Advanced Polygons & 3D Shapes - SAT Math
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Cube A has a volume of 
cubic inches. if each side of Cube B is twice as long as each side of Cube A, then what is the volume of Cube B?
Cube A has a volume of 
The relationships between side lengths and area for two dimensional figures, and side lengths and volume for three dimensional figures can be confusing. It is very easy to think that if the lengths all double, the area and volume should follow suit, but if every dimension is doubled in a 2-dimensional shape (a square or rectangle, where you're talking about area) the area is multiplied by 
(the square of the change for the sides) and in a 3-dimensional shape (a cube, sphere, or box, where you're talking about volume) the volume is multiplied by 8 (the cube of the change of the sides).
Suppose that the length of each side of the smaller cube is 
. That would mean that the volume would be 
If you double that length across all dimensions, then in calculating the length, width, and depth you'd multiply:
This simplifies to 
. And note the relationship between the larger and smaller cubes: 
, the volume of the larger cube, 
. So with the given problem, if the volume of the smaller cube is 
, then the volume of the larger cube is 8 times that, so the correct answer is 
.
The relationships between side lengths and area for two dimensional figures, and side lengths and volume for three dimensional figures can be confusing. It is very easy to think that if the lengths all double, the area and volume should follow suit, but if every dimension is doubled in a 2-dimensional shape (a square or rectangle, where you're talking about area) the area is multiplied by 
Suppose that the length of each side of the smaller cube is 
This simplifies to 
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Michael plans to decorate a rectangular wooden box (pictured above) by painting all exterior sides but the top, which he plans to keep open. What is the minimum number of square inches of paint needed?
Michael plans to decorate a rectangular wooden box (pictured above) by painting all exterior sides but the top, which he plans to keep open. What is the minimum number of square inches of paint needed?
This problem asks you to find the surface area for 
sides of the box, since the top side will not have area. You should then determine the dimensions of each side that you'll be using.
For the left and right sides, the measurement will be 
square inches, and since you'll have two of those sides you'll multiply by 
to have 
square inches of sides.
The front and back will measure 
square inches, and since you'll have two of those sides you should multiply by 
to have 
square inches of front/back.
Then you'll need to account for the bottom, which measures 
square inches.
So your total calculation is 
square inches.
This problem asks you to find the surface area for 
For the left and right sides, the measurement will be 
The front and back will measure 
Then you'll need to account for the bottom, which measures 
So your total calculation is 
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A right cylinder soda can has a height of 
and a radius of 
as pictured above. What is the total surface area of the cylinder?
A right cylinder soda can has a height of 
The surface area of a cylinder has three components: the area of the top, the area of the bottom, and the area of the "side."
The areas of the top and bottom are classic circles, so you'll use 
to calculate. Here the radius is 
, so each circle will have a volume of 
. One important key here is to remember to multiply that by 2 to account for both circles. So combined, the top and bottom have an area of 
.
For the "side," it is important to think conceptually about what constitutes that area. If you were to unroll the circular nature of the cylinder, the side would form a rectangle. Quite clearly the height will be the same as the height of the cylinder, but what about the length? The length is the circumference of the circle, the distance along the top (or bottom) for the material to stretch exactly around the circle.
Circumference is 
, so here that's 
. Multiply that by the height of 
and you have 
as the area of the side. So your area is now 
, which sums to 
.
The surface area of a cylinder has three components: the area of the top, the area of the bottom, and the area of the "side."
The areas of the top and bottom are classic circles, so you'll use 
For the "side," it is important to think conceptually about what constitutes that area. If you were to unroll the circular nature of the cylinder, the side would form a rectangle. Quite clearly the height will be the same as the height of the cylinder, but what about the length? The length is the circumference of the circle, the distance along the top (or bottom) for the material to stretch exactly around the circle.
Circumference is 
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A cube with a volume of 
cubic inches is inscribed within a sphere such that all 
vertices of the cube are on the sphere. What is the circumference of the sphere, in inches?
A cube with a volume of 
Importantly here, the greatest distance between two points in the cube (from one corner to the opposite corner) will equal the diameter of the sphere. Because the cube is perfectly inscribed within the circle, a line that travels through the center of the cube will travel through the center of the sphere, and if it touches two corners of the cube then it's touching the outside of the sphere, satisfying the definition of the diameter.
With that, your goal should be to use the volume of the cube to determine the diameter of the sphere. This can be done quickly if you know the rule for the greatest distance in a rectangular box: 
. Here since length, width, and height are all the same, 
, you have a quick calculation:
.
Since the circumference of a circle can be expressed as 
, your answer is simply 
.
Importantly here, the greatest distance between two points in the cube (from one corner to the opposite corner) will equal the diameter of the sphere. Because the cube is perfectly inscribed within the circle, a line that travels through the center of the cube will travel through the center of the sphere, and if it touches two corners of the cube then it's touching the outside of the sphere, satisfying the definition of the diameter.
With that, your goal should be to use the volume of the cube to determine the diameter of the sphere. This can be done quickly if you know the rule for the greatest distance in a rectangular box: 
Since the circumference of a circle can be expressed as 
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A rectangular aquarium is 
feet high, 
feet long, and 
feet wide. If the aquarium is 
full of water, how many cubic feet of water are in the aquarium?
A rectangular aquarium is 
The volume of a rectangular box is Length × Width × Height. Here you're given those three dimensions as 
, but then told that the volume of water is only 
of the total. So your calculation is 
, which comes out to 
.
The volume of a rectangular box is Length × Width × Height. Here you're given those three dimensions as 
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If the width, depth and length of a rectangle box were each decreased by 
, by what percent would the volume of the box decrease?
If the width, depth and length of a rectangle box were each decreased by 
An important lesson from this problem involves the relationship between length and volume. Since volume is three-dimensional and length is only one-dimensional, when you reduce the length of all sides in a 3-D shape, you have to account for that change along all three dimensions. In this case, the box is scaled down by a linear factor of 
, so its volume scales down by a factor of 
.
So the new box is 
the volume of the old box, meaning that it decreased in volume by 
. 
expressed as a percentage is 
.
Alternatively, you could avoid the abstraction by choosing your own numbers and playing out the scenario that way. Imagine a 
cubical box (volume 
) being scaled down to a 
cubical box (volume 
). You'd go from a volume of 
to a volume of 
, losing 
of the volume, again equating to an 
reduction.
An important lesson from this problem involves the relationship between length and volume. Since volume is three-dimensional and length is only one-dimensional, when you reduce the length of all sides in a 3-D shape, you have to account for that change along all three dimensions. In this case, the box is scaled down by a linear factor of 
So the new box is 
Alternatively, you could avoid the abstraction by choosing your own numbers and playing out the scenario that way. Imagine a 
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A chocolate box has a long triangular shape and the ends of the box form a 90-degree angle with the rest of the box. The triangular-shaped end piece is an equilateral triangle, the length of the box is 
inches, and the volume is 
. What is the value of 
in inches?
A chocolate box has a long triangular shape and the ends of the box form a 90-degree angle with the rest of the box. The triangular-shaped end piece is an equilateral triangle, the length of the box is 
The volume of the box is Base * Height, where Height is the length of the box and Base is the area of the triangular face. Thus, 
. The area of the base, then, is 
. Now recall that the area of an equilateral triangle is 
.
So 
. and 
and 
, which is the value of 
.
The volume of the box is Base * Height, where Height is the length of the box and Base is the area of the triangular face. Thus, 
So 
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A rectangular label with an area of 
square inches. is wrapped around a can that is 
inches tall, such that the label exactly covers the outside of the can excluding the top and the bottom. What is the volume of the can, in cubic inches?
A rectangular label with an area of 
The surface area of a cylinder is calculated as a top, a bottom, and the outside. The outside is measured as the circumference * height, or 
. That will be the same size as the label. Therefore, 
. We know that 
, therefore, 
, and 
.
The volume of the cylinder is calculated as 
. Make sure that you square the full thing, 
. You will end up with 
.
The surface area of a cylinder is calculated as a top, a bottom, and the outside. The outside is measured as the circumference * height, or 
The volume of the cylinder is calculated as 
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if a cube has a volume of 
, what is its total surface area?
if a cube has a volume of 
The volume of a cube is 
(that's why you call 
to the third power "
-cubed"), where 
represents the length of one of the sides. The surface area is 
, or more conceptually the sum of the areas of each of the six faces (top, bottom, front, back, left, right) of the cube.
So if the volume of a cube is 
, that means that 
, so 
.
Plugging that into the surface area formula, you have 
which is 
.
The volume of a cube is 
So if the volume of a cube is 
Plugging that into the surface area formula, you have 
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A trophy shop has been commissioned to create spherical trophies for a soccer tournament. Each spherical soccer trophy has a radius of 1 inch, while an actual regulation soccer ball has a radius of 4 inches. Which of the following expresses the ratio of the volume of the spherical trophy to the volume of the actual soccer ball?
A trophy shop has been commissioned to create spherical trophies for a soccer tournament. Each spherical soccer trophy has a radius of 1 inch, while an actual regulation soccer ball has a radius of 4 inches. Which of the following expresses the ratio of the volume of the spherical trophy to the volume of the actual soccer ball?
When looking to find the relationship between the volumes of the two spherical figures in the question stem, we’ll want to keep in mind the formula for the volume of a sphere:
So, if we plug each radius into this formula, we’ll arrive at:
Thus, the ratio between the two volumes is:
Since the 
component cancels in both the numerator and the denominator, the ratio of the volumes is 1:64. Be sure to arrange the ratio in the order asked for in the question stem! Additionally, you’ll want to keep in mind that when a change in scale is applied to multiple dimensions, the scale changes exponentially! We can actually save some time on questions like this if we recognize this relationship.
For instance, here, the “scale factor,” or change in scale from the smaller radius to the larger radius is 4, but since the relationship we’re looking for compares volume to volume, we need to account for the fact that the scale factor has been taken to the third power for the three dimensional measurement at hand. Thus, the ratio of the volumes will be 1 to 43, or 1 to 64.
When looking to find the relationship between the volumes of the two spherical figures in the question stem, we’ll want to keep in mind the formula for the volume of a sphere:
So, if we plug each radius into this formula, we’ll arrive at:
Thus, the ratio between the two volumes is:
Since the 
For instance, here, the “scale factor,” or change in scale from the smaller radius to the larger radius is 4, but since the relationship we’re looking for compares volume to volume, we need to account for the fact that the scale factor has been taken to the third power for the three dimensional measurement at hand. Thus, the ratio of the volumes will be 1 to 43, or 1 to 64.
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What is the sum of the angles of an octagon?
What is the sum of the angles of an octagon?
The sum of the interior angles of an n-sided polygon can be calculated as:
Which you can test for yourself: a triangle (3 sides) has 180 degrees: 
.
A rectangle has 360 degrees. 
So in this case, where 
, you'd calculate as:
.
The sum of the interior angles of an n-sided polygon can be calculated as:
Which you can test for yourself: a triangle (3 sides) has 180 degrees: 
A rectangle has 360 degrees.
So in this case, where 
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If 
represents the measure of an interior angle of a regular octagon and 
represents the measure of an interior angle of a regular pentagon, what is the ratio 
?
If 
The total interior angle of a polygon with 
sides equals 
. Thus, the total interior angle of an octagon is 
and the total interior angle of a pentagon is 
. A regular polygon has the property that all interior angles are congruent, so an interior angle of a regular octagon is 
. Similarly, the interior angle of a pentagon is 
. 
is equivalent to 
.
For quicker math-by-hand, recognize that you're being asked about a ratio. If you set up the measure of one angle of a regular octagon as 
(the total measure divided by the number of angles) and the measure of one angle of a regular pentagon as 
, then notice that with a 180 term multiplied in the numerator of each portion of the ratio, the 180s can factor out. Then you're just taking the ratio of 
to 
, which nets quickly to 
.
The total interior angle of a polygon with 
For quicker math-by-hand, recognize that you're being asked about a ratio. If you set up the measure of one angle of a regular octagon as 
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In a particular n-sided polygon, all sides and all angles are congruent. If each angle measures 144 degrees, what is the value of 
?
In a particular n-sided polygon, all sides and all angles are congruent. If each angle measures 144 degrees, what is the value of 
In order to effectively tackle this question, you’ll want to be comfortable with the fact that the sum of the interior angles of any given polygon will always equal 
, where n is the number of sides in that polygon. This should be fairly straightforward to prove, as we can always break up any given polygon into two fewer triangles than the number of sides of the polygon, and the sum of the interior angles of a triangle is 180 degrees.
In this example, we’re being asked to think about that formula a little differently. We’ve been told that each angle measures 144 degrees, so the sum of the interior angles should be equal to 
, where n is the number of sides. We can also express the sum of the interior angles as 
, as we just mentioned. So, all we need to do from here is set those two expressions equal to one another, since all we’ve done is said the same thing two different ways!
So, if
, we can distribute to arrive at
and get all our knowns to one side and all our unknowns to the other side to find that
If we divide both sides of our equation by 36, we can see that
.
In order to effectively tackle this question, you’ll want to be comfortable with the fact that the sum of the interior angles of any given polygon will always equal 
In this example, we’re being asked to think about that formula a little differently. We’ve been told that each angle measures 144 degrees, so the sum of the interior angles should be equal to 
So, if
and get all our knowns to one side and all our unknowns to the other side to find that
If we divide both sides of our equation by 36, we can see that
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The following is a regular pentagon. What is the measure of angle 
?
The following is a regular pentagon. What is the measure of angle 
This question tasks us to identify and apply knowledge of the sum of interior angles of a polygon, as well as the regular nature of our shape to solve for individual angles. The sum of the interior angles of a triangle is 
, where n is the number of sides in the polygon. So, the sum of the interior angles of a pentagon is 
. Thus, each individual angle is 
.
If we know that angle EDC is 
. Since triangle EDC is an isosceles triangle made up of the two identical sides ED and CD, angles DEC and ECD must be the same and are each 
. Since AED is 108°, and CED is 
, AEC, labeled 
in our diagram, must be 
.
This question tasks us to identify and apply knowledge of the sum of interior angles of a polygon, as well as the regular nature of our shape to solve for individual angles. The sum of the interior angles of a triangle is 
If we know that angle EDC is 
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A regular polygon with interior angles of 
each has how many more sides than a regular polygon with interior angles of 
each?
A regular polygon with interior angles of 
To answer this question, we’ll need to use our understanding of the interior angles of a regular polygon. Since the sum of the interior angles of a polygon is as follows:
Where 
is the number of sides and, thus, the number of vertices in the polygon, if all angles of the polygon are the same (as is the case in a regular polygon), each angle will be 
, since the 
each angle.
So, if 
, the number of sides must be 5, since:
(*notice - at this step, we’ve just expressed the sum of the angles two different ways, as (each angle)*(number of angles), and using the formula where the sum of the angles is 
. We could have begun our work at this step, if this setup feels more intuitive to you!)
Similarly, if each angle in a regular polygon is 
, that polygon must be 8-sided, since:
So, an 8-sided polygon has 3 more sides than a 5 sided polygon. (Once you’ve completed the math - be sure to answer the question you’ve been asked!)
To answer this question, we’ll need to use our understanding of the interior angles of a regular polygon. Since the sum of the interior angles of a polygon is as follows:
Where 
So, if 
Similarly, if each angle in a regular polygon is 
So, an 8-sided polygon has 3 more sides than a 5 sided polygon. (Once you’ve completed the math - be sure to answer the question you’ve been asked!)
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What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?
What is the measure, in degrees, of each interior angle of a regular convex polygon that has twelve sides?
The sum of the interior angles, in degrees, of a regular polygon, is given by the formula 
, where 
is the number of sides. The problem concerns a polygon with twelve sides, so we will let 
. The sum of the interior angles in this polygon would be 
.
Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide 
by 
, which gives us 
.
The answer is 
.
The sum of the interior angles, in degrees, of a regular polygon, is given by the formula 
Because the polygon is regular (meaning its sides are all congruent), all of the angles have the same measure. Thus, if we divide the sum of the measures of the angles by the number of sides, we will have the measure of each interior angle. In short, we need to divide 
The answer is 
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In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?
In the figure above, polygon ABDFHGEC is a regular octagon. What is the measure, in degrees, of angle FHI?
Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180-degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.
The sum of the interior angles in a regular polygon is given by the formula 
, where 
is the number of sides in the polygon. An octagon has eight sides, so the sum of the angles of the octagon is 
. Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8. Therefore, each angle in the polygon has a measure of 
. This means that angle FHG has a measure of 135-degrees.
Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180-degrees because the two angles form a line and are supplementary. We can write the following equation:
Measure of FHG + measure of FHI = 180
135 + measure of FHI = 180
Subtract 135 from both sides.
Measure of FHI = 45 degrees.
The answer is 45.
Angle FHI is the supplement of angle FHG, which is an interior angle in the octagon. When two angles are supplementary, their sum is equal to 180-degrees. If we can find the measure of each interior angle in the octagon, then we can find the supplement of angle FHG, which will give us the measure of angle FHI.
The sum of the interior angles in a regular polygon is given by the formula 
Now that we know the measure of angle FHG, we can find the measure of FHI. The sum of the measures of FHG and FHI must be 180-degrees because the two angles form a line and are supplementary. We can write the following equation:
Measure of FHG + measure of FHI = 180
135 + measure of FHI = 180
Subtract 135 from both sides.
Measure of FHI = 45 degrees.
The answer is 45.
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What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?
What is the average (arithmetic mean) of all 15 interior angles of a quadrilateral, pentagon, and hexagon?
The 4 angles of a quadrilateral add to 360
The 5 angles of a pentagon add to 540
The 6 angles of a hexagon add to 720
The 4 angles of a quadrilateral add to 360
The 5 angles of a pentagon add to 540
The 6 angles of a hexagon add to 720
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Each interior angle of a regular polygon has measure 
. How many sides does the polygon have?
Each interior angle of a regular polygon has measure
The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures 
, then each exterior angle, which is supplementary to an interior angle, measures
The measures of the exterior angles of a polygon, one per vertex, total 
; in a regular polygon, they are congruent, so if there are 
such angles, each measures 
. Since the number of vertices is equal to the number of sides, if we set this equal to 
and solve for 
, we will find the number of sides.
Multiply both sides by 
:
The polygon has 72 vertices and, thus, 72 sides.
The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures
The measures of the exterior angles of a polygon, one per vertex, total
Multiply both sides by
The polygon has 72 vertices and, thus, 72 sides.
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A regular polygon has a measure of 
for each of its internal angles. How many sides does it have?
A regular polygon has a measure of
To determine the measure of the angles of a regular polygon use:
Thus, 
To determine the measure of the angles of a regular polygon use:
Thus,
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