Question

Write an expression in simplest radical form INTERMS OF PI that represents the difference between Leilani's approximated área and Desmond's actual area
Please show the steps

Answer 1

Answer:

$9(2\sqrt{3}-\pi)\ units^2$

Step-by-step explanation:

step 1

Find Leilani's approximated área

we know that

The approximate area of the circle is approximately the area of the six equilateral triangles

The formula to calculate the length side of a regular hexagon given the radius of a inscribed circle is equal to

$a=\frac{2r\sqrt{3}}{3}$

where

a is the length side of a regular hexagon

r is the radius of the inscribed circle

we have

$r=3\ units$

substitute

$a=\frac{2(3)\sqrt{3}}{3}$

$a=2\sqrt{3}\ units$

Find the area of six equilateral triangles

$A=6[\frac{1}{2}(r)(a)]$

simplify

$A=3(r)(a)$

we have

$r=3\ units\\a=2\sqrt{3}\ units$

substitute

$A=3(3)(2\sqrt{3})\\A=18\sqrt{3}\ units^2$

step 2

Find Desmond's actual area

we know that

The area of a circle is equal to

$A=\pi r^{2}$

we have

$r=3\ units$

substitute

$A=\pi (3)^{2}$

$A=9\pi\ units^2$

step 3

Find the difference between Leilani's approximated área and Desmond's actual area

$(18\sqrt{3}-9\pi)\ units^2$

simplify

$9(2\sqrt{3}-\pi)\ units^2$