# The base of a triangular pyramid is a triangle with corners at #(6 ,2 )#, #(3 ,5 )#, and #(4 ,2 )#. If the pyramid has a height of #9 #, what is the pyramid's volume?

The volume is

This response consists of two main sections:

Section 1: The area of the triangular base is equal to half of the height measured from one vertex to the other.

Section 2: The pyramid's volume is equal to one-third the base's area times its height.

Let's get to the math now.

By signing up, you agree to our Terms of Service and Privacy Policy

Volume of pyramid is

By signing up, you agree to our Terms of Service and Privacy Policy

The volume of the triangular pyramid can be calculated using the formula:

[ \text{Volume} = \frac{1}{3} \times \text{Area of Base} \times \text{Height} ]

First, we need to find the area of the base triangle using the coordinates given. Then, we can calculate the volume using the given height.

Using the coordinates provided, the base triangle has vertices at (6, 2), (3, 5), and (4, 2). We can use the formula for the area of a triangle given its vertices:

[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| ]

Substituting the coordinates, we get:

[ \text{Area} = \frac{1}{2} |6(5 - 2) + 3(2 - 2) + 4(2 - 5)| = \frac{1}{2} |12 + 0 - 6| = \frac{1}{2} |6| = 3 ]

Now, we can calculate the volume:

[ \text{Volume} = \frac{1}{3} \times \text{Area of Base} \times \text{Height} = \frac{1}{3} \times 3 \times 9 = 9 ]

Therefore, the volume of the triangular pyramid is 9 cubic units.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #66 # and the height of the cylinder is #5 #. If the volume of the solid is #45 pi#, what is the area of the base of the cylinder?
- Two corners of a triangle have angles of #(3 pi ) / 8 # and # pi / 6 #. If one side of the triangle has a length of #8 #, what is the longest possible perimeter of the triangle?
- A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #3 #, its base has sides of length #5 #, and its base has a corner with an angle of # pi/4 #. What is the pyramid's surface area?
- The radius of a circle is 2 inches. What is the circle's area?
- The base of a triangular pyramid is a triangle with corners at #(4 ,2 )#, #(3 ,7 )#, and #(5 ,3 )#. If the pyramid has a height of #5 #, what is the pyramid's volume?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7