 0.86, -0.50)), locate( 0.35, 0.4, R), arc ( 0.0, 0.0, 0.35, 0.35, 330, 30), arc ( 0.0, 0.0, 0.41, 0.41, 330, 30), locate( 0.9, 0.44, a/2), locate( 0.9, -0.06, a/2), locate( 0.9, 0.05, D), arc ( 0.0, 0.0, 0.48, 0.48, 330, 360), locate( 0.41, 0.90, a), locate( 0.90, -0.46, A1), locate( 0.90, 0.65, A2), locate(-0.08, 1.16, A3), locate(-1.02, 0.65, A4), locate(-1.04, -0.46, A5), locate(-0.08, -1.00, A6) )}}} <B>Figure 1b</B>. To the <B>Theorem 1</B> </TD> </TR> </TABLE>of the given regular polygon. Let {{{R}}} be the radius of this circle. Connect the center of the polygon with its vertices by the straight segments. These straight segments divide the interior of the polygon in <B>n</B> isosceled congruent triangles. In each of these triangles, the altitude is the median and the angle bisector simultaneously. Therefore, the measure of the altitude is {{{R*cos(2pi/(2n))}}} = {{{R*cos(pi/n)}}}. The base of each triangle is {{{2R*sin(pi/n)}}} in accordance with the lesson <A HREF=http://www.algebra.com/algebra/homework/Polygons/The-side-length-of-a-regular-polygon-via-the-radius-of-the-circumscribed-circle.lesson>The side length of a regular polygon via the radius of the circumscribed circle</A>. Hence, the area of each isosceles triangle is {{{R^2*cos(pi/n)*sin(pi/n)}}}, and the area of the entire polygon is {{{nR^2}}}.{{{cos(pi/n)}}}.{{{sin(pi/n)}}}. It proves the first half of the formula (1). To prove the second half of the formula (1), use the formula {{{sin(beta)*cos(beta)}}} = {{{(sin(2beta))/2}}} from <B>Trigonometry</B> to replace {{{cos(pi/n)}}}.{{{sin(pi/n)}}} by {{{1/2}}}.{{{sin(2pi/n)}}}. Or, alternatively, simply use the formula  {{{S}}} = {{{1/2}}}.{{{a*b*sin(alpha)}}} for the area of each isosceles triangle with {{{a}}} = {{{R}}}, {{{b}}} = {{{R}}} and {{{alpha}}} = {{{2pi/n}}}. <H3>Example 1</H3>The area of a regular triangle equals {{{S}}} = {{{3}}}.{{{R^2}}}.cos(60°).sin(60°) = {{{3}}}.{{{R^2}}}.{{{1/2}}}.{{{sqrt(3)/2}}} = {{{3sqrt(3)/4}}}{{{R^2}}}, where {{{R}}} is the radius of the circumscribed circle. <H3>Example 2</H3>The area of a square equals {{{S}}} = {{{4}}}.{{{R^2}}}.cos(45°).sin(45°) = {{{4}}}.{{{R^2}}}.{{{sqrt(2)/2}}}.{{{sqrt(2)/2}}} = {{{2R^2}}}, where {{{R}}} is the radius of the circumscribed circle. <H3>Example 3</H3>The area of a regular hexagon equals {{{S}}} = {{{1/2}}}{{{6}}}.{{{R^2}}}.sin(60°) = {{{3}}}.{{{R^2}}}.{{{sqrt(3)/2}}} = {{{3sqrt(3)/2}}}{{{R^2}}}, where {{{R}}} is the radius of the circumscribed circle. <H3>Problem 1</H3>Find the area of a regular triangle inscribed in the circle of the radius of 10 cm. <B>Solution</B> Use the formula of the <B>Theorem 1</B> above. You have {{{S}}} = {{{3}}}.{{{R^2}}}.cos(60°).sin(60°) = {{{3}}}.{{{10^2}}}.{{{1/2}}}.{{{sqrt(3)/2}}} = {{{3sqrt(3)/4}}}.{{{100}}} = {{{75}}}.{{{1.732}}} = {{{129.90}}} (approximately). <B>Answer</B>. The area of the regular triangle is {{{129.90}}} {{{cm^2}}} (approximately). <H3>Problem 2</H3>Find the area of a regular pentagon inscribed in the circle of the radius of 10 cm. <B>Solution</B> Use the formula of the <B>Theorem 1</B> above. You have {{{S}}} = {{{1/2}}}.{{{n}}}.{{{R^2}}}.{{{sin(2pi/n)}}} = {{{1/2}}}.{{{5}}}.{{{100}}}.{{{sin(2pi/5)}}} = {{{250}}}.sin(72°) = {{{250}}}.{{{0.951}}} = 237.76 {{{cm^2}}} (approximately). <B>Answer</B>. The area of the regular pentagon is 237.76 {{{cm^2}}} (approximately). . <H3>Problem 3</H3>Find the area of a regular hexagon inscribed in the circle of the radius of 10 cm. <B>Solution</B> Use the formula of the <B>Theorem 1</B> above. You have {{{S}}} = {{{1/2}}}.{{{n}}}.{{{R^2}}}.{{{sin(2pi/n)}}} = {{{1/2}}}.{{{6}}}.{{{100}}}.{{{sin(2pi/6)}}} = {{{300}}}.sin(60°) = {{{300}}}.{{{sqrt(3)/2}}} = 259.81 {{{cm^2}}} (approximately). <B>Answer</B>. The area of the regular hexagon is 259.81 {{{cm^2}}} (approximately). My other lessons on the area of polygons in this site are - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-n-sided-polygon-circumscribed--about-a-circle.lesson>Area of n-sided polygon circumscribed about a circle</A> and - <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 <B>Area and surface area</B> of the section <B>Geometry</B>, and - <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 <B>Geometry</B> of the section <B>Word problems</B>. To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
