Problem 54 - Algebra and Trigonometry

Question

Find the area of an equilateral triangle with side of length 10 .

Step-by-Step Solution

Answer

The area of the triangle is \(25\sqrt{3}\) or approximately 43.3 square units.

Step 1: Understanding the Problem

The problem asks us to find the area of an equilateral triangle, where all three sides are 10 units long. In an equilateral triangle, all sides and angles are equal. To find its area, we will use the formula for the area of an equilateral triangle given the side length \(a\): \[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]

Step 2: Substitute the Side Length into the Formula

Given that the side length \(a = 10\), we will substitute this value into the formula for the area: \[ \text{Area} = \frac{\sqrt{3}}{4} \times 10^2 \]

Step 3: Calculate \(10^2\)

First, calculate \(10^2\), which involves multiplying 10 by itself: \[ 10^2 = 100 \]

Step 4: Multiply by the Fraction

Now multiply the result by the fraction \(\frac{\sqrt{3}}{4}\): \[ \text{Area} = \frac{\sqrt{3}}{4} \times 100 \] \[ \text{Area} = 25\sqrt{3} \]

Step 5: Approximate the Area

To find an approximate decimal value, compute \(25\sqrt{3}\) using \(\sqrt{3} \approx 1.732\): \[ 25 \times 1.732 = 43.3 \]

Key Concepts

Triangle Geometry

An equilateral triangle is a fascinating shape in triangle geometry. Its key feature is that all three sides are identical in length and its three interior angles are equal too, each being precisely 60 degrees. This uniformity and balance make equilateral triangles one of the simplest yet most stable geometric figures. Triangle geometry often explores properties like angle measures, symmetry, and congruency.

In terms of symmetry, equilateral triangles exhibit rotational symmetry, meaning they look the same after a rotation of 120 degrees around the center. This symmetry gives equilateral triangles a unique aesthetic and structural quality, making them appear in various fields, from architecture to nature.

• All sides are equal.
• All angles measure 60 degrees.
• It has rotational symmetry.
• Used extensively in different applications for stability.

Geometric Formulas

Geometric formulas are essential in solving triangle-related problems, such as determining areas and perimeters. For an equilateral triangle, the formula to find its area is especially useful and unique, where side length is denoted by \(a\). This special formula is:

• Area = \(\frac{\sqrt{3}}{4} a^2\)

This equation arises from the properties of equilateral triangles and mathematical derivations involving angles and ratios. It utilizes the constant \(\sqrt{3}\), stemming from the inherent geometric configuration of equal sides and angles.

Mastery of geometric formulas enables solving diverse problems, like calculating perimeters using the simple formula:

• Perimeter = \(3a\)

Familiarity with these equations allows students to efficiently tackle a wide range of geometric exercises.

Area Calculation

Area calculation for an equilateral triangle involves plugging the side length into the specific formula for this type of triangle. Start by substituting the given side length into the area formula. With \(a = 10\) from our problem, the area formula becomes:

• Area = \(\frac{\sqrt{3}}{4} \times 10^2\)

Calculating further involves determining \(10^2\) which results in 100. Once you have \(a^2\), replace it into the formula giving:

• Area = \(\frac{\sqrt{3}}{4} \times 100\)

The next step is multiplying 100 by \(\frac{\sqrt{3}}{4}\), which simplifies to \(25\sqrt{3}\). This process reflects the straightforward yet intriguing nature of area calculations in triangles.

For practical purposes, converting \(25\sqrt{3}\) into a decimal approximation produces approximately 43.3, providing a tangible number often used in real-world contexts.

Equilateral Triangle Properties

Equilateral triangles possess distinct properties that are interesting and useful in various applications. These properties stem from all sides being equal and all angles being 60 degrees, forming a perfectly balanced figure. The inherent symmetry of equilateral triangles makes them robust in construction and design, where even weight distribution is desired.

• Equal sides and angles create a perfect balance.
• They have three lines of symmetry.
• They are often used in tiling and structural engineering.

Another notable property is that the altitude, median, and angle bisector from any vertex are all identical, simplifying complex calculations. This unique trait points to why understanding equilateral triangle properties is fundamental in geometry education.

Recognizing these properties not only simplifies calculations but also aids in deeper geometric insights.