Pyramids - Math

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Question

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

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Answer

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.

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Question

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?

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Answer

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 $15^{2}=225$

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 .

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Question

Find the surface area of the following pyramid.

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Answer

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:

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

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Question

Find the surface area of the following pyramid.

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Answer

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:

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

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Question

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

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Answer

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

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Question

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?

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Answer

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

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Question

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

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Answer

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 $2\ ft$ .

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Question

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

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Answer

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

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Question

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

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Answer

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

.

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Question

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

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Answer

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:

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

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

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

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Question

Find the volume of the following pyramid.

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Answer

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:

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

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

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Question

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?

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Answer

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

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Question

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

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Answer

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.

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Question

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 reveal answer

Answer

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 $15^{2}=225$

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 .

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Question

Find the surface area of the following pyramid.

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Answer

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:

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

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Question

Find the surface area of the following pyramid.

Tap to reveal answer

Answer

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:

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

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Question

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

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Answer

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

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Question

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

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Answer

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 $2\ ft$ .

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Question

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

Tap to reveal answer

Answer

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

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Question

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

Tap to reveal answer

Answer

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

.

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