Question native wiljohn, a student:

find the area the a 5 sharp star in the circle with a radius of 12.517


Even if you desire your 5-pointed star to be REGULAR, there will certainly be two feasible answers to your problem. Let"s start with what we mean by a continuous PENTAGRAM, then look at the feasible ways to compute its area.

Begin v a suggest A ~ above a circle of radius r. (You can set r = 12.517 in ~ the end. For now, any kind of fixed number r will certainly do.) specify the 2nd vertex to it is in the point B ~ above the circle so that the angle BO provides with AO is $2 \times \large \frac3605 = 144$ degrees. Similarly, edge COB = DOC = EOD = AOE =144 degrees. This 5-pointed star is regular since every side (such as AB) has actually the exact same length and the angle between nearby sides (such as abdominal muscle and BC) room equal (to 36 degrees).


The means many mathematicians think of the area is to take it a straightforward component -- right here the triangle BOA -- and use the to tile the object who area us wish to compute. Us then merely compute the area of the initial triangle and also multiply through 5 to acquire the area that the pentagram: The elevation of the triangle is $r \times \cos 72$ while its basic is $2 \times r \times \sin 72.$ The area the the generating triangle is therefore

\<\mboxAREA(BOA)= \frac12 \times \mboxbase \times \mboxheight = r^2\times \sin 72 \times \cos 72.\>

Our first answer to your difficulty is therefore $5 \times r^2 \times \sin 72 \times \cos 72.$

You will have noticed that utilizing this rotation an approach for obtaining the area method that the central pentagon region (labeled PQRST and colored eco-friendly in the figure) has been count twice. It renders sense to compute the area this means because that"s the way our pentagram to be generated. That is also feasible to develop the star number by starting with the consistent green pentagon and also sticking a yellow triangle "hat" (such together SRA) on every of its 5 edges. Yet there continues to be the question of the form that triangle hat. It deserve to be tall and thin (with an area as huge as we wish to make it), or it have the right to be short and fat (with an area close come zero). To make the star regular, we define its political parties to be expansions of the sides of the beginning pentagon. Now we can easily determine the area the the star figure (which we currently agree to consist that a pentagon and also five triangles) by subtracting the area of the eco-friendly pentagon native the area the the pentagram we computed above.

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The pentagon is created of 5 congruent isosceles triangles such as OPQ. The height of this triangle we know to be $r \times \cos 72;$ its base because of this is $2 \times r \times \cos 72 \times \tan 36.$ The area is

\< \mboxAREA(OPQ) = r^2 \times (\cos 72)^2 \times \tan 36.\>

The area that the star number is therefore obtained by individually $5 \times r^2 \times (\cos 72)^2 \times \tan 36$ from $5 \times r^2 \times \sin 72 \times \cos 72.$


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