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Chapter 8: Problem 29

The volume of a triangular pyramid is given by the formula \(V=\frac{1}{3} B h,\) where \(B\) represents the area of the triangular base and \(h\) is the height of the pyramid. Find the volume of a triangular pyramid whose height is given and whose base has the coordinates shown. Assume units are in m. $$h=6 \mathrm{m} ; \text { vertices }(3,5),(-4,2), \text { and }(-1,6)$$

Short Answer

Expert verified
The volume of the triangular pyramid is 19 m鲁.

Step by step solution

01

Identify the given values

We are given the height of the triangular pyramid, \( h = 6 \) m. The vertices of the triangular base are (3,5), (-4,2), and (-1,6). We need to calculate the area \( B \) of this base triangle.
02

Use the area formula for a triangle with coordinates

The area \( B \) of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) can be calculated using the formula:\[B = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\]Substitute the given points into this formula:\[B = \frac{1}{2} \left| 3(2 - 6) + (-4)(6 - 5) + (-1)(5 - 2) \right|\]
03

Calculate the base area

Carry out the arithmetic inside the formula:\[B = \frac{1}{2} \left| 3(-4) + (-4)(1) + (-1)(3) \right|\]\[\Rightarrow B = \frac{1}{2} \left| -12 - 4 - 3 \right| = \frac{1}{2} \left| -19 \right| = \frac{1}{2} \times 19 = 9.5 \text{ m}^2\]
04

Calculate the volume of the pyramid

Now that we have the base area, we can use it to find the volume of the pyramid using the formula:\[ V = \frac{1}{3} B h \]Substitute the known values \( B = 9.5 \text{ m}^2 \) and \( h = 6 \text{ m} \):\[ V = \frac{1}{3} \times 9.5 \times 6 \]\[V = \frac{1}{3} \times 57 = 19 \text{ m}^3 \]
05

Conclude with the final answer

So, the volume of the triangular pyramid is \( 19 \text{ m}^3 \).

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Key Concepts

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

Base Area of Triangle Using Coordinates
The base of the triangular pyramid is a triangle with vertices given in coordinate form. To find the area of this triangle, we will use the formula for the area of a triangle given its vertices on a coordinate plane. This method is particularly useful when the exact shape or tilt of the triangle isn't immediately clear.
By using coordinates, you can calculate the area efficiently without having to draw the triangle on paper.
To do this, we implement the following formula:
  • If your vertices are \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) then the area (B) is: \[B = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
  • This formula helps in finding the area even when the coordinates are negative or mixed positive and negative, as long as they're on a flat plane.
Through this formula, the triangle's base area is derived purely from its vertex coordinates, a clever use of algebra and geometry together.
Triangle Area Formula
The triangle area formula is an essential component of geometry, allowing us to compute the area of a triangle using non-standard measurements, such as coordinates. This is particularly helpful when dealing with triangles that exist on a coordinate plane rather than presented in typical geometric configurations like equilateral or isosceles. Use of the formula:
  • Apply the coordinates of the triangle's vertices into this structured equation.
  • The structured approach simplifies calculation by breaking down the contributions of each vertex.
For instance, substituting our coordinates (3,5), (-4,2), and (-1,6) into this formula enables us to compute the triangle's base area without additional measuring tools or sketching, which ultimately provides a clean and precise base area following simple arithmetic.
Calculation of Volume
Finding the volume of a triangular pyramid involves using the area of its base and its height. For a triangular pyramid, the volume is calculated using the formula \[ V = \frac{1}{3}B h \] where \( B \) is the area of the triangular base, and \( h \) is the vertical height extending from the base to the apex of the pyramid.
Here's how this works:
  • First, determine the area of the base using the triangle area formula from its vertices.
  • Multiply the base area by the vertical height of the pyramid.
  • Finally, divide the result by three, as it is a property of pyramids that a third of the space occupied relates back to having both a base and an apex.
By carefully following these steps, the volume can be deduced correctly, taking into account all necessary components.
Step by Step Problem Solving
Problem-solving in mathematics often involves breaking down the problem into smaller, more manageable parts. In the case of determining the volume of a triangular pyramid:
  • Start by identifying given values, such as vertices for the base and the pyramid's height.
  • Use the vertices to calculate the base area employing the triangle area formula.

  • Then, apply the volume formula for pyramids using the computed base area and known height.
This step-by-step approach clarifies the process, ensuring each part is solved before proceeding to the next. It emphasizes the importance of solving problems in a logical sequence, facilitating easier understanding and reducing errors. The key is methodically following each step with the associated formula before moving to the subsequent calculation.

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