Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(p\) denote the perimeter of an equilateral triangle. Find a formula for \(A(p)\), the area of such a triangle.

Step-by-Step Solution

Answer

The formula is \( A(p) = \frac{\sqrt{3}}{36}p^2 \).

Step 1: Understand the relationship between perimeter and side length

An equilateral triangle has all sides equal in length. Let's denote the side length as \( s \). Since the perimeter \( p \) is the sum of all sides, we have \( p = 3s \).

Step 2: Express side length in terms of perimeter

Rearranging the equation from Step 1, we find the side length in terms of the perimeter: \( s = \frac{p}{3} \).

Step 3: Use the formula for the area of an equilateral triangle

The formula for the area \( A \) of an equilateral triangle with side length \( s \) is \( A = \frac{\sqrt{3}}{4}s^2 \).

Step 4: Substitute the expression for \( s \) in the area formula

Substitute \( s = \frac{p}{3} \) into the area formula: \( A = \frac{\sqrt{3}}{4}\left(\frac{p}{3}\right)^2 \).

Step 5: Simplify the expression for the area

Simplify \( A = \frac{\sqrt{3}}{4}\left(\frac{p}{3}\right)^2 \) to find \( A = \frac{\sqrt{3}}{4} \cdot \frac{p^2}{9} \). Simplify further to obtain \( A = \frac{\sqrt{3}}{36}p^2 \).

Key Concepts

Perimeter-to-Side Length Relationship

Understanding how the perimeter of a shape relates to its side length is crucial. In an equilateral triangle, all three sides are equal. This means the perimeter is simply three times the length of one side. Let's use the letter \( s \) to denote the side length. The formula for the perimeter \( p \) of an equilateral triangle is:

• \( p = 3s \)

This equation tells you that by dividing the perimeter by three, you can find the length of one side. This relationship is fundamental when solving problems that involve equilateral triangles because it allows you to switch between perimeter and side length whenever necessary.
Knowing how to rearrange this relationship—\( s = \frac{p}{3} \)—allows you to find the side length if you're given only the perimeter. It's a simple yet powerful concept that makes dealing with equilateral triangle problems much easier.

Equilateral Triangle Properties

Equilateral triangles are unique due to their symmetry and equal sides. They have several interesting properties that can help you solve various problems. Here are a few:

• All three sides are of equal length.
• All three interior angles are 60 degrees since the total sum of angles in any triangle is 180 degrees.
• The area of an equilateral triangle can be calculated using the side length \( s \), through the formula: \( A = \frac{\sqrt{3}}{4}s^2 \).
• Each median of an equilateral triangle is also an altitude, angle bisector, and perpendicular bisector due to its symmetry.

These properties not only give the equilateral triangle a special position in geometry but also make calculations and deductions involving it much easier. Whether you're finding angles, altitudes, or areas, knowing that all sides and angles are equal greatly simplifies the process.
This understanding of equilateral triangles helps provide the necessary insights when tackling more complex problems, such as those involving perimeter and area.

Perimeter of a Triangle

The perimeter of any triangle, including an equilateral one, is the sum of its three sides. For non-equilateral triangles—where sides can differ—the perimeter is calculated as:

• \( p = a + b + c \)

where \( a \), \( b \), and \( c \) represent the side lengths.
In an equilateral triangle, however, since all sides are equal (\( s \)), the formula simplifies to:

• \( p = 3s \)

Calculating the perimeter is often the first step in figuring out other properties, such as the area.
This is because once you calculate or are given the perimeter, you can easily find the side length, which is the key to finding other measurements. An understanding of this foundational concept is critical when you're doing any geometric calculations involving triangles. Remember, whether dealing with one triangle or a series, the perimeter can guide you to all other important figures and formulas.