Question

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

Short answer

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

Step-by-step solution

Express side length in terms of perimeter

Let the side length of the equilateral triangle be \( s \). Since the perimeter \( p \) is the sum of all three equal sides, we have \( p = 3s \). Thus, the side length \( s \) can be expressed as \( s = \frac{p}{3} \).

Use formula for the area of an equilateral triangle

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

Substitute side length expression into the area formula

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

Simplify the expression

Simplify the expression:\[A = \frac{\sqrt{3}}{4} \cdot \frac{p^2}{9} = \frac{\sqrt{3} p^2}{36}\]

Conclude with the formula for the area

We have derived that the area \( A(p) \) of an equilateral triangle with perimeter \( p \) is given by:\[A(p) = \frac{\sqrt{3} p^2}{36}\]

Key concepts

Area Formula

To find the area of an equilateral triangle, we utilize a specific formula that is distinct from other types of triangles. An equilateral triangle is unique because all its sides are equal in length. The area formula for an equilateral triangle with side length \( s \) is expressed as:\[A = \frac{\sqrt{3}}{4}s^2\]This formula comes from the fact that an equilateral triangle can be divided into two 30-60-90 right triangles, allowing us to use trigonometry and geometry to derive the formula.
It's crucial to understand that this formula works perfectly for equilateral triangles due to their symmetry and equal angles.To use this formula effectively, simply determine the side length \( s \) of the triangle, then substitute it into the formula to compute the area. For problems involving the perimeter instead of side length, further steps are necessary to express \( s \) in terms of the perimeter value, as shown in the given exercise.

Perimeter

The concept of perimeter is fundamental in geometry and extremely important when dealing with equilateral triangles. The perimeter of a geometric figure is the total length around a shape. In an equilateral triangle, since all sides are equal, the perimeter \( p \) can be calculated as:\[p = 3s\]Here, \( s \) represents the length of one side of the triangle. To solve problems where the perimeter is given, we need to manipulate the formula to find the side length. This can be done by rearranging the formula to find that \( s = \frac{p}{3} \).
With this expression for the side, we can then use it to find other properties of the triangle, such as its area, using the area formula.

Triangular Geometry

Triangular geometry is a fascinating area of mathematics that explores various types of triangles, with equilateral triangles being one of the simplest yet most intriguing. An equilateral triangle is defined by having all three sides of equal length and all three internal angles equal to 60 degrees. This symmetrical nature gives rise to several interesting properties:

• Uniform side lengths make calculations straightforward.
• All angles are equal, leading to uniform angle measurements of 60 degrees each.
• These properties lead to unique formulas and relationships such as the area and height.

The geometry of equilateral triangles allows for an understanding of how simple ratios and relationships can describe complex shapes. This helps in solving various geometric problems, making equilateral triangles a crucial part of the study in triangular geometry.

Mathematical Derivation

The mathematical derivation of formulas is a critical skill in understanding how solutions are developed. In the case of the area of an equilateral triangle, derivation involves using known geometrical properties to express one quantity in terms of another.To derive the area formula \( A(p) = \frac{\sqrt{3}p^2}{36} \), we start by understanding how the side length \( s \) is related to the perimeter \( p \):\[s = \frac{p}{3}\]Using this relation, the area formula \( A = \frac{\sqrt{3}}{4}s^2 \) is modified by substituting \( s \) with \( \frac{p}{3} \), which involves some algebraic manipulation and simplification:\[A = \frac{\sqrt{3}}{4}\left(\frac{p}{3}\right)^2 = \frac{\sqrt{3} p^2}{36}\]Each step in the derivation requires careful attention to express geometrical concepts algebraically, which helps us understand and derive solutions for any given perimeter \( p \). This approach not only provides answers but also deepens comprehension of mathematical relationships.