The perimeter of a regular hexagon is 48 inches. What is the number of square inches in the positive difference between the areas of the circumscribed and the inscribed circles of the hexagon? Express your answer in terms of pi.

Question

Avery Allen · Accepted Answer

#color(blue)("Diff. in area between Circumscribed and Inscribed circles "#

#color(green)(A_d = pi R^2 - pi r^2 = 36 pi - 27 pi = 9pi " sq inch"#

Perimeter of regular hexagon #P = 48 "inch"#

Side of hexagon #a = P / 6 = 48 / 6 = 6 " inch"#

Regular hexagon consists of 6 equilateral triangles of side a each.

Inscribed circle : Radius #r = a / (2 tan theta), theta = 60/2 = 30^@ #

#r = 6 / (2 tan (30)) = 6 / (2 (1/sqrt3)) = 3 sqrt 3 " inch"#

#"Area of inscribed circle " A_r = pi r^2 = pi (3 sqrt3)^2 = 27 pi " sq inch"#

#"Radius of circumscribed circle " R = a = 6 " inch"#

#"Area of circumscribed circle "A_R = pi R^2 = pi 6^2 = 36 pi " sq inch"#

#"Diff. in area between Circumscribed and Inscribed circles "#

#A_d = pi R^2 - pi r^2 = 36 pi - 27 pi = 9pi " sq inch"#

Avery Allen · Answer

undefined The positive difference between the areas of the circumscribed and inscribed circles of a regular hexagon can be found using the formula:

$$ \text{Difference} = \pi \left( R^2 - r^2 \right) $$

where $ R $ is the radius of the circumscribed circle and $ r $ is the radius of the inscribed circle.

For a regular hexagon with perimeter $ P $, the radius of the circumscribed circle $ R $ is given by:

$$ R = \frac{P}{2\sqrt{3}} $$

And the radius of the inscribed circle $ r $ is given by:

$$ r = \frac{P}{6} $$

Substituting the values:

$$ R = \frac{48}{2\sqrt{3}} $$
$$ r = \frac{48}{6} $$

Calculate the difference:

$$ \text{Difference} = \pi \left( \left( \frac{48}{2\sqrt{3}} \right)^2 - \left( \frac{48}{6} \right)^2 \right) $$