Pyramids - HiSET

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Question

What is the area of the base of the pyramid with volume $12 ;\textup{m}^3$ and height $1;\textup{m}$ ?

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Answer

The formula for the volume of a pyramid is

$V = \frac{lwh}{3}$

The height of the pyramid is $1;\textup{m}$ , so

.

The volume of the pyramid is $12 ;\textup{m}^3$ .

Thus,

$12=\frac{lw}{3}$

so

$lw=12\times3=36$ .

Note, the area of the base of the pyramid is

.

Thus,

.

Hence,

$A = 36 ;\textup{m}^2$

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Question

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?

Tap to reveal answer

Answer

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 $B = s^{2}$ , and

$V =s^{2}h$

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

$V '=\frac{1}{3} B'h'$ .

Since its base is also a square, if we let be the length of one side, then $B' = s'^{2}$ , and

$V '=\frac{1}{3} s'^{2}h'$ .

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:

$V '=\frac{1}{3} (1.2s )^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.2^{2} \cdot s ^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.44 \cdot 1.2 \cdot s ^{2} h$

$=0.576 s ^{2} h$

We can substitute , the volume of the prism, for $s^{2}h$ . This yields

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

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Question

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

Give its volume.

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Answer

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

$A = \frac{s^{2}\sqrt{3}}{4}$ ;

the total area of the base is six times this.

Substituting 6 for , the area of each triangle is

$A = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9 \sqrt{3}$

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

$B = 6 A =6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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

$V = \frac{1}{3}Bh$ .

Set $B = 54 \sqrt{3}$ and ;

$V = \frac{1}{3} (54\sqrt{3}) (6) = 108 \sqrt{3}$

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Question

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

Give the volume of the pyramid.

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Answer

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

$V = \frac{1}{3}Bh$ .

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

$B = \frac{1}{2}d_{1}d_{2} = \frac{1}{2} (48) (48) = 1,152$

Substituting for and , we get

$V = \frac{1}{3} (1,152)(10) = 3,840$ .

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Question

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

Tap to reveal answer

Answer

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 $B = \frac{1}{2}c^{2}$ .

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

Setting $B = \frac{1}{2}c^{2}$ and in the formula for the volume of a pyramid:

$V = \frac{1}{3} Bh$

$V = \frac{1}{3}\cdot \frac{1}{2}c^{2} \cdot c= \frac{1}{6}c^{3}$ , the correct response.

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Question

What is the area of the base of the pyramid with volume $12 ;\textup{m}^3$ and height $1;\textup{m}$ ?

Tap to reveal answer

Answer

The formula for the volume of a pyramid is

$V = \frac{lwh}{3}$

The height of the pyramid is $1;\textup{m}$ , so

.

The volume of the pyramid is $12 ;\textup{m}^3$ .

Thus,

$12=\frac{lw}{3}$

so

$lw=12\times3=36$ .

Note, the area of the base of the pyramid is

.

Thus,

.

Hence,

$A = 36 ;\textup{m}^2$

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Question

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?

Tap to reveal answer

Answer

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 $B = s^{2}$ , and

$V =s^{2}h$

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

$V '=\frac{1}{3} B'h'$ .

Since its base is also a square, if we let be the length of one side, then $B' = s'^{2}$ , and

$V '=\frac{1}{3} s'^{2}h'$ .

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:

$V '=\frac{1}{3} (1.2s )^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.2^{2} \cdot s ^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.44 \cdot 1.2 \cdot s ^{2} h$

$=0.576 s ^{2} h$

We can substitute , the volume of the prism, for $s^{2}h$ . This yields

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

← Didn't Know|Knew It →

Question

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

Give its volume.

Tap to reveal answer

Answer

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

$A = \frac{s^{2}\sqrt{3}}{4}$ ;

the total area of the base is six times this.

Substituting 6 for , the area of each triangle is

$A = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9 \sqrt{3}$

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

$B = 6 A =6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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

$V = \frac{1}{3}Bh$ .

Set $B = 54 \sqrt{3}$ and ;

$V = \frac{1}{3} (54\sqrt{3}) (6) = 108 \sqrt{3}$

← Didn't Know|Knew It →

Question

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

Give the volume of the pyramid.

Tap to reveal answer

Answer

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

$V = \frac{1}{3}Bh$ .

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

$B = \frac{1}{2}d_{1}d_{2} = \frac{1}{2} (48) (48) = 1,152$

Substituting for and , we get

$V = \frac{1}{3} (1,152)(10) = 3,840$ .

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Question

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

Tap to reveal answer

Answer

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 $B = \frac{1}{2}c^{2}$ .

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

Setting $B = \frac{1}{2}c^{2}$ and in the formula for the volume of a pyramid:

$V = \frac{1}{3} Bh$

$V = \frac{1}{3}\cdot \frac{1}{2}c^{2} \cdot c= \frac{1}{6}c^{3}$ , the correct response.

← Didn't Know|Knew It →

Question

What is the area of the base of the pyramid with volume $12 ;\textup{m}^3$ and height $1;\textup{m}$ ?

Tap to reveal answer

Answer

The formula for the volume of a pyramid is

$V = \frac{lwh}{3}$

The height of the pyramid is $1;\textup{m}$ , so

.

The volume of the pyramid is $12 ;\textup{m}^3$ .

Thus,

$12=\frac{lw}{3}$

so

$lw=12\times3=36$ .

Note, the area of the base of the pyramid is

.

Thus,

.

Hence,

$A = 36 ;\textup{m}^2$

← Didn't Know|Knew It →

Question

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?

Tap to reveal answer

Answer

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 $B = s^{2}$ , and

$V =s^{2}h$

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

$V '=\frac{1}{3} B'h'$ .

Since its base is also a square, if we let be the length of one side, then $B' = s'^{2}$ , and

$V '=\frac{1}{3} s'^{2}h'$ .

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:

$V '=\frac{1}{3} (1.2s )^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.2^{2} \cdot s ^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.44 \cdot 1.2 \cdot s ^{2} h$

$=0.576 s ^{2} h$

We can substitute , the volume of the prism, for $s^{2}h$ . This yields

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

← Didn't Know|Knew It →

Question

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

Give its volume.

Tap to reveal answer

Answer

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

$A = \frac{s^{2}\sqrt{3}}{4}$ ;

the total area of the base is six times this.

Substituting 6 for , the area of each triangle is

$A = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9 \sqrt{3}$

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

$B = 6 A =6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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

$V = \frac{1}{3}Bh$ .

Set $B = 54 \sqrt{3}$ and ;

$V = \frac{1}{3} (54\sqrt{3}) (6) = 108 \sqrt{3}$

← Didn't Know|Knew It →

Question

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

Give the volume of the pyramid.

Tap to reveal answer

Answer

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

$V = \frac{1}{3}Bh$ .

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

$B = \frac{1}{2}d_{1}d_{2} = \frac{1}{2} (48) (48) = 1,152$

Substituting for and , we get

$V = \frac{1}{3} (1,152)(10) = 3,840$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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 $B = \frac{1}{2}c^{2}$ .

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

Setting $B = \frac{1}{2}c^{2}$ and in the formula for the volume of a pyramid:

$V = \frac{1}{3} Bh$

$V = \frac{1}{3}\cdot \frac{1}{2}c^{2} \cdot c= \frac{1}{6}c^{3}$ , the correct response.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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 $B = \frac{1}{2}c^{2}$ .

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

Setting $B = \frac{1}{2}c^{2}$ and in the formula for the volume of a pyramid:

$V = \frac{1}{3} Bh$

$V = \frac{1}{3}\cdot \frac{1}{2}c^{2} \cdot c= \frac{1}{6}c^{3}$ , the correct response.

← Didn't Know|Knew It →

Question

What is the area of the base of the pyramid with volume $12 ;\textup{m}^3$ and height $1;\textup{m}$ ?

Tap to reveal answer

Answer

The formula for the volume of a pyramid is

$V = \frac{lwh}{3}$

The height of the pyramid is $1;\textup{m}$ , so

.

The volume of the pyramid is $12 ;\textup{m}^3$ .

Thus,

$12=\frac{lw}{3}$

so

$lw=12\times3=36$ .

Note, the area of the base of the pyramid is

.

Thus,

.

Hence,

$A = 36 ;\textup{m}^2$

← Didn't Know|Knew It →

Question

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?

Tap to reveal answer

Answer

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 $B = s^{2}$ , and

$V =s^{2}h$

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

$V '=\frac{1}{3} B'h'$ .

Since its base is also a square, if we let be the length of one side, then $B' = s'^{2}$ , and

$V '=\frac{1}{3} s'^{2}h'$ .

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:

$V '=\frac{1}{3} (1.2s )^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.2^{2} \cdot s ^{2} \cdot 1.2h$

$=\frac{1}{3} \cdot 1.44 \cdot 1.2 \cdot s ^{2} h$

$=0.576 s ^{2} h$

We can substitute , the volume of the prism, for $s^{2}h$ . This yields

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

← Didn't Know|Knew It →

Question

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

Give its volume.

Tap to reveal answer

Answer

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

$A = \frac{s^{2}\sqrt{3}}{4}$ ;

the total area of the base is six times this.

Substituting 6 for , the area of each triangle is

$A = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9 \sqrt{3}$

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

$B = 6 A =6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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

$V = \frac{1}{3}Bh$ .

Set $B = 54 \sqrt{3}$ and ;

$V = \frac{1}{3} (54\sqrt{3}) (6) = 108 \sqrt{3}$

← Didn't Know|Knew It →

Question

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

Give the volume of the pyramid.

Tap to reveal answer

Answer

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

$V = \frac{1}{3}Bh$ .

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

$B = \frac{1}{2}d_{1}d_{2} = \frac{1}{2} (48) (48) = 1,152$

Substituting for and , we get

$V = \frac{1}{3} (1,152)(10) = 3,840$ .

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