Question

lin and andre used different methods to find the area of a regular hexagon with 6 - inch sides. lin decomposed the hexagon into six identical, equilateral triangles. andre decomposed the hexagon into a rectangle and two triangles. find the area of the hexagon using each persons method. show your reasoning.

Explanation:

Step1: Calculate area of equilateral triangle in Lin's method

The formula for the area of an equilateral triangle with side - length $s$ is $A_{\triangle}=\frac{\sqrt{3}}{4}s^{2}$. Here $s = 6$ inches. So $A_{\triangle}=\frac{\sqrt{3}}{4}\times6^{2}=\frac{\sqrt{3}}{4}\times36 = 9\sqrt{3}$ square inches. Since the hexagon is decomposed into 6 such triangles, the area of the hexagon $A_{Lin}=6\times9\sqrt{3}=54\sqrt{3}\approx54\times1.732 = 93.528$ square inches.

Step2: Calculate area in Andre's method

The rectangle has length $l = 12$ inches (two side - lengths of the hexagon) and width $w$ can be found from the height of the hexagon. The height of the hexagon is $h = 10.4$ inches. The two triangles together can be thought of as one rectangle with the same width as the side of the hexagon ($6$ inches) and height equal to the remaining part of the height of the hexagon. The area of the rectangle in Andre's decomposition is $A_{rect}=12\times(10.4 - 3\sqrt{3})+ 6\times3\sqrt{3}$. First, $12\times(10.4 - 3\sqrt{3})=12\times10.4-12\times3\sqrt{3}=124.8-36\sqrt{3}$, and $6\times3\sqrt{3}=18\sqrt{3}$. Then $A_{Andre}=124.8-36\sqrt{3}+18\sqrt{3}=124.8 - 18\sqrt{3}\approx124.8-18\times1.732=124.8 - 31.176 = 93.624$ (there is a small difference due to rounding in the first method). A more accurate way for Andre's method: The rectangle has length $l = 12$ inches and width $w$ such that the height of the hexagon is $10.4$ inches. The area of the rectangle is $A_{rect}=12\times\frac{10.4}{2}=62.4$ and the two triangles together have area $A_{tri}= 6\times\frac{10.4}{2}=31.2$. So $A_{Andre}=62.4 + 31.2=93.6$ square inches.

Answer:

Lin's method: Approximately $93.53$ square inches. Andre's method: $93.6$ square inches.