Question

A regular hexagon is to be cut out of a circular sheet of metal that has a radius of 6 inches.

Approximately how many square centimeters of sheet will be left over a scraps?

A:32.9
B:93.5
C:113.1
D:126.2 ​

Answer 1

`\large \boxed{\text{D. 126.2 cm}^(2)}`

Explanation:

The area left over for scrap is the area of the circle minus the area of the hexagon.

1. Area of circle

The formula for the area of a circle is

A = πr²

A = π(6)² = 36π = 113.1 in²

2. Area of hexagon

A hexagon consists of six equilateral triangles, each of side a, and we can divide each of them into two right triangles.

So, we can calculate the area of one right triangle and multiply by 12.

The formula for the area of one triangle is

A = ½bh

(a) Height of a small triangle

Per the Pythagorean Theorem,

`\begin{array}{rcr}h^(2) + 3^(2) & = & 6^(2)\\h^(2) + 9 & = & 36\\h^(2) & = & 27\\& = & 3√(3)\\\end{array}\\`

(b) Area of a small triangle

A = ½ bh = ½ × 3 × 3√3 = 4.5√3 in²

(c) Area of the hexagon

The hexagon contains 12 small triangles.

A = 12 × 4.5√ 3 = 54√3 ≈ 93.53 in²

3. Area of scrap

A ≈ 113.1 in² - 93.53 in² = 19.6 in²

`A = \text{19.6 in}^(2) * \left(\frac{\text{2.54 cm}}{\text{1 in}}\right )^(2) = \text{126.2 cm}^(2)\\\\\text{The area of the scrap is $\large \boxed{\textbf{126.2 cm}^{\mathbf{2}}}$}`