Prisms - ACT Math
Card 0 of 15
A right, rectangular prism has has a length of 
, a width of 
, and a height of 
. What is the length of the diagonal of the prism?
A right, rectangular prism has has a length of
First we must find the diagonal of the prism's base (
). This can be done by using the Pythagorean Theorem with the length (
) and width (
):
Therefore, the diagonal of the prism's base is 
. We can then use this again in the Pythagorean Theorem, along with the height of the prism (
), to find the diagonal of the prism (
):
Therefore, the length of the prism's diagonal is 
.
First we must find the diagonal of the prism's base (
Therefore, the diagonal of the prism's base is
Therefore, the length of the prism's diagonal is
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What is the diagonal of a rectangular prism with a height of 4, width of 4 and height of 6?
What is the diagonal of a rectangular prism with a height of 4, width of 4 and height of 6?
In order to solve this problem, it's helpful to visualize where the diagonal is within the prism.
In this image, the diagonal is the pink line. By noting how it relates to the blue and green lines, we can observe how the pink line is connected and creates a right triangle. This very quickly becomes a problem that employs the Pythagorean theorem.
The goal is essentially to find the hypotenuse of this sketched-in right triangle; however, only one of the legs is given: the green line, the height of the prism. The blue line can be solved for by understanding that it is the measurement of the diagonal of a 4x4 square.
Either using trig functions or the rules for a special 45/45/90 triangle, the blue line measures out to be 
.
The rules for a 45/45/90 triangle: both legs are "
" and the hypotenuse is "
". Keep in mind, this is is only for isosceles right triangles.
Now that both legs are known, we can solve for the hypotenuse (diagonal).
In order to solve this problem, it's helpful to visualize where the diagonal is within the prism.
In this image, the diagonal is the pink line. By noting how it relates to the blue and green lines, we can observe how the pink line is connected and creates a right triangle. This very quickly becomes a problem that employs the Pythagorean theorem.
The goal is essentially to find the hypotenuse of this sketched-in right triangle; however, only one of the legs is given: the green line, the height of the prism. The blue line can be solved for by understanding that it is the measurement of the diagonal of a 4x4 square.
Either using trig functions or the rules for a special 45/45/90 triangle, the blue line measures out to be
The rules for a 45/45/90 triangle: both legs are "
Now that both legs are known, we can solve for the hypotenuse (diagonal).
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Find the diagonal of a right rectangular prism if the length, width, and height are 3,4, and 5, respectively.
Find the diagonal of a right rectangular prism if the length, width, and height are 3,4, and 5, respectively.
Write the diagonal formula for a rectangular prism.
Substitute and solve for the diagonal.
Write the diagonal formula for a rectangular prism.
Substitute and solve for the diagonal.
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If the dimensions of a right rectangular prism are 1 yard by 1 foot by 1 inch, what is the diagonal in feet?
If the dimensions of a right rectangular prism are 1 yard by 1 foot by 1 inch, what is the diagonal in feet?
Convert the dimensions into feet.
The new dimensions of rectangular prism in feet are: 
Write the formula for the diagonal of a right rectangular prism and substitute.
Convert the dimensions into feet.
The new dimensions of rectangular prism in feet are:
Write the formula for the diagonal of a right rectangular prism and substitute.
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The length of a box is 3 times the width. Which of the following gives the length (L inches) in terms of the width (W inches) of the box?
The length of a box is 3 times the width. Which of the following gives the length (L inches) in terms of the width (W inches) of the box?
When reading word problems, there are certain clues that help interpret what is going on. The word “is” generally means “=” and the word “times” means it will be multiplied by something. Therefore, “the length of a box is 3 times the width” gives you the answer: L = 3 x W, or L = 3W.
When reading word problems, there are certain clues that help interpret what is going on. The word “is” generally means “=” and the word “times” means it will be multiplied by something. Therefore, “the length of a box is 3 times the width” gives you the answer: L = 3 x W, or L = 3W.
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The width of a box, in inches, is 5 inches less than three times its length. Which of the following equations gives the width, W inches, in terms of the length, L inches, of the box?
The width of a box, in inches, is 5 inches less than three times its length. Which of the following equations gives the width, W inches, in terms of the length, L inches, of the box?
We notice the width is “5 inches less than three times its width,” so we express W as being three times its width (3L) and 5 inches less than that is 3L minus 5. In this case, W is the dependent and L is the independent variable.
W = 3L - 5
We notice the width is “5 inches less than three times its width,” so we express W as being three times its width (3L) and 5 inches less than that is 3L minus 5. In this case, W is the dependent and L is the independent variable.
W = 3L - 5
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Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of 
. Sturgis knows that the exhibit will be 
long and go 
back into the wall.
What will the height of the new exhibit be?
Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of
What will the height of the new exhibit be?
This sounds like a geometry problem, so start by drawing a picture so that you know exactly what you are dealing with.
Because we are dealing with rectangular prisms and volume, we will need the following formula:
Or
We are solving for height, so you can begin by rearranging the equation to get 
by itself:
Then, plug in our knowns (
, 
and 
)
Here is the problem worked out with a corresponding picture:
This sounds like a geometry problem, so start by drawing a picture so that you know exactly what you are dealing with.
Because we are dealing with rectangular prisms and volume, we will need the following formula:
Or
We are solving for height, so you can begin by rearranging the equation to get
Then, plug in our knowns (
Here is the problem worked out with a corresponding picture:
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Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of 
. Sturgis knows that the exhibit will be 
long and go 
back into the wall.
If three-quarters of the exhibit's volume will be water, how high up the wall will the water come?
Sturgis is in charge of designing a new exhibit in the shape of a rectangular prism for a local aquarium. The exhibit will hold alligator snapping turtles and needs to have a volume of
If three-quarters of the exhibit's volume will be water, how high up the wall will the water come?
The trickiest part of this question is the wording. This problem is asking for the height of the water in the exhibit if the exhibit is three-quarters full. We can find this at least two different ways.
- The longer way requires that we begin by finding three quarters of the total volume:
Now we go back to our volume equation, and since we are again looking for height, we want it solved for 
:
Becomes
- The easier way requires that we recognize a key detail. If we take three-quarters of the volume without changing our length or width, our new height will just be three-quarters of the total height. We can solve for the total height of the exhibit by using the volume equation and rearranging it to solve for
:
At this point, we can substitute in our given values and solve for 
:
So, the total height of the exhibit is 
. We can now easily solve for three-quarters of the total height:
The trickiest part of this question is the wording. This problem is asking for the height of the water in the exhibit if the exhibit is three-quarters full. We can find this at least two different ways.
- The longer way requires that we begin by finding three quarters of the total volume:
Now we go back to our volume equation, and since we are again looking for height, we want it solved for
Becomes
- The easier way requires that we recognize a key detail. If we take three-quarters of the volume without changing our length or width, our new height will just be three-quarters of the total height. We can solve for the total height of the exhibit by using the volume equation and rearranging it to solve for
At this point, we can substitute in our given values and solve for
So, the total height of the exhibit is
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What is the surface area of a rectangular brick with a length of 12 in, a width of 8 in, and a height of 6 in?
What is the surface area of a rectangular brick with a length of 12 in, a width of 8 in, and a height of 6 in?
The formula for the surface area of a rectangular prism is given by:
SA = 2LW + 2WH + 2HL
SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)
SA = 2(96) + 2(48) + 2(72)
SA = 192 + 96 + 144
SA = 432 in2
216 in2 is the wrong answer because it is off by a factor of 2
576 in3 is actually the volume, V = L * W * H
The formula for the surface area of a rectangular prism is given by:
SA = 2LW + 2WH + 2HL
SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)
SA = 2(96) + 2(48) + 2(72)
SA = 192 + 96 + 144
SA = 432 in2
216 in2 is the wrong answer because it is off by a factor of 2
576 in3 is actually the volume, V = L * W * H
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David wants to paint the walls in his bedroom. The floor is covered by a 
carpet. The ceiling is 
tall. He selects a paint that will cover 
per quart and 
per gallon. How much paint should he buy?
David wants to paint the walls in his bedroom. The floor is covered by a
Find the surface area of the walls: SAwalls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.
This gives a total surface area of 416 ft2. One gallon covers 300 ft2, and each quart covers 75 ft2, so we need 1 gallon and 2 quarts of paint to cover the walls.
Find the surface area of the walls: SAwalls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.
This gives a total surface area of 416 ft2. One gallon covers 300 ft2, and each quart covers 75 ft2, so we need 1 gallon and 2 quarts of paint to cover the walls.
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A box is 5 inches long, 5 inches wide, and 4 inches tall. What is the surface area of the box?
A box is 5 inches long, 5 inches wide, and 4 inches tall. What is the surface area of the box?
The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.
Since the six faces consider of three sets of pairs, we can set up the equation as:
Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.
Substitute the values from the question to solve.
The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.
Since the six faces consider of three sets of pairs, we can set up the equation as:
Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.
Substitute the values from the question to solve.
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Solve for the volume of a prism that is 4m by 3m by 8m.
Solve for the volume of a prism that is 4m by 3m by 8m.
The volume of the rectangle
so we plug in our values and obtain
.
The volume of the rectangle
so we plug in our values and obtain
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A rectangular box has two sides with the following lengths:
and 
If it possesses a volume of 
, what is the area of its largest side?
A rectangular box has two sides with the following lengths:
If it possesses a volume of 
The volume of a rectangular prism is found using the following formula:
If we substitute our known values, then we can solve for the missing side.
Divide both sides of the equation by 12.
We now know that the missing length equals 7 centimeters.
This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.
The volume of a rectangular prism is found using the following formula:
If we substitute our known values, then we can solve for the missing side.
Divide both sides of the equation by 12.
We now know that the missing length equals 7 centimeters.
This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.
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A box's length is twice as long as its width. Its height is the sum of its length and its width. What is the volume of this box if its length is 10 units?
A box's length is twice as long as its width. Its height is the sum of its length and its width. What is the volume of this box if its length is 10 units?
The formula for the volume of a rectangular prism is 
, where "
" is volume, "
" is length, "
" is width and "
" is height.
We know that 
and 
. By rearranging 
, we get 
. Substituting 
into the volume equation for 
and 
into the same equation for 
, we get the following:
units cubed
The formula for the volume of a rectangular prism is
We know that
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A rectangular prism has the following dimensions:
Length: 
Width: 
Height: 
Find the volume.
A rectangular prism has the following dimensions:
Length:
Width:
Height:
Find the volume.
Given that the dimensions are: 
, 
, and 
and that the volume of a rectangular prism can be given by the equation:
, where 
is length, 
is width, and 
is height, the volume can be simply solved for by substituting in the values.
This final value can be approximated to 
.
Given that the dimensions are:
This final value can be approximated to
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