Problem 15 - Algebra 2 and Trigonometry

Question

Find the exact value of the area of an equilateral triangle if the length of one side is 40 meters.

Step-by-Step Solution

Answer

The area of the equilateral triangle is \(400\sqrt{3}\) square meters.

Step 1: Understand the formula for the area of an equilateral triangle

The formula for finding the area of an equilateral triangle is \( A = \frac{a^2\sqrt{3}}{4}\) where \(a\) represents the length of a side of the triangle. This formula arises from the fact that an equilateral triangle can be split into two 30-60-90 right triangles.

Step 2: Substitute the given value into the formula

Since the side length \(a\) of the equilateral triangle is 40 meters, we substitute \(a = 40\) into the formula: \[A = \frac{40^2\sqrt{3}}{4}\]

Step 3: Calculate the squared term

First, calculate \(40^2\): \(40 \times 40 = 1600\). Hence, \[A = \frac{1600\sqrt{3}}{4}\]

Step 4: Simplify the expression

Divide 1600 by 4, as per the formula: \(1600 \div 4 = 400\). This gives us the expression \[A = 400\sqrt{3}\].

Key Concepts

Area Formula for Equilateral Triangle

An equilateral triangle is a special kind of triangle where all three sides are equal in length, and each angle measures 60 degrees.
Because of this symmetry, there is a straightforward formula to calculate the area:

• Area, denoted as \( A \), is calculated using \( A = \frac{a^2\sqrt{3}}{4} \)
• In this formula, \( a \) represents the length of one side of the triangle.
• This formula is derived from the properties of a 30-60-90 triangle, which we'll discuss more in a later section.

This simple area formula makes it quick and efficient to find the area of any equilateral triangle if you know the length of a side.
For instance, with a side length of 40 meters, you plug it directly into the formula to find the area without needing any additional measurements.

Geometry Problem Solving

Geometry problem solving often involves recognizing patterns and using formulas strategically.
For an equilateral triangle, knowing one side length allows you to figure out the entire area using the specialized area formula.

• Start with the formula \( A = \frac{a^2\sqrt{3}}{4} \)
• Substitute the known side length into the formula.
• Simplify the expression step by step, ensuring that calculations like squaring the side length and dividing by 4 are done carefully.

By methodically applying the formula and carefully managing calculations, you solve for the area efficiently. This structured approach enhances understanding and efficiency in tackling similar geometry problems.

30-60-90 Triangle Properties

A 30-60-90 triangle is a special type of right triangle with specific angle measures that are always 30 degrees, 60 degrees, and 90 degrees.
Here's how these properties help in understanding the structure of an equilateral triangle:

• An equilateral triangle can be divided into two 30-60-90 triangles by drawing a perpendicular from one vertex to the opposite side.
• This perpendicular acts as the height, splitting the base into two equal parts.
• The ratio of the sides opposite the 30°, 60°, and 90° angles is always 1:√3:2

These properties are fundamental in deriving the formula for the area of an equilateral triangle.
They also help in understanding the triangle's structure and can assist in many other geometry problems involving right triangles.