Chapter 5.4, Problem 22E

(a)

To determine

To find:the apothem of the hexagon.

Answer to Problem 22E

  $\text{apothem }\approx \text{ 2}\text{.77}$

Explanation of Solution

Given information: Geometry: A regular hexagon is inscribed in a circle with diameter 6.4 centimeters.

Calculation:

Draw a regular hexagon inscribed in a circle with diameter 6.4 cm.

Using the formula $\frac{\left(n-2\right)\times 180}{n}$ where n is the number of sides:

  $\frac{\left(8-2\right)\times 180}{8}=120$

Since the radius bisects the angle:

120 / 2 = 60

Splitting the triangle in half will give us a right triangle, with hypotenuse 3.2 cm, leg 1.6 cm and an unknown leg or the apothem.

Option 1: Use the Pythagorean Theorem

Using the radius $\left(3.2\right)$ and the leg $\left(1.6\right),$ we can find the apothem.

  $\begin{array}{l}{c}^2=a^2+b^2\\ {3.2}^2={1.6}^2+b^2\\ b^2=10.24-2.56\\ b\approx 2.77\end{array}$

Option 2: Use Sine

Since we have an angle and the hypotenuse, we can use sine to figure out the apothem, which is the side opposite to the ${60}^{\circ }$ angle.

  $\begin{array}{l}\sin {60}^{\circ }=\frac{\text{apothem}}{3.2}\\ \sin {60}^{\circ }\times 3.2=\text{apothem}\\ \text{apothem}\approx \text{2}\text{.77}\end{array}$

Option 3: use cosine

Since the apothem is the adjacent side of the ${30}^{\circ }$ angle and we have the angle and the hypotenuse, we can use cosine to figure out the apothem.

  $\cos {30}^{\circ }=\frac{\text{apothem}}{3.2}$

  $\text{apothem}=\cos {30}^{\circ }\times 3.2$

  $\text{apothem}\approx 2.77$

(b)

To determine

To find: The length of a side of the hexagon.

Answer to Problem 22E

  $\text{length of side = 3}\text{.2 cm}$

Explanation of Solution

Given information: Geometry: A regular hexagon is inscribed in a circle with diameter 6.4 centimeters.

Calculation:

  $x\text{ is }$ length of a side

  $\begin{array}{l}\sin {30}^{\circ }=\frac{\frac{x}{2}}{3.2}\\ \sin {30}^{\circ }=\frac{x}{6.4}\\ x=6.4\sin {30}^{\circ }\\ x=3.2\text{ cm}\end{array}$

Thus, the length of the side of the hexagon is 3.2 cm.

(c)

To determine

To find: The perimeter of the hexagon.

Answer to Problem 22E

  $\text{Perimeter = 19}\text{.2 cm}$

Explanation of Solution

Given information: Geometry: A regular hexagon is inscribed in a circle with diameter 6.4 centimeters.

Calculation:

Total sides = 6

One side length = 3.2

Perimeter = 3.2(6) = 19.2

Thus, the perimeter of the hexagon is 19.2 cm.

(d)

To determine

To find: The area of the polygon, if the area of a regular polygon equals one half times the perimeter of the polygon times the apothem.

Answer to Problem 22E

  $\text{Area of polygon = 26}{\text{.6 cm}}^2$

Explanation of Solution

Given information: Geometry: A regular hexagon is inscribed in a circle with diameter 6.4 centimeters.

Calculation:

  $\begin{array}{l}\text{Area}=\frac{1}{2}\left(\text{perimeter}\right)\left(\text{apothem}\right)\\ \text{Area}=\frac{1}{2}\left(19.2\right)\left(2.77\right)\\ \text{Area}\simeq 26.6{\text{ cm}}^2\end{array}$