Find the volume of each regular pyramid with the given...

Find the volume of each regular pyramid with the given dimensions. Round to the nearest tenth, if necessary. $$\begin{array}{|c|c|c|c|}\hline \text { Number of sides of base } & \text { Base edge length } & \text { Height } & \text { Volume }\\\hline 3 & 10 \mathrm{ft} & 6 \mathrm{ft} & \\\hline\end{array}$$

Find the volume of each regular pyramid with the given dimensions. Round to the nearest tenth, if necessary.
$$\begin{array}{|c|c|c|c|}\hline \text { Number of sides of base } & \text { Base edge length } & \text { Height } & \text { Volume }\\\hline 3 & 10 \mathrm{ft} & 6 \mathrm{ft} & \\\hline\end{array}$$

Key Concepts

Volume of a Pyramid

The volume of a pyramid is found using the formula V = (1/3) × (Base Area) × (Height). This formula applies to any pyramid, regardless of the shape of its base, and it comes from the fact that the pyramid occupies one-third of the volume of a prism with the same base area and height.

Area of a Regular Polygon Base

The area of a regular polygon, which is a polygon with all sides and angles equal, can be calculated using the formula A = (n × s²) / (4 × tan(?/n)), where n is the number of sides and s is the length of each side. This formula is crucial when dealing with pyramids whose bases are regular polygons because it allows computation of the base area needed in the volume formula.

Transcript

00:04 Hi, we're looking at question 28 that asks us to find the volume of a pyramid with a base that has three sides.

00:14 So basically they're telling us it's a triangular pyramid.

00:17 And so to do that, we're going to use the formula volume equals one third area of the base times the height, which we use for all pyramids.

00:26 And to find the capital b for the area of the base, we know that that's a triangle.

00:31 So we have, it's actually an equilateral triangle because they tell us that it's a regular pyramid and they tell us that the length of each side is 10.

00:43 So we need to be able to figure out the area of that triangle.

00:47 Well, to find the area of a triangle, you need one half base times height, that's the formula.

00:52 So base of the triangle times the height of the triangle.

00:55 And that's why i'm doing it over here is it often gets confusing with the bs and the hs and so forth.

01:00 So because we're talking about capital b, we're just talking about this two -dimensional triangle.

01:04 And so knowing that each of the sides is 10, our base then would be 10.

01:10 We are actually going to have to use something we learned earlier to figure out our height.

01:15 Since this is an equilateral triangle, we know that this angle would be 60 degrees.

01:21 And we know that this side would get cut in half, so this will be five.

01:25 So we know that there's that relationship between the short leg, the long leg, and the hypotenuse that we learned with 13...

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