Question 1050794: The arc of a spider web is 1/4 of a circle and 20 in. in length. Calculate area of web in square in.
Unsure how to solve. Non-homework question.
Thanks.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the short answer is this.
if the length of the arc is 1/4 times the circumference of the circle, then the area of the sector created by the arc is 1/4 times the area of the circle.
you would find the circumference of the circle and then find the radius and then find the area of the circle and then find the area of the sector of the circle.
you are given that the length of the arc is 20 inches and that the length of the arc is 1/4 times the length of the circumference of the circle.
this means the circumference of the circle is 80 inches because 4 *20 = 80.
the formula for the circumference of a circle is c = 2 * pi * r
c = 80, so the formula becomes 80 = 2 * pi * r.
solve for r to get r = 40 / pi.
the formula for the area of a circle is a = pi * r^2.
when r = 40 / pi, this formula becomes a = pi * (40 / pi)^2.
simplify this to get a = pi * 40^2 / pi^2.
simplify further to get a = pi * 1600 / pi^2.
simplify further to get a = 1600 / pi.
take 1/4 of this to get the area of the sector is 400 / pi.
the central angle of the sector of the circle created by the arc would be 90 degrees.
that's because the circumference of a circle is equal to 360 when measured in degrees and the central angle of the circle is equal to 360 degrees as well.
if the length of the arc is 1/4 * the circumference of the circle, then the number of degrees of that arc is 1/4 * 360 = 90 degrees.
the central angle of the arc is therefore also equal to 90 degrees.
the formula for the length of the arc of a sector that is 90 degrees is:
length of arc = 90 / 360 * circumference of circle.
the formula for the area of a sector that is 90 degrees is:
area of sector = 90 / 360 * area of circle.
if the length of the arc is 1/4 times the circumference of the circle, then the area of the sector created by the arc is 1/4 times the area of the circle.
this reference might help to visualize it.
http://www.mathopenref.com/arcsectorarea.html
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