# 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?

Volume of pyramid

First to find the area of the triangular base.

If three sides are known, area of the triangle is given by the formula

where s is the semi perimeter of the triangular base, a,b and c the sides of the base.

Using distance formula we can find the sides.

Semi perimeter

Area of triangular base

Volume of pyramid

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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 find the area of the base triangle using its coordinates. We can use the formula for the area of a triangle given its vertices:

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

Given the coordinates: ( (x_1, y_1) = (4, 2) ) ( (x_2, y_2) = (3, 7) ) ( (x_3, y_3) = (5, 3) )

Substituting the coordinates into the formula, we get:

[ \text{Area} = \frac{1}{2} \times |4(7 - 3) + 3(3 - 2) + 5(2 - 7)| ] [ \text{Area} = \frac{1}{2} \times |4(4) + 3(1) + 5(-5)| ] [ \text{Area} = \frac{1}{2} \times |16 + 3 - 25| ] [ \text{Area} = \frac{1}{2} \times |-6| ] [ \text{Area} = \frac{1}{2} \times 6 ] [ \text{Area} = 3 ]

Now, we have the base area and the height of the pyramid. Substituting into the volume formula:

[ V = \frac{1}{3} \times 3 \times 5 ] [ V = \frac{1}{3} \times 15 ] [ V = 5 ]

So, the volume of the triangular pyramid is ( 5 ) cubic units.

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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.

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