The base of a triangular pyramid is a triangle with corners at #(2 ,5 )#, #(1 ,6 )#, and #(2 ,8 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(2 ,5  )#, #(1  ,6   )#, and #(2  ,8  )#.  If the pyramid has a height of #4  #, what is the pyramid's volume?

Jayden Jackson · Accepted Answer

Volume of a pyramid is # 2 # cubic unit.

Volume of a pyramid is #1/3*#base area #*#hight. #(x_1,y_1)=(2,5) ,(x_2,y_2)=(1,6),(x_3,y_3)=(2,8) , h=4# Area of Triangle is #A_b = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|# #A_b = |1/2(2(6−8)+1(8−5)+2(5−6))|# or #A_b = |1/2(-4+3-2)| = | -3/2| =3/2#sq.unit Volume of a pyramid is #1/3*A_b*h = 1/cancel3 *cancel3/2*4 = 2 # cubic.unit [Ans]

Autumn Armstrong · Answer

undefined To find the volume of the triangular pyramid, we use the formula:

$$ V = \frac{1}{3} \times \text{base area} \times \text{height} $$

First, we need to calculate the area of the base triangle using the coordinates given for its vertices. We can use the formula for the area of a triangle given its vertices.

Let's denote the vertices as $ A(2, 5) $, $ B(1, 6) $, and $ C(2, 8) $.

The formula for the area of a triangle with vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ is:

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

Let's calculate the area of the base triangle using the given coordinates:

$$ \text{Area} = \frac{1}{2} |2(6 - 8) + 1(8 - 5) + 2(5 - 6)| $$

$$ \text{Area} = \frac{1}{2} |-4 + 3 - 2| $$

$$ \text{Area} = \frac{1}{2} \times 3 $$

$$ \text{Area} = \frac{3}{2} $$

Now, we have the area of the base triangle and the height of the pyramid. Let's substitute these values into the formula for the volume of the pyramid:

$$ V = \frac{1}{3} \times \frac{3}{2} \times 4 $$

$$ V = \frac{1}{3} \times 6 \times 4 $$

$$ V = 8 $$

Therefore, the volume of the triangular pyramid is 8 cubic units.

Jayden Jackson · Answer

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

$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

First, we need to find the area of the base triangle using the coordinates provided. We can use the formula for the area of a triangle given its vertices.

Then, we can substitute the base area and the height into the volume formula to find the volume of the pyramid.

Given the coordinates of the base triangle as (2,5), (1,6), and (2,8), the base triangle's area is 3 square units.

Substituting the base area and height into the volume formula:

$$ V = \frac{1}{3} \times 3 \times 4 = 4 $$

So, the volume of the triangular pyramid is 4 cubic units.