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. lins method andres method find the area of the hexagon using each persons method. show your reasoning.

Explanation:

Step1: Calculate area using Lin's method

For an equilateral triangle with side - length \(a = 6\) inches, the area formula of an equilateral triangle is \(A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}\). Substitute \(a = 6\) into the formula: \(A_{\triangle}=\frac{\sqrt{3}}{4}\times6^{2}=\frac{\sqrt{3}}{4}\times36 = 9\sqrt{3}\) square - inches. Since the hexagon is composed of 6 such equilateral triangles, the area of the hexagon \(A_{Lin}=6\times A_{\triangle}=6\times9\sqrt{3}=54\sqrt{3}\approx54\times1.732 = 93.528\) square - inches.

Step2: Calculate area using Andre's method

The rectangle in Andre's decomposition has length \(l = 6\) inches and width \(w = 10.4\) inches. The area of the rectangle \(A_{rect}=l\times w=6\times10.4 = 62.4\) square - inches. The two triangles together can be considered as one rectangle with the same height as the "extra" part of the hexagon's height and base equal to the side - length of the hexagon. The "extra" height of the hexagon that forms the triangles is \(h=\frac{10.4 - 6}{2}=2.2\) inches. The two triangles together have an area \(A_{triangles}=6\times2.2 = 13.2\) square - inches. The area of the hexagon \(A_{Andre}=A_{rect}+A_{triangles}=62.4 + 13.2=93.528\) square - inches.

Answer:

Using Lin's method, the area is approximately \(93.528\) square inches. Using Andre's method, the area is approximately \(93.528\) square inches.