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Chapter 11: Problem 12

Find the area of each figure. An equilateral triangle with perimeter 18

Short Answer

Expert verified
The area of the equilateral triangle with a perimeter of 18 is \(9\sqrt{3}\) square units.

Step by step solution

01

Determine Side Length of the Equilateral Triangle

As an equilateral triangle has equal sides, calculate the length of one side by dividing the total perimeter by the number of sides. Here, the perimeter is given as 18, and the triangle has three sides: \(Side = Perimeter / 3 = 18 / 3 = 6\)
02

Calculate the Area of the Equilateral Triangle

Now, using the formula for the area of an equilateral triangle, which is given by \(\sqrt{3}/4 * side^2\), plug in the calculated side length: \(Area = \sqrt{3}/4 * 6^2\)
03

Compute the Final Value

Perform the operations: Square the side length (6^2 = 36), multiply by \(\sqrt{3}/4\), and simplify to the smallest radical form if needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilateral Triangle
An equilateral triangle is a type of triangle where all three sides are of equal length. This property also means all the angles in an equilateral triangle are equal, each measuring 60 degrees. The symmetry in equilateral triangles makes them a fascinating study in geometry because their properties allow for simplified calculations of areas and perimeters. This uniformity gives rise to specific formulas that can solve various geometric problems with ease.
Area Calculation
To calculate the area of an equilateral triangle, there's a specific formula to use. This formula is:
  • Area = \( \frac{\sqrt{3}}{4} \times \text{side}^2 \)
The formula relies on knowing the length of one side. Remember, the side lengths are equal, which simplifies calculations. By squaring the side length and multiplying by \( \frac{\sqrt{3}}{4} \), the area can be directly obtained. This formula is a result of applying trigonometric identities to the geometry of the triangle.
If you're visualizing this, picture dividing the triangle into two 30-60-90 right triangles to derive this relationship.
Perimeter
The perimeter of a shape is simply the total length around the shape. For an equilateral triangle, calculating the perimeter is straightforward because all sides are equal.
Simply sum the length of all three sides, or multiply one side's length by three.
  • Perimeter = 3 \( \times \) side
In this exercise, you've seen that given the total perimeter, dividing it by three gives the length of one side. This relationship is unique to polygons with equal sides, like equilateral triangles, and simplifies calculations considerably.
Geometric Formulas
Geometric formulas are essential tools in solving many problems related to shapes and figures. For equilateral triangles, these formulas simplify understanding complex geometric relationships.
  • Side length from perimeter: \( \text{side} = \frac{\text{perimeter}}{3} \)
  • Area of an equilateral triangle: \( \frac{\sqrt{3}}{4} \times \text{side}^2 \)
These formulas are derived from fundamental principles of geometry. They showcase the power of mathematical relationships and symmetry. Such formulas not only provide answers but also deepen the understanding of geometric concepts. They enable comparisons and adaptations to similar problems, leading to a stronger grasp of mathematics overall.

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