Problem 56 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The area is \(25\sqrt{3} .\)

Step 1: Understanding the Formula for the Area of an Equilateral Triangle

To find the area of an equilateral triangle, we can use the formula: \[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]where \(s\) is the length of a side. For this problem, \(s = 10 .\)

Step 2: Substituting the Side Length into the Formula

Now, we will substitute \(s = 10\) into the formula. This gives us: \[ \text{Area} = \frac{\sqrt{3}}{4} \times 10^2 \] which simplifies to:\[ \text{Area} = \frac{\sqrt{3}}{4} \times 100 .\]

Step 3: Calculating the Area

Now let’s calculate the numerical value:\[ \text{Area} = \frac{\sqrt{3}}{4} \times 100 = 25 \times \sqrt{3} .\]So, the area of the triangle is \(25\sqrt{3} .\)

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, and consequently, all three angles are equal as well. Each angle in an equilateral triangle is always 60 degrees. This triangle is not just symmetrical but also a fundamental shape in geometry due to its simplicity and uniqueness.
In such triangles, many other geometrical properties come into play. For instance, the height can be calculated using the Pythagorean theorem because an equilateral triangle can be split into two equal right triangles by dropping a perpendicular from a vertex to the midpoint of the opposite side.

• The equal sides make solving for unknowns straightforward, as you only need one measurement to define the triangle completely.
• The equilateral triangle serves as a basis for many mathematical and real-world problems, especially when dealing with uniform designs and tessellations.

Understanding equilateral triangles helps build a foundation for more complex geometrical studies and solving various geometric problems.

Triangle Geometry

Triangle geometry is a wide-ranging field encompassing various types of triangles and their properties. Among these, the equilateral triangle stands out because it represents pure symmetry. In a greater context, triangles are essential in geometry as they are the simplest polygon and any polygon can be divided into triangles. This makes them a key unit of measure in geometry.
Triangles follow some basic rules, most notably the Triangle Sum Theorem, which states that all interior angles of a triangle add up to 180 degrees. This is true for any triangle, not just equilateral ones. But in an equilateral triangle, this is simply three 60-degree angles.

• Besides angles, triangles are classified by their sides: isosceles (two equal sides), scalene (all sides different), and equilateral (all sides equal).
• Understanding these classifications provides insight into solving for unknown features within geometric shapes.

Once the basic properties of triangles are mastered, moving on to more advanced concepts, like the calculation of area and center points, becomes manageable and fun.

Geometric Formulas

Geometric formulas are the backbone of finding areas, perimeters, and other properties of shapes. For an equilateral triangle, these formulas simplify due to its symmetry. The formula for the area of an equilateral triangle is unique because it includes the square root of 3, indicating its connection to the geometric properties of these triangles.
The specific formula used is: \[\text{Area} = \frac{\sqrt{3}}{4} s^2\]where \(s\) is the side length. This formula is derived from the fact that when you drop a perpendicular from one of the vertices, you create two 30-60-90 right triangles which are primitive and help in deriving the \(\sqrt{3}\) factor.

• Understanding this formula allows for quick calculations in problems involving equilateral triangles, just like the given exercise.
• Substituting the side length into this formula gives a straightforward method to solve for the area, making complex geometrical analysis accessible even with basic mathematics.

Mastery of geometric formulas empowers students to navigate various mathematical challenges effectively.