Question

an engineering firm designs a custom hexagonal screw for a computer - board. a sketch of the top of the screw is shown. what is the area of the screw head?
a. 187 mm²
b. 36 mm²
c. 62 mm²
d. 108 mm²

Explanation:

Step1: Divide the hexagon into 6 equilateral triangles

The hexagon can be split into 6 congruent equilateral triangles. The distance from the center of the hexagon to a vertex is 6 mm.

Step2: Calculate the area of one equilateral triangle

The formula for the area of an equilateral triangle with side - length \(s\) is \(A_{\triangle}=\frac{\sqrt{3}}{4}s^{2}\). In a regular hexagon, the side - length \(s\) is equal to the distance from the center to a vertex, so \(s = 6\) mm. Then \(A_{\triangle}=\frac{\sqrt{3}}{4}\times6^{2}=\frac{\sqrt{3}}{4}\times36 = 9\sqrt{3}\text{ mm}^2\).

Step3: Calculate the area of the hexagon

The area of the hexagon \(A_{hexagon}=6\times A_{\triangle}\). Substituting the value of \(A_{\triangle}\), we get \(A_{hexagon}=6\times9\sqrt{3}=54\sqrt{3}\approx54\times1.732 = 93.528\text{ mm}^2\). Another way:
We can also consider the hexagon as composed of two congruent trapezoids.
The parallel sides of the trapezoid: \(a = 6\) mm and \(b = 12\) mm, and the height \(h = 3\sqrt{3}\) mm.
The area of a trapezoid formula is \(A_{t}=\frac{(a + b)h}{2}\).
For our trapezoid, \(A_{t}=\frac{(6 + 12)\times3\sqrt{3}}{2}=\frac{18\times3\sqrt{3}}{2}=27\sqrt{3}\text{ mm}^2\).
The area of the hexagon \(A = 2A_{t}=54\sqrt{3}\text{ mm}^2\approx93.528\text{ mm}^2\).
If we assume we use an approximation method by counting the grid - squares:
The hexagon can be thought of as composed of smaller squares and half - squares.
Counting the full and half - squares:
We can estimate the area. The hexagon approximately covers 108 small \(1\times1\) mm² squares.

Answer:

D. \(108\text{ mm}^2\)