Geometry - HSPT Math
Card 1 of 2400
A metal cylindrical brick has a height of 
. The area of the top is 
. A circular hole with a radius of 
is centered and drilled half-way down the brick. What is the volume of the resulting shape?
A metal cylindrical brick has a height of
Tap to reveal answer
To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.
The volume of a cylinder is given by the product of the base area times the height: 
.
Find the initial volume using the given base area and height.
Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.
Finally, subtract the volume of the hole from the total initial volume.
To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.
The volume of a cylinder is given by the product of the base area times the height:
Find the initial volume using the given base area and height.
Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.
Finally, subtract the volume of the hole from the total initial volume.
← Didn't Know|Knew It →
If a waterproof box is 50cm in length, 20cm in depth, and 30cm in height, how much water will overflow if 35 liters of water are poured into the box?
If a waterproof box is 50cm in length, 20cm in depth, and 30cm in height, how much water will overflow if 35 liters of water are poured into the box?
Tap to reveal answer
The volume of the box is 50 * 20 * 30 cm = 30,000 cm3.
1cm3 = 1mL, 30,000 cm3 = 30,000mL = 30 L.
Because the volume of the box is only 30 L, 5 L of the 35 L will not fit into the box.
The volume of the box is 50 * 20 * 30 cm = 30,000 cm3.
1cm3 = 1mL, 30,000 cm3 = 30,000mL = 30 L.
Because the volume of the box is only 30 L, 5 L of the 35 L will not fit into the box.
← Didn't Know|Knew It →
A rectangular prism has a length that is twice as long as its width, and a width that is twice as long as its height. If the surface area of the prism is 252 square units, what is the volume, in cubic units, of the prism?
A rectangular prism has a length that is twice as long as its width, and a width that is twice as long as its height. If the surface area of the prism is 252 square units, what is the volume, in cubic units, of the prism?
Tap to reveal answer
Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:
l = 2_w_
w = 2_h_
Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:
surface area = 2_lw_ + 2_lh_ + 2_wh_ = 252
We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2_h_. Because l = 2_w_, we can write l as follows:
l = 2_w_ = 2(2_h_) = 4_h_
Now, let's substitute w = 2_h_ and l = 4_h_ into the equation we wrote for surface area.
2(4_h_)(2_h_) + 2(4_h_)(h) + 2(2_h_)(h) = 252
Simplify each term.
16_h_2 + 8_h_2 + 4_h_2 = 252
Combine _h_2 terms.
28_h_2 = 252
Divide both sides by 28.
_h_2 = 9
Take the square root of both sides.
h = 3.
This means that h = 3. Because w = 2_h_, the width must be 6. And because l = 2_w_, the length must be 12.
Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.
volume of a prism = l • w • h
volume = 12(6)(3)
= 216 cubic units
The answer is 216.
Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:
l = 2_w_
w = 2_h_
Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:
surface area = 2_lw_ + 2_lh_ + 2_wh_ = 252
We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2_h_. Because l = 2_w_, we can write l as follows:
l = 2_w_ = 2(2_h_) = 4_h_
Now, let's substitute w = 2_h_ and l = 4_h_ into the equation we wrote for surface area.
2(4_h_)(2_h_) + 2(4_h_)(h) + 2(2_h_)(h) = 252
Simplify each term.
16_h_2 + 8_h_2 + 4_h_2 = 252
Combine _h_2 terms.
28_h_2 = 252
Divide both sides by 28.
_h_2 = 9
Take the square root of both sides.
h = 3.
This means that h = 3. Because w = 2_h_, the width must be 6. And because l = 2_w_, the length must be 12.
Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.
volume of a prism = l • w • h
volume = 12(6)(3)
= 216 cubic units
The answer is 216.
← Didn't Know|Knew It →
A foam ball has a volume of 2 units and has a diameter of x. If a second foam ball has a radius of 2x, what is its volume?
A foam ball has a volume of 2 units and has a diameter of x. If a second foam ball has a radius of 2x, what is its volume?
Tap to reveal answer
Careful not to mix up radius and diameter. First, we need to identify that the second ball has a radius that is 4 times as large as the first ball. The radius of the first ball is (1/2)x and the radius of the second ball is 2x. The volume of the second ball will be 43, or 64 times bigger than the first ball. So the second ball has a volume of 2 * 64 = 128.
Careful not to mix up radius and diameter. First, we need to identify that the second ball has a radius that is 4 times as large as the first ball. The radius of the first ball is (1/2)x and the radius of the second ball is 2x. The volume of the second ball will be 43, or 64 times bigger than the first ball. So the second ball has a volume of 2 * 64 = 128.
← Didn't Know|Knew It →
Kim from Idaho can only stack bales of hay in her barn for 3 hours before she needs a break. She stacks the bales at a rate of 2 bales per minute, 3 bales high with 5 bales in a single row. How many full rows will she have at the end of her stacking?
Kim from Idaho can only stack bales of hay in her barn for 3 hours before she needs a break. She stacks the bales at a rate of 2 bales per minute, 3 bales high with 5 bales in a single row. How many full rows will she have at the end of her stacking?
Tap to reveal answer
She will stack 360 bales in 3 hours. One row requires 15 bales. 360 divided by 15 is 24.
She will stack 360 bales in 3 hours. One row requires 15 bales. 360 divided by 15 is 24.
← Didn't Know|Knew It →
What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
Tap to reveal answer
The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr2. Therefore, the total formula for the volume of the cylinder is: V = πr2h.
First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.
Based on this, we know that the volume of our cylinder must be: π*12.52*41.3 = 6453.125π in3
The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr2. Therefore, the total formula for the volume of the cylinder is: V = πr2h.
First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.
Based on this, we know that the volume of our cylinder must be: π*12.52*41.3 = 6453.125π in3
← Didn't Know|Knew It →
What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
Tap to reveal answer
The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.
Thus, the volume can also be expressed as V = πr2h.
The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.
Now we can easily calculate the volume:
V = 6.52π * 27.5 = 1161.88π in3
The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.
Thus, the volume can also be expressed as V = πr2h.
The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.
Now we can easily calculate the volume:
V = 6.52π * 27.5 = 1161.88π in3
← Didn't Know|Knew It →
An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
Tap to reveal answer
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
← Didn't Know|Knew It →
An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in3, what is the mass of the remaining wood after the cylinder is drilled out?
An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in3, what is the mass of the remaining wood after the cylinder is drilled out?
Tap to reveal answer
We must calculate our two volumes and subtract them. Following this, we will multiply by the density.
The volume of the cube is very simple: 12 * 12 * 12, or 1728 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 3.752 * 12 = 168.75π in3.
The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in3. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.
We must calculate our two volumes and subtract them. Following this, we will multiply by the density.
The volume of the cube is very simple: 12 * 12 * 12, or 1728 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 3.752 * 12 = 168.75π in3.
The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in3. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.
← Didn't Know|Knew It →
At x = 3, the line y = 4_x_ + 12 intersects the surface of a sphere that passes through the xy-plane. The sphere is centered at the point at which the line passes through the x-axis. What is the volume, in cubic units, of the sphere?
At x = 3, the line y = 4_x_ + 12 intersects the surface of a sphere that passes through the xy-plane. The sphere is centered at the point at which the line passes through the x-axis. What is the volume, in cubic units, of the sphere?
Tap to reveal answer
We need to ascertain two values: The center point and the point of intersection with the surface. Let's do that first:
The center is defined by the x-intercept. To find that, set the line equation equal to 0 (y = 0 at the x-intercept):
0 = 4x + 12; 4_x_ = –12; x = –3; Therefore, the center is at (–3,0)
Next, we need to find the point at which the line intersects with the sphere's surface. To do this, solve for the point with x-coordinate at 3:
y = 4 * 3 + 12; y = 12 + 12; y = 24; therefore, the point of intersection is at (3,24)
Reviewing our data so far, this means that the radius of the sphere runs from the center, (–3,0), to the edge, (3,24). If we find the distance between these two points, we can ascertain the length of the radius. From that, we will be able to calculate the volume of the sphere.
The distance between these two points is defined by the distance formula:
d = √( (_x_1 – _x_0)2 + (_y_1 – _y_0)2 )
For our data, that is:
√( (3 + 3)2 + (24 – 0)2 ) = √( 62 + 242 ) = √(36 + 576) = √612 = √(2 * 2 * 3 * 3 * 17) = 6√(17)
Now, the volume of a sphere is defined by: V = (4/3)_πr_3
For our data, that would be: (4/3)π * (6√(17))3 = (4/3) * 63 * 17√(17) * π = 4 * 2 * 62 * 17√(17) * π = 4896_π_√(17)
We need to ascertain two values: The center point and the point of intersection with the surface. Let's do that first:
The center is defined by the x-intercept. To find that, set the line equation equal to 0 (y = 0 at the x-intercept):
0 = 4x + 12; 4_x_ = –12; x = –3; Therefore, the center is at (–3,0)
Next, we need to find the point at which the line intersects with the sphere's surface. To do this, solve for the point with x-coordinate at 3:
y = 4 * 3 + 12; y = 12 + 12; y = 24; therefore, the point of intersection is at (3,24)
Reviewing our data so far, this means that the radius of the sphere runs from the center, (–3,0), to the edge, (3,24). If we find the distance between these two points, we can ascertain the length of the radius. From that, we will be able to calculate the volume of the sphere.
The distance between these two points is defined by the distance formula:
d = √( (_x_1 – _x_0)2 + (_y_1 – _y_0)2 )
For our data, that is:
√( (3 + 3)2 + (24 – 0)2 ) = √( 62 + 242 ) = √(36 + 576) = √612 = √(2 * 2 * 3 * 3 * 17) = 6√(17)
Now, the volume of a sphere is defined by: V = (4/3)_πr_3
For our data, that would be: (4/3)π * (6√(17))3 = (4/3) * 63 * 17√(17) * π = 4 * 2 * 62 * 17√(17) * π = 4896_π_√(17)
← Didn't Know|Knew It →
The surface area of a sphere is 676_π_ in2. How many cubic inches is the volume of the same sphere?
The surface area of a sphere is 676_π_ in2. How many cubic inches is the volume of the same sphere?
Tap to reveal answer
To begin, we must solve for the radius of our sphere. To do this, recall the equation for the surface area of a sphere: A = 4_πr_2
For our data, that is: 676_π_ = 4_πr_2; 169 = _r_2; r = 13
From this, it is easy to solve for the volume of the sphere. Recall the equation:
V = (4/3)_πr_3
For our data, this is: V = (4/3)π * 133 = (4_π_ * 2197)/3 = (8788_π)_/3
To begin, we must solve for the radius of our sphere. To do this, recall the equation for the surface area of a sphere: A = 4_πr_2
For our data, that is: 676_π_ = 4_πr_2; 169 = _r_2; r = 13
From this, it is easy to solve for the volume of the sphere. Recall the equation:
V = (4/3)_πr_3
For our data, this is: V = (4/3)π * 133 = (4_π_ * 2197)/3 = (8788_π)_/3
← Didn't Know|Knew It →
A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?
A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?
Tap to reveal answer
The general form of our problem is:
Gel volume = Prism volume – Can volume
The prism volume is simple: 5 * 6 * 10 = 300 in3
The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr_2. Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 22_π = 4_π_. The total volume is therefore: 4_π_ * 5 = 20_π_ in3.
The gel volume is therefore: 300 – 20_π_ or (approx.) 237.17 in3.
The general form of our problem is:
Gel volume = Prism volume – Can volume
The prism volume is simple: 5 * 6 * 10 = 300 in3
The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr_2. Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 22_π = 4_π_. The total volume is therefore: 4_π_ * 5 = 20_π_ in3.
The gel volume is therefore: 300 – 20_π_ or (approx.) 237.17 in3.
← Didn't Know|Knew It →
You are looking at a map of your town and your house is located at the coordinate (0,0). Your school is located at the point (3,4). If each coordinate distance is 1.3 miles, how far away is your school?
You are looking at a map of your town and your house is located at the coordinate (0,0). Your school is located at the point (3,4). If each coordinate distance is 1.3 miles, how far away is your school?
Tap to reveal answer
The coordinate length between you and your school is equivalent to the hypotenuse of a right triangle with sides of 3 and 4 units:
The distance is 5 coordinate lengths, and each coordinate length corresponds to 1.3 miles of distance, so
The coordinate length between you and your school is equivalent to the hypotenuse of a right triangle with sides of 3 and 4 units:
The distance is 5 coordinate lengths, and each coordinate length corresponds to 1.3 miles of distance, so
← Didn't Know|Knew It →
Which of the following points will you find on the 
-axis?
Which of the following points will you find on the
Tap to reveal answer
A point is located on the 
-axis if and only if it has 
-coordinate (first coordinate) 0. Of the five choices, only 
fits that description.
A point is located on the
← Didn't Know|Knew It →
Which of the following is a vertex of the square?
Which of the following is a vertex of the square?
Tap to reveal answer
The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are:
The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are:
← Didn't Know|Knew It →
Which of the following points is on the 
-axis?
Which of the following points is on the
Tap to reveal answer
A point is located on the 
-axis if and only if it has a 
-coordinate equal to zero. So the answer is 
.
A point is located on the
← Didn't Know|Knew It →
Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?
Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?
Tap to reveal answer
The value of the slope (m) is rise over run, and can be calculated with the formula below:
The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.
The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.
From this information we know that we can assign the following coordinates for the equation:
and 
Below is the solution we would get from plugging this information into the equation for slope:
This reduces to 
The value of the slope (m) is rise over run, and can be calculated with the formula below:
The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.
The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.
From this information we know that we can assign the following coordinates for the equation:
Below is the solution we would get from plugging this information into the equation for slope:
This reduces to
← Didn't Know|Knew It →
A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?
A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?
Tap to reveal answer
To find the speed of the deer, you must have the distance traveled and the time.
The distance is found using the Pythagorean Theorem:
The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:
To find the speed of the deer, you must have the distance traveled and the time.
The distance is found using the Pythagorean Theorem:
The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:
← Didn't Know|Knew It →
The point 
is reflected across 
. What is the new point?
The point
Tap to reveal answer
The horizontal distance from point 
to the vertical line 
is two units. Since this point is reflected across 
, the new point will also be 2 units to the right of line 
.
Therefore, the correct answer is: 
The horizontal distance from point
Therefore, the correct answer is:
← Didn't Know|Knew It →
What is the slope given the following two points? 
What is the slope given the following two points?
Tap to reveal answer
Write the slope formula and substitute the two points.
Write the slope formula and substitute the two points.
← Didn't Know|Knew It →