Question

The apothem of a regular polygon is the measure of the line segment from the center of the polygon to the midpoint of one of its sides. A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters. a. Find the radius of the circumscribed circle. b. What is the length of a side of the hexagon? c. What is the perimeter of the hexagon?

Answer 1

Answer:

a. The radius of the circumscribed circle is 5.54 cm.

b. The length of a side of the hexagon is 5.54 cm.

c. The perimeter of the hexagon is 33.24 cm.

Step-by-step explanation:

The first step is to draw a diagram.

Let $a$ be the angle measure formed by a radius and its adjacent apothem.

Because the measure of the central angle of a regular hexagon is 60º, the measure of $a$ is $a=\frac{60^{\circ}}{2}=30^{\circ}$.

a. Now we know the measures of an angle and the side adjacent to the angle, we can find the radius of the circumscribed circle.

We can use the definition of cosine:

$cos(\theta)=\frac{side \:adjacent}{hypotenuse}$

$cos(30)=\frac{4.8}{r}$

We solve for $r$:

$r=\frac{4.8}{cos(30)}=\frac{16\sqrt{3}}{5}\approx 5.54 \:cm$

b. To find the length of a side of the hexagon we use the Pythagorean theorem on the right triangle:

$r^2=a^2+b^2\\5.54^2=4.8^2+b^2\\b^2=7.6516\\b=\sqrt{7.6516}\approx2.77$

We know that the apothem is a line segment from the center of a regular polygon to the midpoint of a side. Therefore, the length of a side of the hexagon is $length = 2b = 2(2.77)= 5.54 \:cm$

c. The perimeter is 6 times the length of side, so the perimeter is

$P=6\cdot l=6\cdot 5.54=33.24 \:cm$