HiSET: Math : Pyramids

Example Questions

Example Question #1 : Pyramids

What is the area of the base of the pyramid with volume  and height ?

Explanation:

The formula for the volume of a pyramid is

The height of the pyramid is , so

.

The volume of the pyramid is .

Thus,

so

.

Note, the area of the base of the pyramid is

.

Thus,

.

Hence,

Example Question #2 : Pyramids

A pyramid with a square base is inscribed inside a right cone with radius 24 and height 10.

Give the volume of the pyramid.

None of the other choices gives the correct response.

Explanation:

The volume of a pyramid with height  and a base of area can be determined using the formula

.

The height of the inscribed pyramid is equal to that of the cone, so we can set . The base of the pyramid is a square inscribed inside a circle, so the length of each diagonal of the square is equal to the diameter of the circle. See the figure below, which shows the bases of the pyramid and the cone.

The circle has radius 24, so its diameter - and the lengths of the diagonals - is twice this, or 48.

The area of a square - which is also a rhombus - is equal to half the product of the lengths of its diagonals, so

Substituting for and , we get

.

Example Question #3 : Pyramids

The base of a right pyramid with height 6 is a regular hexagon with sides of length 6.

Give its volume.

Explanation:

The regular hexagonal base can be divided by its diameters into six equilateral triangles, as seen below:

Each of the triangles has as its sidelength that of the hexagon. If we let this common sidelength be , each of the triangles has area

;

the total area of the base is six times this.

Substituting 6 for , the area of each triangle is

The total area of the base is six times this, or

The volume of a pyramid with height  and a base of area can be determined using the formula

.

Set and ;

Example Question #4 : Pyramids

A right pyramid and a right rectangular prism both have square bases. The base of the pyramid has sides that are 20% longer than those of the bases of the prism; the height of the pyramid is 20% greater than that of the prism.

Which of the following is closest to being correct?

The volume of the pyramid is 33.3% less than that of the prism.

The volume of the pyramid is 74.4% less than that of the prism.

The volume of the pyramid is 82.9% less than that of the prism.

The volume of the pyramid is 61.6% less than that of the prism.

The volume of the pyramid is 42.4% less than that of the prism.

The volume of the pyramid is 42.4% less than that of the prism.

Explanation:

The volume of a right prism with height  and bases of area can be determined using the formula

.

Since its base is a square, if we let be the length of one side, then , and

The volume of a right pyramid with height  and a base of area  can be determined using the formula

.

Since its base is also a square, if we let be the length of one side, then , and

.

The height of the pyramid is 20% greater than the height of the prism - this is 120% of , so . Similarly, the length of a side of the base of the pyramid is 20% greater than that of a base of the prism, so . Substitute in the pyramid volume formula:

We can substitute , the volume of the prism, for . This yields

The volume of the pyramid is equal to 57.6 % of that of the prism, or, equivalently, less.

Example Question #5 : Pyramids

A triangular pyramid in coordinate space has its vertices at the origin, , , and . In  terms of , give its volume.

Explanation:

The pyramid in question can be seen in the diagram below:

This pyramid can be seen as having as its base the triangle on the -plane with vertices at the origin, , and ; this is a right triangle with two legs of length , so its area is half their product, or .

The altitude (perpendicular to the base) is the segment from the origin to , which has length (the height of the pyramid) .

Setting   and  in the formula for the volume of a pyramid:

, the correct response.