Solid Geometry - SAT Math

Card 0 of 1496

Question

A right pyramid with a square base has a height that is twice the length of one edge of the base. If the height of the pyramid is 6 meters, find the volume of the pyramid.

Answer

If the height, which is twice the length of the base edges, measures 6 meters, then each base edge must measure 3 meters.

Since the base is a square, the area of the base is 3 x 3 = 9.

Therefore the volume of the right pyramid is V = (1/3) x area of the base x height = 1/3(9)(6) = 18.

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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?

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

Find the volume of the pyramid shown below:

Answer

The formula for the area of a pyramid is $\frac{lwh}{3}$ . In this case, the length is , the width is , and the height is .

$8\times6\times9=432$ and $439\div3=144$ .

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Question

Figure not drawn to scale

In the pyramid above, the base is a square. The distance between points C and D is 6 inches and the length of side b is 5 inches. What is the volume of this pyramid?

Answer

To find the volume of a pyramid, you need to use the equation below:

$Volume=\frac{(area, of, the, base)(height)}{3}$

To find the height (shown by the yellow line), we can draw a right triangle using the yellow line, blue line and side b (5 inches). Because the hypotenuse is 5 inches, using the common Pythagorean 3-4-5 triple. The blue line is 3 inches and the yellow line (height) is 4 inches. Also, to find side a, we can use the blue line (3 inches) and half of the red line (3 inches) and the Pythagorean Theorum.

$\sqrt{18}, in=side, a$

Because the base is a square, the area of the base is equal to the square of side a:

Now we plug in these values to find the volume:

$Volume=\frac{(18)(4)}{3}$

$Volume=\frac{(72)}{3}$

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Question

Calculate the volume of the rectangular pyramid with height , base width , and base length

Answer

The volume of a rectangular pyramid with height , base width , and base length is given by

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

For this pyramid, , , and To calculate its volume, substitute the values for , , and into the formula:

$V=\frac{1}{3}(6)(6)(4)=\frac{144}{3}=48$

Therefore, the volume of the given rectangular pyramid is

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Question

If a cube is 3” on all sides, what is the length of the diagonal of the cube?

Answer

General formula for the diagonal of a cube if each side of the cube = s

Use Pythagorean Theorem to get the diagonal across the base:

s2 + s2 = h2

And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:

h2 + s2 = d2

s2 + s2 + s2 = d2

3 * s2 = d2

d = √ (3 * s2) = s √3

So, if s = 3 then the answer is 3√3

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Question

A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube?

Answer

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.

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Question

What is the length of the diagonal of a cube with volume of 512 in3?

Answer

The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:

_s_3 = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)

Or for our simpler case:

d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

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Question

What is the length of the diagonal of a cube with volume of 1728 in3?

Answer

The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:

_s_3 = 1728, or (taking the cube root of both sides), s = 12.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):

d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)

Or, for our simpler case:

d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (12)2 + (12)2 + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)

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Question

What is the length of the diagonal of a cube with surface area of 294 in2?

Answer

The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6_s_2, where s is the length of the cube. For our data, this is:

6_s_2 = 294

_s_2 = 49

(taking the square root of both sides) s = 7

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √((_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)

Or for our simpler case:

d = √((x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (7)2 + (7)2 + (7)2) = √( 49 + 49 + 49) = √(49 * 3) = 7√(3)

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Question

A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?

Answer

The volume of a rectangular prism is $length\cdot width\cdot height$ .

We are told that the shortest edge is 3. Let us call this the height.

We now have , or .

Now we replace variables by known values:

$192=2lw+6l+6w = 2(48) + 6(\frac{48}{w})+6w$

Now we have:

$96 = \frac{288}{w}+6w \Rightarrow 96w = 288 + 6w^2 \Rightarrow 16w=48+w^2$

$\Rightarrow w^2-16w+48=0\Rightarrow (w-4)(w-12)=0$

We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:

$\sqrt{3^2+4^2+12^2}=\sqrt{9+16+144} = \sqrt{169} = 13$

If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:

$\sqrt{3^2+4^2}=\sqrt{25}=5$

We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).

This diagonal is then $\sqrt{5^2+12^2}=\sqrt{169}=13$ .

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Question

A cone has a base circumference of 77_π_ in and a height of 2 ft. What is its approximate volume?

Answer

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2_πr_, or, for our values 77_π_ = 2_πr_. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr_2_h

For our values this would be:

V = (1/3)π * 38.52 * 24 = 8 * 1482.25_π_ = 11,858π in3

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Question

An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?

Answer

The volume of a cone is given by the formula V = (πr2)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr2)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr2)/(3w) * (1/60) = π(r2)(h)/(180w)

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Question

A cone has a base radius of 13 in and a height of 6 in. What is its volume?

Answer

The basic form for the volume of a cone is:

V = (1/3)πr_2_h

For this simple problem, we merely need to plug in our values:

V = (1/3)π_132 6 = 169 * 2_π* = 338_π_ in3

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Question

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Answer

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

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Question

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Answer

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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Question

Find the volume of a cone with a radius of and a height of .

Answer

Write the formula to find the volume of a cone.

$V=\frac{1}{3}\pi r^2 h$

Substitute the known values and simplify.

$V=\frac{1}{3}\pi (10)^2 (90)= \frac{1}{3}\pi (100)(90)= \pi (100)(30)=3000\pi$

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Question

Find the area of a cone whose radius is 4 and height is 3.

Answer

To solve, simply use the formula for the area of a cone. Thus,

$V=\frac{4}{3}\pi{r^2}h=\frac{4}{3}\pi4^23=4\pi*16=64\pi$

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Question

Find the volume of a cone with radius 3 and height 5.

Answer

To solve, simply use the formula for the volume of a cone. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi{3^2}5=\frac{1}{3}\pi95=15\pi$

To remember the formula for volume of a cone, it helps to break it up into it's base and height. The base is a circle and the height is just h. Now, just multiplying those two together would give you the formula of a cylinder (see problem 3 in this set). So, our formula is going to have to be just a portion of that. Similarly to volume of a pyramid, that fraction is one third.

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Question

The volume of a right circular cone is $9\pi$ . If the cone's height is equal to its radius, what is the radius of the cone?

Answer

The volume of a right circular cone with radius and height is given by:

$V = \pi r^2 \frac{h}{3}$

Since the height of this cone is equal to its radius, we can say:

$V = \pi r^2 \frac{r}{3} = \frac{1}{3}\pi r^3$

Now, we can substitute our given volume into the equation and solve for our radius.

$9\pi = \frac{1}{3}\pi r^3$

$27\pi = \pi r^3$

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