Problem 69 - Calculus: One and Several Variables

Question

Express the area of an equilateral triangle as a function of the length of a side.

Step-by-Step Solution

Answer

The area of an equilateral triangle as a function of the length of a side \( s \) is \( s^2\sqrt{3}/4 \).

Step 1: Define the variables

Let \( s \) be the length of a side in the equilateral triangle.

Step 2: Find the height of the triangle

By drawing a line from one corner to the midpoint of the opposite side (this line is called the 'altitude' or 'height'), an equilateral triangle can be split into two 30-60-90 right triangles. The height (h) of the triangle can be found using Pythagorean theorem (as we define the half-side as \( s/2 \) and the hypotenuse as \( s \)): \( h = \sqrt{s^2 - (s/2)^2} = \sqrt{3s^2/4} = s\sqrt{3}/2 \).

Step 3: Find the area of the triangle

The formula to find the area (A) of a triangle is A = 1/2 * base * height. In an equilateral triangle, all sides are the same length, so the base is also \( s \). So we can substitute \( s \) and \( s\sqrt{3}/2 \) into the formula to get: \( A = 1/2 * s * (s\sqrt{3}/2) = s^2\sqrt{3}/4 \).

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle. It has three equal sides and three equal angles. Each angle in an equilateral triangle measures 60 degrees.

• When any triangle has all sides equal, it is also 'equiangular' – all angles are equal.
• This unique property makes equilateral triangles easy to work with using symmetry.

Another useful aspect is that splitting an equilateral triangle down the middle gives two 30-60-90 right triangles.
This property is handy for calculating various triangle attributes, such as height. Understanding equilateral triangles paves the way to calculate more complex geometry.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles. It states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). The formula is written as:
\[ a^2 + b^2 = c^2 \]

• For our equilateral triangle, when split, each half becomes a 30-60-90 triangle.
• In this setup, the hypotenuse is the triangle's side \( s \), and one leg is \( s/2 \), the base of the split triangle.
• This helps us derive the height \( h \) as: \( h = \sqrt{s^2 - (s/2)^2} \).

Mastery of the Pythagorean Theorem allows us to confidently find unknown lengths within triangles.

Triangle Area Formula

Finding the area of any triangle involves a straightforward formula: the area \( A \) is half the product of the base and height: \( A = \frac{1}{2} \, \text{base} \, \times \, \text{height} \). In an equilateral triangle:

• We use one side as the base, \( s \).
• The height, derived using the Pythagorean Theorem, is \( s\sqrt{3}/2 \).

By substituting these into the formula, the area of an equilateral triangle can be neatly expressed as:
\[ A = \frac{s^2\sqrt{3}}{4} \]
This formula allows quick computation of the area once any side length is known.