Chapter 11.5, Problem 19PSC

a.

To determine

To calculate:Area of hexagon.

Answer to Problem 19PSC

The area of hexagon is $450\sqrt{3}$

Explanation of Solution

Given: The span of regular hexagon is $30$

Concept used: Here, we are using the formula

Area of hexagon = $\frac{3\sqrt{3}}{2}\times a^2$

Calculation: Here, Tan ${60}^{\circ }=\frac{P}{B}$ (here, “ $P$ ” is perpendicular and “ $B$ ” is base)

  $\sqrt{3}=\frac{15}{B}$

  $B=10\sqrt{3}$

So that, Area of hexagon = $\frac{3\sqrt{3}}{2}\times 100\times 3$

  → $450\sqrt{3}$

Conclusion: Hence, the area of hexagon is $450\sqrt{3}$

b.

To determine

To calculate: The span of a regular hexagon.

Answer to Problem 19PSC

The span of a regular hexagon is $8$

Explanation of Solution

Given: The area of a regular hexagon is $32\sqrt{3}$

Formula used: Here, the formula is

Area of hexagon = $\frac{3\sqrt{3}}{2}\times a^2$

Calculation: $32\sqrt{3}=\frac{3\sqrt{3}}{2}\times a^2$

  $\frac{64}{3}=a^2$

  $a=\frac{8}{\sqrt{3}}$

And, Tan ${60}^{\circ }=\frac{P}{B}$ (here, “ $P$ ” is perpendicular and “ $B$ ” is base)

  $\sqrt{3}=\frac{P}{4}\sqrt{3}$

  $P=4$

So, span will be $8$

Conclusion: Hence, the span of a regular hexagon is $8$

c.

To determine

To calculate:Formula for the area of a regular hexagon.

Answer to Problem 19PSC

The formula for the area of a regular hexagon is $\frac{\sqrt{3}}{2}{\left(span\right)}^2$

Explanation of Solution

Given: Hexagon with a span $s$

Calculation: For area we need to find the side with span

So, Tan ${60}^{\circ }=\frac{\frac{1}{2}\times span}{\frac{1}{2}\times side}$

  $side=\frac{span}{\sqrt{3}}$

Hence, area = $\frac{3\sqrt{3}}{2}\times \frac{{\left(span\right)}^2}{3}$

  → $\frac{\sqrt{3}}{2}{\left(span\right)}^2$

Conclusion: Hence, the formula for the area of a regular hexagon is $\frac{\sqrt{3}}{2}{\left(span\right)}^2$