Calculating The Height Of An Equilateral Triangle Base In A Pyramid

Understanding the geometry of three-dimensional shapes, especially pyramids, often involves working with the properties of their two-dimensional components. In the context of a solid oblique pyramid with an equilateral triangle as its base, finding the height of this triangular base is a fundamental step. This article delves into how to calculate the height of such an equilateral triangle, providing a detailed explanation and step-by-step solution. Whether you're a student tackling geometry problems or simply interested in mathematical concepts, this guide will offer a clear and comprehensive understanding of the process.

Problem Statement

We are given a solid oblique pyramid with an equilateral triangle as its base. The base edge length of this triangle is 18 inches. Our task is to determine the height of this triangular base. The options provided are:

A. $9 \sqrt{2} \text{ in}$. B. $9 \sqrt{3} \text{ in}$. C. $18 \sqrt{2} \text{ in}$. D. $18 \sqrt{3} \text{ in}$.

Understanding Equilateral Triangles

Before we dive into the solution, it’s crucial to understand the properties of an equilateral triangle. An equilateral triangle is a triangle in which all three sides are of equal length, and all three angles are equal (each being 60 degrees). This symmetry allows us to use specific formulas and theorems to find various dimensions, including the height.

Key Properties of Equilateral Triangles

1. Equal Sides: All three sides have the same length.
2. Equal Angles: All three angles are 60 degrees.
3. Altitude as Median: The altitude (height) of an equilateral triangle bisects the base, creating two congruent right-angled triangles.
4. Altitude as Angle Bisector: The altitude also bisects the angle at the vertex from which it is drawn.

Method 1: Using the Pythagorean Theorem

One of the most straightforward methods to find the height of an equilateral triangle is by using the Pythagorean Theorem. Since the altitude bisects the base, we can form a right-angled triangle with the altitude as one side, half the base as another side, and the original side of the equilateral triangle as the hypotenuse.

Steps to Apply the Pythagorean Theorem

1. Visualize the Right Triangle: Imagine the equilateral triangle with the altitude drawn from one vertex to the midpoint of the opposite side. This altitude divides the equilateral triangle into two congruent right-angled triangles.

2. Identify the Sides:

   • Hypotenuse: The side of the equilateral triangle (18 inches).
   • Base: Half the base of the equilateral triangle (18 inches / 2 = 9 inches).
   • Height: The altitude, which we want to find (let’s call it h).

3. Apply the Pythagorean Theorem: The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, this is represented as:

   $a^2 + b^2 = c^2$

   Where c is the hypotenuse, and a and b are the other two sides.

4. Substitute the Values:

   In our case, the equation becomes:

   $9^2 + h^2 = 18^2$

5. Solve for h:

   • Calculate the squares:

     $81 + h^2 = 324$

   • Subtract 81 from both sides:

     $h^2 = 324 - 81$

     $h^2 = 243$

   • Take the square root of both sides:

     $h = \sqrt{243}$

6. Simplify the Square Root:

   • Factor 243:

     $243 = 81 \times 3$

   • Simplify the square root:

     $h = \sqrt{81 \times 3}$

     $h = \sqrt{81} \times \sqrt{3}$

     $h = 9 \sqrt{3}$

Result from the Pythagorean Theorem

Therefore, the height of the equilateral triangle is $9 \sqrt{3}$ inches.

Method 2: Using Trigonometry

Another elegant method to find the height of an equilateral triangle involves trigonometry. Specifically, we can use trigonometric ratios in the right-angled triangle formed by the altitude.

Steps to Apply Trigonometry

1. Visualize the Right Triangle: As before, imagine the equilateral triangle with the altitude drawn. This forms two right-angled triangles.

2. Identify the Angle: Each angle in an equilateral triangle is 60 degrees. The altitude bisects the angle at the vertex, creating a 30-degree angle in the right-angled triangle.

3. Choose the Trigonometric Ratio: We can use the sine function, which relates the opposite side (height h) to the hypotenuse (18 inches).

   $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

4. Substitute the Values:

   • $\theta = 60$ degrees
   • Opposite side = h
   • Hypotenuse = 18 inches

   So, the equation becomes:

   $\sin(60^\circ) = \frac{h}{18}$

5. Solve for h:

   • We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$

   • Substitute this value into the equation:

     $\frac{\sqrt{3}}{2} = \frac{h}{18}$

   • Multiply both sides by 18:

     $h = 18 \times \frac{\sqrt{3}}{2}$

     $h = 9 \sqrt{3}$

Result from Trigonometry

Using trigonometry, we also find that the height of the equilateral triangle is $9 \sqrt{3}$ inches.

Method 3: Using the Formula for Equilateral Triangle Height

There's a direct formula to calculate the height of an equilateral triangle, which can save time if you remember it. This formula is derived from either the Pythagorean Theorem or trigonometry and is given by:

$h = \frac{s \sqrt{3}}{2}$

Where h is the height and s is the side length of the equilateral triangle.

Steps to Apply the Formula

1. Identify the Side Length: In our case, the side length s is 18 inches.

2. Substitute the Value:

   $h = \frac{18 \sqrt{3}}{2}$

3. Simplify:

   $h = 9 \sqrt{3}$

Result from the Formula

Using the formula, we again find that the height of the equilateral triangle is $9 \sqrt{3}$ inches.

Conclusion

In conclusion, we have explored three different methods to find the height of the equilateral triangle base of the given pyramid: the Pythagorean Theorem, trigonometry, and the direct formula. All three methods converge on the same answer: the height of the equilateral triangle is $9 \sqrt{3}$ inches. Therefore, the correct answer is:

B. $9 \sqrt{3} \text{ in}$

Understanding these methods not only helps in solving specific problems but also enhances your grasp of geometrical principles and problem-solving strategies. Whether you prefer the intuitive approach of the Pythagorean Theorem, the elegance of trigonometry, or the efficiency of a direct formula, you now have a comprehensive toolkit for tackling similar geometry problems.

Final Answer

The final answer is $\boxed{9 \sqrt{3} \text{ in}}$