Problem 8 · AMC 8 Stretch
Stretch
Geometry & Measurement
considering-extreme-casesvisual-representation
A regular hexagon (6 equal sides) has side length \(4\). The apothem — the distance from the center straight out to the middle of a side — is about \(3.46\). Find the area by cutting the hexagon into triangles. (Round to the nearest tenth.)
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Answer: About 41.5 square units
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Hint 1 of 4
Draw lines from the center of the hexagon to each corner. How many triangles do you get, and are they all the same?
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Hint 2 of 4
Each triangle has a base equal to one side of the hexagon (\(4\)) and a height equal to the apothem (\(3.46\)). What is the area of ONE triangle?
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Hint 3 of 4
Add up all \(6\) triangles. Then notice: \(6\) bases together equal the whole perimeter.
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Approach: Cut into triangles; area = half perimeter times apothem
- Draw segments from the center to all \(6\) corners. This cuts the hexagon into \(6\) identical triangles, each with base \(= 4\) (one side) and height \(= 3.46\) (the apothem hits the side at a right angle).
- Area of one triangle \(= \frac{1}{2} \times 4 \times 3.46 = 6.92\). All \(6\) triangles give \(6 \times 6.92 = 41.52\) square units.
- Look at the structure: \(6 \times \left(\frac{1}{2} \times \text{side} \times \text{apothem}\right) = \frac{1}{2} \times (6 \times \text{side}) \times \text{apothem}\). But \(6 \times \text{side}\) is the whole perimeter, so Area \(= \frac{1}{2} \times (\text{perimeter}) \times (\text{apothem})\) for any regular polygon.
- Here \(\frac{1}{2} \times 24 \times 3.46 \approx 41.5\) square units. (Extreme case: a circle has apothem \(= r\), giving \(\frac{1}{2}(2\pi r)(r) = \pi r^2\).)
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