Problem 11 - 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

The perimeter is \( P(x) = 3x \) and the area is \( A(x) = \frac{\sqrt{3}}{4} x^2 \).

Step 1: Identify the formulas needed

To express the area and perimeter of an equilateral triangle as a function of its side length, we need to know the formulas for the perimeter and area of a triangle. For an equilateral triangle, all sides are equal, hence the perimeter is simply three times the side length. For the area, use the formula for the area of an equilateral triangle: \( \text{Area} = \frac{\sqrt{3}}{4} x^2 \), where \( x \) is the side length.

Step 2: Calculate the perimeter

To calculate the perimeter of the equilateral triangle, use the formula: \( \text{Perimeter} = 3x \). This formula is derived from the fact that all three sides of the triangle have the same length \( x \).

Step 3: Calculate the area

To express the area as a function of the side length \( x \), use the formula for the area of an equilateral triangle: \( \text{Area} = \frac{\sqrt{3}}{4} x^2 \). This formula relies on the trigonometric properties of an equilateral triangle, in which all angles are 60 degrees and it can be geometrically derived.

Step 4: Write the final expressions

Now that we have the formulas, we can write down the expressions for both the area and the perimeter of the equilateral triangle. The perimeter is: \( P(x) = 3x \). The area is: \( A(x) = \frac{\sqrt{3}}{4} x^2 \). Both are expressed in terms of \( x \), the side length.

Key Concepts

Area of Equilateral Triangle

The area of an equilateral triangle can be derived using a specific formula that involves the side length. An equilateral triangle is unique because all its sides and angles are equal. In mathematics, the area is commonly a measure of the space enclosed within a 2D shape. To calculate it for an equilateral triangle, we use the formula:
\[ A(x) = \frac{\sqrt{3}}{4} x^2 \]
Here, \( x \) represents the length of one side of the triangle.
This formula comes from its geometric properties, where every angle in an equilateral triangle is 60 degrees. By dividing the triangle into two right triangles, you can apply basic trigonometry to arrive at this result. The height can be found using Pythagorean theorem, and from there, the area computation becomes straightforward as the area is half the product of the base and the height.
The beauty of this formula lies in its simplicity, efficiently capturing the balanced symmetry of the equilateral triangle.

Perimeter of Equilateral Triangle

Understanding how to compute the perimeter of an equilateral triangle is crucial in geometry. The perimeter is the total distance around the triangle’s outer boundary.
For an equilateral triangle, which has all sides equal, the calculation is straightforward:
\[ P(x) = 3x \]
To compute this, you simply multiply the length of one side, \( x \), by 3. This is because the sum of all sides gives you the perimeter, and in an equilateral triangle, each side is identical.
Using this formula helps in quickly finding the perimeter without needing to measure each side separately. It simplifies problems involving equilateral triangles and underscores the relationship between its side lengths and perimeter.

Geometric Formulas

Geometric formulas serve as the backbone of solving problems in geometry, providing essential relationships between various properties of shapes. For equilateral triangles, two primary formulas are frequently used: those for area and perimeter.
To recap:

• The area formula, \( A(x) = \frac{\sqrt{3}}{4} x^2 \), gives us the measure of space inside the triangle.
• The perimeter formula, \( P(x) = 3x \), gives us the measure of the distance around the triangle.

Both formulas highlight the core principle that equilateral triangles have symmetrical properties. This symmetry simplifies otherwise complex calculations, reducing them to simple equations depending only on the side length.
Geometric formulas not only guide students through their math homework but also instill a deeper understanding of mathematical symmetry and its applications.