Chapter 11.5, Problem 15PSB

a.

To determine

To calculate: Area of triangle

Answer to Problem 15PSB

The area of triangle is $108\sqrt{3}$

Explanation of Solution

Given:

The radius of the circle is $12$

Concept used:

Here, we are using the formula

Area of triangle = $\frac{abc}{4R}$

Calculation:

Here, let sides of triangle be x

Circumradius = $12$ cm

Area of triangle = $\frac{abc}{4R}$

  → $\frac{x^3}{48}$ ......... $\left(1\right)$

Now, area of equilateral triangle = $\frac{\sqrt{3}}{4}{a}^2$

  $\frac{\sqrt{3}}{4}{x}^2=\frac{x^3}{48}$

  $\frac{48\sqrt{3}}{4}=x$

  $x=12\sqrt{3}$ putting this value in equation $\left(1\right)$

  → $\frac{x^3}{48}$

  → $\frac{{\left(12\sqrt{3}\right)}^3}{48}$

  → $108\sqrt{3}$

Conclusion: Hence, the area of triangle is $108\sqrt{3}$

b.

To determine

To calculate: Area of the square.

Answer to Problem 15PSB

Area of the square is $288$

Explanation of Solution

Given:

The radius of the circle is $12$

Formula used:

Here, the formula is

Area = $\frac{1}{2}\times d^2$

Calculation: Area = $\frac{1}{2}\times 24\times 24$

  → $288$

Conclusion: Hence, the area of the square is $288$

c.

To determine

To calculate: Area of a hexagon.

Answer to Problem 15PSB

Area of a hexagon is $216\sqrt{3}$

Explanation of Solution

Given:

The radius of the circle is $12$

Formula used:

Here, the formula is

Area of equilateral triangle = $\frac{\sqrt{3}}{4}\times a^2$

Calculation:

Area of equilateral triangle = $\frac{\sqrt{3}}{4}\times a^2$

  → $\frac{\sqrt{3}}{4}\times 12\times 12$

  → $36\sqrt{3}$

And area of hexagon = $36\sqrt{3}\times 6=216\sqrt{3}$

Conclusion: Hence, area of a hexagon is $216\sqrt{3}$