Problem 79 - Algebra for College Students

Question

Find the area of an equilateral triangle, each of whose sides is 18 inches long. Express the area to the nearest square inch.

Step-by-Step Solution

Answer

The area is approximately 140 square inches.

Step 1: Understand the Problem

We need to find the area of an equilateral triangle where each side is 18 inches long. Recall that an equilateral triangle has all sides of the same length and all angles equal to 60 degrees.

Step 2: Recall the Area Formula

The area of an equilateral triangle with side length \( a \) is given by the formula \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \). Here, \( a = 18 \) inches.

Step 3: Substitute and Calculate

Substitute \( a = 18 \) into the formula: \[ \text{Area} = \frac{\sqrt{3}}{4} \times 18^2 \] This becomes \[ \text{Area} = \frac{\sqrt{3}}{4} \times 324 \]

Step 4: Simplify and Compute

Calculate the simplified expression: \[ \text{Area} = 81\sqrt{3} \] Estimate \( \sqrt{3} \approx 1.732 \), so the area becomes \[ 81 \times 1.732 \approx 140.292 \text{ square inches} \]

Step 5: Round the Result

Round 140.292 to the nearest square inch, which results in 140 square inches.

Key Concepts

Geometry Formulas

In geometry, formulas are essential tools. Each shape or figure comes with specific formulas that allow you to calculate properties like area, perimeter, or volume. For triangles, especially equilateral ones, these formulas utilize the unique properties of the shape.

To find the area of an equilateral triangle, the formula is:

• \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \)

Here, \( a \) represents the length of a side. This formula arises from a standard geometric principle: the triangle's height and its properties. Using these specific geometry formulas allows quick calculations, ensuring accuracy.

In any geometry problem, understanding and using the right formula is crucial. This enables you to tackle the problem effectively, ensuring the correct results, just as with our equilateral triangle with sides of 18 inches.

Triangle Properties

Triangles are fascinating due to their diverse properties. An equilateral triangle is a special type where all sides and angles are congruent. Each angle measures exactly 60 degrees.

The equilateral triangle has unique characteristics:

• All sides are equal in length.
• Each angle is 60 degrees.
• The height can be calculated using its side, forming 30-60-90 right triangles.

These properties help simplify calculations, especially in determining dimensions like area.

In geometry, recognizing these properties enables effective problem-solving strategies. It leverages symmetry and regularity, simplifying complex calculations with these predictable features, enhancing learning and application.

Area Calculation

Calculating the area of an equilateral triangle is straightforward with the correct formula. By substituting the given side length into the formula, you can effectively find the area.

Here’s how it works:

• Start with the side length \( a = 18 \) inches.
• Insert \( a \) into the formula \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \).
• Perform the calculation: \( = \frac{\sqrt{3}}{4} \times 18^2 \).
• Compute this to simplify as \( 81\sqrt{3} \), using \( \sqrt{3} \approx 1.732 \).
• This rounds to approximately 140 square inches.

This process demonstrates the efficiency of formulas in solving geometry problems.

Rounding the result gives a neat, manageable figure, making it practical for real-world applications. Ensuring estimates are as accurate as possible aids in precise area calculations.