Pyramids - Math

Card 0 of 64

A pyramid with a square base has height equal to the perimeter of its base. Which is the greater quantity?

(A) Twice the area of its base

(B) The area of one of its triangular faces

Tap to see back →

Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or $4 \times 2 = 8$ . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or

$\frac{1}{2}$ \times 2 \times $\sqrt{65}$ = $\sqrt{65}$

Since twice the area of the base is $8 = $\sqrt{64}$$ , the problem comes down to comparing $\sqrt{64}$ and $\sqrt{65}$ ; the latter, which is (B), is greater.

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

Tap to see back →

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = $x^{2}$; h = 2x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $x^{2}$ \cdot 2x = \frac {2}{3} $x^{3}$$

(b) $B = (2 $x)^{2}$ = $4x^{2}$; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $4x^{2}$ \cdot x = \frac {4}{3} $x^{3}$$

Regardless of , (b) is the greater quantity.

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Tap to see back →

The volume of a pyramid with height and a square base with sidelength is

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

(a) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $1^{2}$ \cdot 4 = $\frac{4}{3}$$

(b) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $2^{2}$ \cdot 1 = $\frac{4}{3}$$

The two pyramids have equal volume.

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Tap to see back →

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

A pyramid with a square base has height equal to the perimeter of its base. Its volume is . In terms of , what is the length of each side of its base?

Tap to see back →

The volume of a pyramid is given by the formula

$V = $\frac{1}{3}$ A h$

where is the area of its base and is its height.

Let be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

and the area of the base is

$A = $s^{2}$$

So the volume formula becomes

$$\frac{1}{3}$ \cdot $s^{2}$ \cdot 4s = V$

Solve for :

$$\frac{4}{3}$ \cdot $s^{3}$ = V$

$$\frac{3}{4}$ \cdot $\frac{4}{3}$ \cdot $s^{3}$ =\frac{3}{4}$ \cdot V$

$s =\sqrt[3]{$\frac{3V}{4}$}$

An architect wants to make a square pyramid and fill it with 12,000 cubic feet of sand. If the base of the pyramid is 30 feet on each side, how tall does he need to make it?

Tap to see back →

Volume of Pyramid = 1/3 * Area of Base * Height

12,000 ft3 = 1/3 * 30ft * 30ft * H

12,000 = 300 * H

H = 12,000 / 300 = 40

H = 40 ft

What is the surface area of a square pyramid with a height of 12 in and a base side length of 10 in?

Tap to see back →

The surface area of a square pyramid can be broken into the area of the square base and the areas of the four triangluar sides. The area of a square is given by:

$A = $s^{2}$ = 100 $in^{2}$$

The area of a triangle is:

$A = $\frac{1}{2}$ bh$

The given height of 12 in is from the vertex to the center of the base. We need to calculate the slant height of the triangular face by using the Pythagorean Theorem:

$SH^{2}$ = $H^{2}$ + $B^{2}$

where and (half the base side) resulting in a slant height of 13 in.

So, the area of the triangle is:

$A = $\frac{1}{2}$ bh = $\frac{1}{2}$ \cdot 10 \cdot 13 = 65 $in^{2}$$

There are four triangular sides totaling $260 $in^{2}$$ for the sides.

The total surface area is thus $360 $in^{2}$$ , including all four sides and the base.

What is the surface are of a pyramid with a square base length of 15 and a slant height (the height from the midpoint of one of the side lengths to the top of the pyramid) of 12?

Tap to see back →

To find the surface area of a pyramid we must add the areas of all five of the shapes creating the pyramid together.

We have four triangles that all have the same area and a square that supports the pyramid.

To find the area of the square we take the side length of 15 and square it

The area of the square is .

To find the area of the triangle we must use the equation for the area of a triangle which is $\frac{base\times height}{2}$

Plug in the slant height 12 as the height of the triangle and use the side length of the square 15 as the base in our equation to get

$$\frac{1}{2}$1512=90$

The area of each triangle is .

We then multiply the area of each triangle by 4 to find the area of all four triangles .

The four triangles have a surface area of .

We add the surface area of the four triangles with the area of the square to get the answer for the surface area of the pyramid which is .

The answer is .

Find the surface area of the following pyramid.

Tap to see back →

The formula for the surface area of a pyramid is:

$SA = 4 $\left(A_{triangle}$\right)+Base$

$SA = 4 $\left($\frac{1}{2}${h_{s}$}w\right)+(lw)$

Where is the length of the slant height, is the width of the base, and is the length of the base

In order to determine the areas of the triangle, you will need to use the Pythagorean Theorem to find the slant height:

Plugging in our values, we get:

$SA = 4 \left($\frac{1}{2}$\cdot (12m)(10m)\right)+((10m)(10m))$

Find the surface area of the following pyramid.

Tap to see back →

The formula for the surface area of a pyramid is:

$SA = (base)+\frac{1}{2}$(perimeter)(slant height)$

$SA = (l)(w) + $\frac{1}{2}$(l+l+w+w)(h_s)$

Where is the length of the base, is the width of the base, and is the slant height

Use the Pythagorean Theorem to find the length of the slant height:

Plugging in our values, we get:

$SA = (12m)(12m) + $\frac{1}{2}$(12m+12m+12m+12m)(8m)$

What is the surface area of a square pyramid with a base side equal to 4 and a slant length equal to 6?

Tap to see back →

The surface area of of a square pyramid can be determined using the following equation:

The volume of a 6-foot-tall square pyramid is 8 cubic feet. How long are the sides of the base?

Tap to see back →

Volume of a pyramid is

$$\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (height)$

Thus:

$8=\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (6)$

$8=2\cdot (Area\ of\ the\ base)$

Area of the base is $4\ $ft^{2}$$ .

Therefore, each side is $2\ ft$ .

What is the sum of the number of vertices, edges, and faces of a square pyramid?

Tap to see back →

A square pyramid has one square base and four triangular sides.

Vertices (where two or more edges come together): 5. There are four vertices on the base (one at each corner of the square) and a fifth at the top of the pyramid.

Edges (where two faces come together): 8. There are four edges on the base (one along each side) and four more along the sides of the triangular faces extending from the corners of the base to the top vertex.

Faces (planar surfaces): 5. The base is one face, and there are four triangular faces that form the top of the pyramid.

Total

What is the volume of a pyramid with a height of and a square base with a side length of ?

Tap to see back →

To find the volume of a pyramid we must use the equation

$Volume: of: pyramid=\frac{1}{3}$(area\ of\ base)(height)$

We must first solve for the area of the square using

$A=(side\ $length^{2}$)$

We plug in and square it to get

We then plug our answer into the equation for the pyramid with the height to get

$V=\frac{1}{3}$(49)(15)$

We multiply the result to get our final answer for the volume of the pyramid

.

Find the volume of the following pyramid. Round the answer to the nearest integer.

Tap to see back →

The formula for the volume of a pyramid is:

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

where is the width of the base, is the length of the base, and is the height of the pyramid.

In order to determine the height of the pyramid, you will need to use the Pythagoream Theorem to find the slant height:

Now we can use the slant height to find the pyramid height, once again using the Pythagoream Theorem:

$A = $\sqrt{119}$m$

Plugging in our values, we get:

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

$V = $\frac{(10m)(10m)(\sqrt{119}$m)}{3}$

$V = $$\frac{100\sqrt{119}$m^3$}{3} \approx $364m^3$$

Find the volume of the following pyramid.

Tap to see back →

The formula for the volume of a pyramid is:

$V = $\frac{1}{3}$ (base)(height)$

$V = $\frac{1}{3}$ (l)(w)(h)$

Where is the length of the base, is the width of the base, and is the height of the pyramid

Use the Pythagorean Theorem to find the length of the slant height:

Now, use the Pythagorean Theorem again to find the length of the height:

$A = 2$\sqrt{7}$m$

Plugging in our values, we get:

$V = $\frac{1}{3}$ (12m)(12m)(2$\sqrt{7}$m)$

$V = $96$\sqrt{7}$m^3$$

A pyramid with a square base has height equal to the perimeter of its base. Which is the greater quantity?

(A) Twice the area of its base

(B) The area of one of its triangular faces

Tap to see back →

Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or $4 \times 2 = 8$ . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or

$\frac{1}{2}$ \times 2 \times $\sqrt{65}$ = $\sqrt{65}$

Since twice the area of the base is $8 = $\sqrt{64}$$ , the problem comes down to comparing $\sqrt{64}$ and $\sqrt{65}$ ; the latter, which is (B), is greater.

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

Tap to see back →

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = $x^{2}$; h = 2x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $x^{2}$ \cdot 2x = \frac {2}{3} $x^{3}$$

(b) $B = (2 $x)^{2}$ = $4x^{2}$; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $4x^{2}$ \cdot x = \frac {4}{3} $x^{3}$$

Regardless of , (b) is the greater quantity.

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Tap to see back →

The volume of a pyramid with height and a square base with sidelength is

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

(a) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $1^{2}$ \cdot 4 = $\frac{4}{3}$$

(b) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $2^{2}$ \cdot 1 = $\frac{4}{3}$$

The two pyramids have equal volume.

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Tap to see back →

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.