Problem 9 - Thomas Calculus

Question

Express the area and perimeter of an equilateral triangle as a function of the triangle's side length \(x\) .

Step-by-Step Solution

Answer

Perimeter: \( P = 3x \); Area: \( A = \frac{\sqrt{3}}{4}x^2 \).

Step 1: Identify the formula for the perimeter

An equilateral triangle has three equal sides. If the side length is \( x \), then the perimeter \( P \) is the sum of the lengths of all three sides. Therefore, the formula for the perimeter is \( P = 3x \).

Step 2: Identify the formula for the area

The area \( A \) of an equilateral triangle with side length \( x \) can be calculated using the formula: \( A = \frac{\sqrt{3}}{4}x^2 \). This formula derives from the formula for the area of a triangle \( A = \frac{1}{2} \times \text{base} \times \text{height} \), where the height is found using 30-60-90 triangle properties.

Key Concepts

Perimeter Calculation

The perimeter of an equilateral triangle is the total distance around its three identical sides. In an equilateral triangle, each side has the same length, which we denote as \( x \). To find the perimeter, you simply add up the lengths of all three sides. This leads to the simple formula for the perimeter of an equilateral triangle: \( P = 3x \). This calculation is straightforward because you're multiplying the side length by three. This multiplication arises because of the triangle's symmetry, meaning that each side contributes equally to the total perimeter. So, if you know the side length of an equilateral triangle, calculating its perimeter is quick and easy. Just multiply the side length by three, and you'll have your answer!

Area Calculation

Finding the area of an equilateral triangle requires a bit more thought because we're interested in the space the triangle occupies. The formula to calculate the area \( A \) is \( A = \frac{\sqrt{3}}{4}x^2 \). This formula comes from a neat trick using a standard formula for the area of a triangle, \( A = \frac{1}{2} \times \text{base} \times \text{height} \). In an equilateral triangle, the height can be found using properties of a 30-60-90 triangle, which is a special right triangle. By splitting the equilateral triangle into two 30-60-90 triangles, you can calculate the height, then substitute it into the area formula.This derivation involves trigonometric properties, yet the resulting formula \( \frac{\sqrt{3}}{4}x^2 \) makes it simple to determine the area using only the side length.

Algebraic Functions

In the context of an equilateral triangle, algebraic functions allow us to express the perimeter and area in terms of the side length \( x \). These functions are expressions where the output is clearly determined by the value of \( x \). 1. **Perimeter Function**: The perimeter function \( P(x) = 3x \) is linear, meaning it graphs as a straight line and indicates that the perimeter grows directly with \( x \). 2. **Area Function**: The area function \( A(x) = \frac{\sqrt{3}}{4}x^2 \) is quadratic. This means its graph will be a parabola. The area increases with the square of \( x \), showing that as the triangle's side length increases, the area grows quite rapidly. Algebraic functions form a crucial part of understanding not just geometry but many other mathematical contexts. They provide a way to succinctly express complex geometric concepts using simple algebraic expressions.