Lesson OVERVIEW of LESSONS on Area of a Circle, a Sector and a Segment of the circle

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<H2>OVERVIEW of LESSONS on Area of a Circle, a Sector and a Segment of the circle</H2>For your convenience, &nbsp;this file contains

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the list of my lessons on area of a circle, &nbsp;area of a sector and area of a segment of the circle in this site,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the major formulas for calculating the area of a circle, &nbsp;area of a sector and area of a segment of the circle, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the list of relevant solved problems.

<H3>The major formulas for calculating the area of a circle, area of a sector and area of a segment of the circle</H3>&nbsp;&nbsp;&nbsp;&nbsp;- The area of a &nbsp;<B><U>circle</U></B>&nbsp; equals &nbsp;{{{pi*r^2}}}, &nbsp;where &nbsp;{{{r}}}&nbsp; is the radius of the circle and &nbsp;{{{pi}}}&nbsp; is the ratio of the circle circumference to its diameter, &nbsp;{{{pi}}} = 3.14 . . . .

<TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- The area &nbsp;{{{S}}}&nbsp; of a &nbsp;<B><U>sector</U></B>&nbsp; of the circle with the radius &nbsp;{{{r}}}&nbsp; and the central angle of the radian measure &nbsp;{{{alpha}}}&nbsp; is equal&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{S}}} = {{{1/2}}}{{{alpha*r^2}}}.

</TD>
  <TD>

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red(line  ( 0.00,  0.00,  0.8090,  0.5811)),
        red(line  ( 0.00,  0.00,  0.8090, -0.5811)),
        red( arc  ( 0, 0,  2.00,   2.00,   328, 36)),
        red( arc  ( 0, 0,  2.05,   2.05,   328, 36)),
             arc  ( 0, 0,  0.5, 0.5, 328, 36),
            locate( 0.25,  0.6, r),
            locate( 0.3,  0.25, alpha)
)}}}

</TD>
 </TR>
</TABLE><TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- The area &nbsp;{{{S}}}&nbsp; of a &nbsp;<B><U>segment of the circle</U></B>&nbsp; with the radius &nbsp;{{{r}}}&nbsp; and the central angle of the radian measure &nbsp;{{{alpha}}}&nbsp; is  equal&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{S}}} = {{{1/2}}}[{{{alpha - sin(alpha)}}}].{{{r^2}}}.

</TD>
  <TD>
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            locate(-0.03, -0.02, O),

line  ( 0.00,  0.00,  0.8090,  0.5811),
            line  ( 0.00,  0.00,  0.8090, -0.5811),
        red( arc  ( 0, 0,  2.00,   2.00,   328, 36)),
        red( arc  ( 0, 0,  2.05,   2.05,   328, 36)),
             arc  ( 0, 0,  0.5, 0.5, 328, 36),
        red(line  ( 0.8090, -0.5811, 0.8090, 0.5811)),
            locate( 0.25,  0.6, r),
            locate( 0.3,  0.25, alpha)
)}}}

</TD>
 </TR>
</TABLE>

<H3>The list of my lessons on area of a circle, area of a sector and area of a segment of the circle</H3>&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-circle.lesson>Area of a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-sector.lesson>Area of a sector</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-segment-of-the-circle.lesson>Area of a segment of the circle</A>

under the current topic &nbsp;<B>Area and surface area</B>&nbsp; of the section &nbsp;<B>Geometry</B>, &nbsp;and

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

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-a-segment-of-the-circle.lesson>Solved problems on area of a segment of the circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-aCircle-aSector-and-aSegment-of-the-circle.lesson>Solved problems on area of a circle, a sector and a segment of the circle</A>

under the topic &nbsp;<B>Geometry</B>&nbsp; of the section &nbsp;<B>Word problems</B>.

<H3>The list of solved problems on area of a circle, area of a sector and area of a segment of the circle</H3>Lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-circle.lesson>Area of a circle</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-a-circle.lesson>Solved problems on area of a circle</A>

<TABLE>
  <TR>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a circle which has the radius of &nbsp;10 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a ring concluded between two concentric circles that have the radii of &nbsp;10 cm &nbsp;and&nbsp; 6 cm.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of the circle which is inscribed in the &nbsp;60�-sector of the circle with the radius of &nbsp;12 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of the circle which is inscribed in the &nbsp;90�-sector of the circle with the radius of &nbsp;10 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a semicircle inscribed in a triangle with the side measures of &nbsp;13 cm, &nbsp;14 cm&nbsp; and &nbsp;15 cm

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in a way that the center and the diameter of the semicircle lie on the side of the measure &nbsp;14 cm&nbsp; of

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the triangle.

</TD>
  <TD>

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line( 0, 0, 3, 0),

arc (0, 0, 0.8, 0.8, 300, 360),

red(circle(1.73, 1, 1))
)}}}

</TD>
  <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 88, 88,  -0.7, 10.3, -0.7, 10.3, 
            arc (0, 0, 20, 20, 270, 360),
            circle (0, 0, 0.1),

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line( 0, 0, 10, 0),

arc (0, 0, 2.0, 2.0, 270, 360),

red(circle(4.12, 4.14, 4.14))
)}}}

</TD>
  <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 100, 100,  -0.5, 4.5, -0.5, 4.5, 
            line( 0.0,  0.0, 4.0,  0.0), 
            line( 0.0,  0.0, 2.5,  4.0),
            line( 2.5,  4.0, 4.0,  0.0),

red(  arc (2.099, 0, 3.560, 3.560, 180, 360))
)}}}

</TD>
 </TR>
</TABLE>

Lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-sector.lesson>Area of a sector</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-a-sector.lesson>Solved problems on area of a sector</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a &nbsp;72�-sector of the circle which has the radius of &nbsp;10 cm.<TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of the figure which is restricted by the contour of a square with the side of &nbsp;10 cm&nbsp; from&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the bottom and from the lateral sides and by a semicircle from the top.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of the figure which is obtained from a square with the side of &nbsp;10 cm&nbsp; after cutting off

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a semicircle of the radius of &nbsp;6 cm. 
 </TD>
  <TD>{{{drawing(  80, 100,  -4.0, 4.0, -3.7, 6.3,

line (-3, -3, 3, -3),
            line (-3, -3,-3,  3),
            line ( 3, -3, 3,  3),

arc    (0, 3, 6.00, 6.00, 180, 360)
)}}}
 </TD>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 80, 100,  -4.0, 4.0, -3.7, 6.3,

line (-3, -3, 3, -3),
            line (-3, -3,-3,  3),
            line ( 3, -3, 3,  3),
            line (-3,  3, 3,  3),

line (-2,  1, 2,  1),
            arc    (0, 1, 4.00, 4.00, 0, 180)
)}}}

</TD>
 </TR>
</TABLE><TABLE>
  <TR>
  <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a figure comprised of a circle and an angle between two tangent lines released from a

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;point outside the circle, &nbsp;if the circle radius is of &nbsp;10 cm and the angle between the tangent lines is of &nbsp;60�.&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a figure restricted by a circle and by two tangent lines released from a point outside

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the circle, &nbsp;if the circle radius is of &nbsp;10 cm and the angle between the tangent lines is of &nbsp;60�.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The interior of the circle is not included to the figure.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a figure restricted by three congruent circles that touch each the other, &nbsp;if the circles

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;radius is of &nbsp;{{{r}}} = 10 cm.

</TD>
  <TD>
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red(arc    (0, 0, 8.0, 8.0,  60, 300)),
        red(arc    (0, 0, 8.1, 8.1,  60, 300)),
            arc    (0, 0, 8.0, 8.0, 300,  60),

red(line( 8, 0, 2,  3.44)),
        red(line( 8, 0, 2, -3.45))
)}}}

</TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 130, 90,  -4.5, 8.5, -4.5, 4.5,

red(arc    (0, 0, 8.0, 8.0, 300,  60)),
        red(arc    (0, 0, 8.1, 8.1, 300,  60)),
            arc    (0, 0, 8.0, 8.0,  60, 300),

red(line( 8, 0, 2,  3.44)),
        red(line( 8, 0, 2, -3.45))
)}}}

</TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 90, 90,  -4.5, 4.5, -4.0, 5.0, 
            circle (-2,   -1, 0.1),

circle ( 2,   -1, 0.1),

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line( -2, -1, 2, -1),
            line( -2, -1, 0,  2.45),
            line(  2, -1, 0,  2.45),

red(arc(-2, -1,    4.00, 4.00, 300, 360)),
        red(arc(-2, -1,    4.05, 4.05, 300, 360)),
            arc(-2, -1,    4.00, 4.00,   0, 300),

red(arc( 2, -1,    4.00, 4.00, 180, 240)),
        red(arc( 2, -1,    4.05, 4.05, 180, 240)),
            arc( 2, -1,    4.00, 4.00, 240, 180),

red(arc( 0,  2.45,    4.00, 4.00,  60, 120)),
        red(arc( 0,  2.45,    4.05, 4.05,  60, 120)),
            arc( 0,  2.45,    4.00, 4.00, 120,  60)
)}}}

</TD>
 </TR>
</TABLE>

Lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Surface-area/Area-of-a-segment-of-the-circle.lesson>Area of a segment of the circle</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-a-segment-of-the-circle.lesson>Solved problems on area of a segment of the circle</A>

<TABLE>
  <TR>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a &nbsp;72�-segment of the circle which has the radius of &nbsp;10 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a figure restricted by two circles that have equal radii of 10 cm, &nbsp;if the distance between&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;their centers is of &nbsp;10 cm&nbsp; also.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a figure restricted by two circles with the centers at the opposite vertices of a square

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and the radii equal to the square side &nbsp;a = 10 cm.

</TD>
  <TD>
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        red(arc( -2.5, 0, 10.14, 10.14, 300,  60)),
            arc( -2.5, 0, 10, 10,  60, 300),

red(arc(  2.5, 0, 10.00, 10.00, 120, 240)),
        red(arc(  2.5, 0, 10.14, 10.14, 120, 240)),
            arc(  2.5, 0, 10, 10, 240, 120),

circle( -2.5, 0, 0.1),
            circle(  2.5, 0, 0.1),

line ( -2.5, 0,  2.5,  0),
            locate(-1.7,   1.8, d=r)
)}}}

</TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 100, 100,  -0.5, 4.5, -0.5, 4.5, 
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line( 0, 0, 4, 4),

red(arc( 4, 0, 8.00, 8.00, 170, 280)),
        red(arc( 4, 0, 7.95, 7.95, 170, 280)),

red(arc( 0, 4, 8.00, 8.00, 350, 100)),
        red(arc( 0, 4, 7.95, 7.95, 350, 100))
)}}}

</TD>
 </TR>
</TABLE>

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-aCircle-aSector-and-aSegment-of-the-circle.lesson>Solved problems on area of a circle, a sector and a segment of the circle</A>

<TABLE>
  <TR>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of the intersection of two circles on a plane, &nbsp;if the circles have the same radius of

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;r = 10 cm&nbsp; and the intersection segments have the central angle &nbsp;{{{alpha}}}&nbsp; of &nbsp;a) 60�; &nbsp;b) 90�; &nbsp;c) 120�.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area covered by the union of two intersecting circles on a plane, &nbsp;if the circles have the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;same radius of &nbsp;r = 10 cm&nbsp; and the intersection segments have the central angle &nbsp;{{{alpha}}}&nbsp; of

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) 60�; &nbsp;b) 90�; &nbsp;c) 120�.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Find the area of a &nbsp;<B>lune</B>&nbsp; shape formed by two intersecting circles on a plane, &nbsp;if the circles have

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the same radius of &nbsp;r = 10 cm&nbsp; and the intersection segment have the central angle &nbsp;{{{alpha}}}&nbsp; of

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a) 60�; &nbsp;b) 90�; &nbsp;c) 120�.

</TD>
  <TD> 
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            arc( -3.53, 0, 10, 10,  45, 315),

red(arc(  3.53, 0, 10.00, 10.00, 135, 225)),
        red(arc(  3.53, 0, 10.14, 10.14, 135, 225)),
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            locate(-2.6,  1.3, alpha),

line ( -3.53, 0,    3.53,  0),

line (  0,    3.53, 0,    -3.53)
)}}}

</TD>
  <TD>

&nbsp;&nbsp;{{{drawing( 157, 100,  -9.0, 9.0, -5.5, 5.5,

arc( -3.53, 0, 10.00, 10.00, 315,  45),
        red(arc( -3.53, 0, 10.00, 10.00,  45, 315)),
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arc(  3.53, 0, 10.00, 10.00, 135, 225),
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            locate(-2.6,  1.3, alpha),

line ( -3.53, 0,    3.53,  0),

line (  0,    3.53, 0,    -3.53)
)}}}

</TD>
  <TD>

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arc(  3.53, 0, 10.00, 10.00, 135, 225),
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circle( -3.53, 0, 0.1),
            circle(  3.53, 0, 0.1),

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line ( -3.53, 0,    3.53,  0),

line (  0,    3.53, 0,    -3.53)
)}}}

</TD>
 </TR>
</TABLE>

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>.