Problem 56 - Precalculus Mathematics for Calculus

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 Problem

We need to find the area of an equilateral triangle where the length of each side is 10 units. An equilateral triangle has all sides of equal length and all internal angles equal to 60 degrees.

Step 2: Formula for Area of Equilateral Triangle

For an equilateral triangle with side length \(a\), the area \(A\) can be calculated using the formula: \[ A = \frac{\sqrt{3}}{4} a^2 \] In this problem, \(a = 10\).

Step 3: Substitute and Calculate

Substitute \(a = 10\) into the area formula for the equilateral triangle: \[ A = \frac{\sqrt{3}}{4} \times 10^2 \] Calculate the result: \[ A = \frac{\sqrt{3}}{4} \times 100 \] \[ A = 25\sqrt{3} \]

Step 4: Final Result

The area of the equilateral triangle with a side length of 10 is \(25\sqrt{3}\).

Key Concepts

Understanding Geometry and Triangles

Geometry is a branch of mathematics that involves the study of shapes, sizes, and the properties of space. It plays a crucial role in understanding different kinds of figures and their dimensions.

When we talk about triangles, they are flat, three-sided shapes with three vertices. Triangles are classified based on their side lengths and angles. An equilateral triangle, for instance, is a special type of triangle where all three sides are of equal length. Additionally, each angle in an equilateral triangle is 60 degrees, making it unique among triangles.

• Geometric figures help us understand spatial relationships.
• Triangles can be distinguished by their side lengths and angles.
• Equilateral triangles have all sides equal and each angle measures 60 degrees.

Understanding the concepts of geometry is essential in calculating areas, such as the area of a triangle, and solving many everyday problems.

What is an Equilateral Triangle?

An equilateral triangle is a simple yet fascinating geometric figure. What sets it apart is the equality of its sides and angles. Because of these properties, equilateral triangles are highly symmetrical figures.

• All three sides in an equilateral triangle are equal.
• All internal angles are 60 degrees, collectively summing up to 180 degrees.
• Due to its symmetry, it has unique properties, like rotational symmetry.

This triangle serves as a perfect example of balance and uniformity in geometry. It is often used in problems requiring symmetrical division or equal distribution.

Using the Triangle Area Formula

The formula for the area of an equilateral triangle is derived from general triangle properties and is particularly simple. When given the side length of an equilateral triangle, calculating its area becomes a straightforward process.

The formula is:\[ A = \frac{\sqrt{3}}{4} a^2 \]where \(a\) represents the side length of the triangle. This formula utilizes the constant \(\sqrt{3}/4\), which is derived from the height and the symmetrical properties of the triangle.

• It applies specifically to equilateral triangles.
• The formula simplifies the calculation of a seemingly complex figure.
• It expresses the area in terms of side length, making it easy for application.

For example, in an equilateral triangle with a side length of 10, substituting into the formula gives:\[ A = \frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3} \]Thus, the area of this triangle is \(25\sqrt{3}\), demonstrating how user-friendly the formula is for quick calculations.