In Exercises 13–16, find the area of the triangle having the given measurements. Round to the nearest square unit. a = 2 meters, b = 2 meters, c = 2 meters

검증된 단계별 안내

Recognize that the given triangle is an equilateral triangle since all sides are equal: \( a = b = c = 2 \) meters.

Use the formula for the area of an equilateral triangle: \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \).

Substitute the side length \( a = 2 \) meters into the formula: \( \text{Area} = \frac{\sqrt{3}}{4} \times 2^2 \).

Simplify the expression: \( \text{Area} = \frac{\sqrt{3}}{4} \times 4 \).

Calculate the area and round to the nearest square unit.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Triangle Area Formulas

The area of a triangle can be calculated using various formulas, depending on the information available. For triangles with known side lengths, Heron's formula is particularly useful. It states that the area can be found using the semi-perimeter and the lengths of the sides. For this triangle, since all sides are equal, the formula simplifies the calculation.

Heron's Formula

Heron's formula allows for the calculation of the area of a triangle when the lengths of all three sides are known. It involves first calculating the semi-perimeter (s = (a + b + c) / 2) and then applying the formula: Area = √(s(s-a)(s-b)(s-c)). This method is particularly effective for non-right triangles, such as the equilateral triangle in this problem.

Equilateral Triangle Properties

An equilateral triangle has all three sides of equal length and all angles measuring 60 degrees. This symmetry simplifies calculations, as the height can be derived from the side length. For an equilateral triangle with side length 'a', the area can also be calculated using the formula: Area = (√3/4) * a², which provides a quick way to find the area without using Heron's formula.