Calculate the area of the equilateral triangle using the formula for the area of a regular polygon, and compare it to Blanca's answer.
The apothem, rounded to the nearest tenth, is ____ units.
The perimeter of the equilateral triangle is ____ units.
Therefore, the area of the equilateral triangle is ____ or approximately 43.5 units$^2$.
The calculated areas are ____
Answer (2)
Calculate the apothem of the equilateral triangle: a = 3 8.7 ≈ 2.9 .
Calculate the perimeter of the equilateral triangle: P = 3 × 10 = 30 .
Calculate the area of the equilateral triangle using the formula A = 2 1 a P : A = 2 1 ( 2.9 ) ( 30 ) = 43.5 .
The calculated area using the regular polygon formula matches Bianca's approximation: 43.5 units 2 .
Explanation
Problem Analysis We are given an equilateral triangle with side length 10. Bianca approximated its area using the formula A = 2 1 bh , where b is the base and h is the height. She found the height to be approximately 8.7, and thus the area to be approximately 43.5 units 2 . We are asked to calculate the area of the same triangle using the formula for the area of a regular polygon and compare the results.
Calculating the Apothem The formula for the area of a regular polygon is A = 2 1 a P , where a is the apothem and P is the perimeter. First, we need to find the apothem of the equilateral triangle. The apothem is the distance from the center of the triangle to the midpoint of a side. In an equilateral triangle, the apothem is one-third of the height. Bianca calculated the height to be approximately 8.7. Therefore, the apothem is approximately 3 8.7 ≈ 2.9 .
Calculating the Perimeter Next, we need to find the perimeter of the equilateral triangle. Since each side has a length of 10, the perimeter is 3 × 10 = 30 .
Calculating the Area Now we can calculate the area of the equilateral triangle using the formula A = 2 1 a P . Substituting the values we found for the apothem and perimeter, we get A = 2 1 ( 2.9 ) ( 30 ) = 43.5 .
Comparison and Conclusion The area calculated using the formula for the area of a regular polygon is 43.5 units 2 , which is the same as Bianca's approximation. Therefore, the apothem, rounded to the nearest tenth, is 2.9 units. The perimeter of the equilateral triangle is 30 units. Therefore, the area of the equilateral triangle is 43.5 units 2 . The calculated areas are the same.
Examples
The formula for the area of a regular polygon is useful in many real-world scenarios. For example, architects can use this formula to calculate the area of a floor plan that is in the shape of a regular polygon. Also, engineers can use this formula to calculate the surface area of a bolt head that is in the shape of a regular polygon. This formula is also useful in calculating the area of stop signs, which are in the shape of a regular octagon. Knowing how to calculate the area of regular polygons is a fundamental skill in geometry and has many practical applications.
The apothem of the equilateral triangle is approximately 2.9 units, the perimeter is 30 units, and the area calculated using the formula for a regular polygon is 43.5 units². This matches Bianca's approximation, confirming the accuracy of the calculations. Therefore, the area of the triangle is consistently found to be 43.5 units².
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