How to find surface area - HSPT Math
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The volume of a sphere is 2304_π_ in3. What is the surface area of this sphere in square inches?
The volume of a sphere is 2304_π_ in3. What is the surface area of this sphere in square inches?
To solve this, we must first begin by finding the radius of the sphere. To do this, recall that the volume of a sphere is:
V = (4/3)_πr_3
For our data, we can say:
2304_π_ = (4/3)_πr_3; 2304 = (4/3)_r_3; 6912 = 4_r_3; 1728 = _r_3; 12 * 12 * 12 = _r_3; r = 12
Now, based on this, we can ascertain the surface area using the equation:
A = 4_πr_2
For our data, this is:
A = 4_π_*122 = 576_π_
To solve this, we must first begin by finding the radius of the sphere. To do this, recall that the volume of a sphere is:
V = (4/3)_πr_3
For our data, we can say:
2304_π_ = (4/3)_πr_3; 2304 = (4/3)_r_3; 6912 = 4_r_3; 1728 = _r_3; 12 * 12 * 12 = _r_3; r = 12
Now, based on this, we can ascertain the surface area using the equation:
A = 4_πr_2
For our data, this is:
A = 4_π_*122 = 576_π_
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A sphere has its center at the origin. A point on its surface is found on the x-y axis at (6,8). In square units, what is the surface area of this sphere?
A sphere has its center at the origin. A point on its surface is found on the x-y axis at (6,8). In square units, what is the surface area of this sphere?
To find the surface area, we must first find the radius. Based on our description, this passes from (0,0) to (6,8). This can be found using the distance formula:
62 + 82 = _r_2; _r_2 = 36 + 64; _r_2 = 100; r = 10
It should be noted that you could have quickly figured this out by seeing that (6,8) is the hypotenuse of a 6-8-10 triangle (which is a multiple of the "easy" 3-4-5).
The rest is easy. The surface area of the sphere is defined by:
A = 4_πr_2 = 4 * 100 * π = 400_π_
To find the surface area, we must first find the radius. Based on our description, this passes from (0,0) to (6,8). This can be found using the distance formula:
62 + 82 = _r_2; _r_2 = 36 + 64; _r_2 = 100; r = 10
It should be noted that you could have quickly figured this out by seeing that (6,8) is the hypotenuse of a 6-8-10 triangle (which is a multiple of the "easy" 3-4-5).
The rest is easy. The surface area of the sphere is defined by:
A = 4_πr_2 = 4 * 100 * π = 400_π_
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A rectangular prism has a volume of 70 m3. If the length, width, and height of the prism are integers measured in meters, which of the following is NOT a possible measure of the surface area of the prism measured in square meters?
A rectangular prism has a volume of 70 m3. If the length, width, and height of the prism are integers measured in meters, which of the following is NOT a possible measure of the surface area of the prism measured in square meters?
Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is important to know the factors of the volume. 70 = (2)(5)(7). This implies that each of these factors (and only these factors with the exception of 1) will show up in the three dimensions exactly once. This creates precisely the following five possibilities:
2, 5, 7
SA = 2((2)(5)+(2)(7)+(5)(7)) = 118
1, 7, 10
SA = 2((1)(7)+(1)(10)+(7)(10)) = 174
1, 5, 14
SA = 2((1)(5)+(1)(14)+(5)(14)) = 178
1, 2, 35
SA = 2((1)(2)+(1)(35)+(2)(35)) = 214
1, 1, 70
SA = 2((1)(1)+(1)(70)+(1)(70)) = 282
Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is important to know the factors of the volume. 70 = (2)(5)(7). This implies that each of these factors (and only these factors with the exception of 1) will show up in the three dimensions exactly once. This creates precisely the following five possibilities:
2, 5, 7
SA = 2((2)(5)+(2)(7)+(5)(7)) = 118
1, 7, 10
SA = 2((1)(7)+(1)(10)+(7)(10)) = 174
1, 5, 14
SA = 2((1)(5)+(1)(14)+(5)(14)) = 178
1, 2, 35
SA = 2((1)(2)+(1)(35)+(2)(35)) = 214
1, 1, 70
SA = 2((1)(1)+(1)(70)+(1)(70)) = 282
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A cube has a surface area of 10m2. If a cube's sides all double in length, what is the new surface area?
A cube has a surface area of 10m2. If a cube's sides all double in length, what is the new surface area?
The equation for surface area of the original cube is 6s2. If the sides all double in length the new equation is 6(2s)2 or 6 * 4s2. This makes the new surface area 4x that of the old. 4x10 = 40m2
The equation for surface area of the original cube is 6s2. If the sides all double in length the new equation is 6(2s)2 or 6 * 4s2. This makes the new surface area 4x that of the old. 4x10 = 40m2
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The three sides of a rectangular box all have integer unit lengths. If each of the side lengths is greater than one unit, and if the volume of the box is 182 cubic units, what is the surface area of the box in square units?
The three sides of a rectangular box all have integer unit lengths. If each of the side lengths is greater than one unit, and if the volume of the box is 182 cubic units, what is the surface area of the box in square units?
Let's call the side lengths of the box l, w, and h. We are told that l, w, and h must all be integer lengths greater than one. We are also told that the volume of the box is 182 cubic units.
Since the volume of a rectangular box is the product of its side lengths, this means that lwh must equal 182.
(l)(w)(h) = 182.
In order to determine possible values of l, w, and h, it would help us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than 1.
Let's factor 182. Because 182 is even, it is divisible by 2.
182 = 2(91).
91 is equal to the product of 7 and 13.
Thus, 182 = 2(7)(13).
This means that the lengths of the box must be 2, 7, and 13 units.
In order to find the surface area, we can use the following formula:
surface area = 2lw + 2lh + 2hw.
surface area = 2(2)(7) + 2(2)(13) + 2(7)(13)
= 28 + 52 + 182
= 262 square units.
The answer is 262.
Let's call the side lengths of the box l, w, and h. We are told that l, w, and h must all be integer lengths greater than one. We are also told that the volume of the box is 182 cubic units.
Since the volume of a rectangular box is the product of its side lengths, this means that lwh must equal 182.
(l)(w)(h) = 182.
In order to determine possible values of l, w, and h, it would help us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than 1.
Let's factor 182. Because 182 is even, it is divisible by 2.
182 = 2(91).
91 is equal to the product of 7 and 13.
Thus, 182 = 2(7)(13).
This means that the lengths of the box must be 2, 7, and 13 units.
In order to find the surface area, we can use the following formula:
surface area = 2lw + 2lh + 2hw.
surface area = 2(2)(7) + 2(2)(13) + 2(7)(13)
= 28 + 52 + 182
= 262 square units.
The answer is 262.
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A sphere is perfectly contained within a cube that has a surface area of 726 square units. In square units, what is the surface area of the sphere?
A sphere is perfectly contained within a cube that has a surface area of 726 square units. In square units, what is the surface area of the sphere?
To begin, we must determine the dimensions of the cube. To do this, recall that the surface area of a cube is made up of six squares and is thus defined as: A = 6_s_2, where s is one of the sides of the cube. For our data, this gives us:
726 = 6_s_2; 121 = _s_2; s = 11
Now, if the sphere is contained within the cube, that means that 11 represents the diameter of the sphere. Therefore, the radius of the sphere is 5.5 units. The surface area of a sphere is defined as: A = 4_πr_2. For our data, that would be:
A = 4_π_ * 5.52 = 30.25 * 4 * π = 121_π_
To begin, we must determine the dimensions of the cube. To do this, recall that the surface area of a cube is made up of six squares and is thus defined as: A = 6_s_2, where s is one of the sides of the cube. For our data, this gives us:
726 = 6_s_2; 121 = _s_2; s = 11
Now, if the sphere is contained within the cube, that means that 11 represents the diameter of the sphere. Therefore, the radius of the sphere is 5.5 units. The surface area of a sphere is defined as: A = 4_πr_2. For our data, that would be:
A = 4_π_ * 5.52 = 30.25 * 4 * π = 121_π_
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A solid sphere is cut in half to form two solid hemispheres. What is the ratio of the surface area of one of the hemispheres to the surface area of the entire sphere before it was cut?
A solid sphere is cut in half to form two solid hemispheres. What is the ratio of the surface area of one of the hemispheres to the surface area of the entire sphere before it was cut?
The surface area of the sphere before it was cut is equal to the following:
surface area of solid sphere = 4_πr_2, where r is the length of the radius.
Each hemisphere will have the following shape:
In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.
area of flat part of hemisphere = _πr_2
The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4_πr_2.
area of curved part of hemisphere = (1/2)4_πr_2 = 2_πr_2
The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.
total surface area of hemisphere = _πr_2 + 2_πr_2 = 3_πr_2
Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.
ratio = (3_πr_2)/(4_πr_2) = 3/4
The answer is 3/4.
The surface area of the sphere before it was cut is equal to the following:
surface area of solid sphere = 4_πr_2, where r is the length of the radius.
Each hemisphere will have the following shape:
In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.
area of flat part of hemisphere = _πr_2
The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4_πr_2.
area of curved part of hemisphere = (1/2)4_πr_2 = 2_πr_2
The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.
total surface area of hemisphere = _πr_2 + 2_πr_2 = 3_πr_2
Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.
ratio = (3_πr_2)/(4_πr_2) = 3/4
The answer is 3/4.
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A right rectangular prism has dimensions of 3 x 5 x 20. What is its surface area?
A right rectangular prism has dimensions of 3 x 5 x 20. What is its surface area?
There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 3 x 5, a face that is 5 x 20 and a face that is 3 x 20. To think this through, imagine that the front face is 3 x 5, the right side is 5 x 20, and the top is 3 x 20. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).
Therefore, we know we have the following areas for the faces of our prism:
2 * 3 * 5 = 30
2 * 5 * 20 = 200
2 * 3 * 20 = 120
Add these to get the total surface area:
30 + 200 + 120 = 350
There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 3 x 5, a face that is 5 x 20 and a face that is 3 x 20. To think this through, imagine that the front face is 3 x 5, the right side is 5 x 20, and the top is 3 x 20. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).
Therefore, we know we have the following areas for the faces of our prism:
2 * 3 * 5 = 30
2 * 5 * 20 = 200
2 * 3 * 20 = 120
Add these to get the total surface area:
30 + 200 + 120 = 350
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A right rectangular prism has dimensions of 12.4 x 2.3 x 33. What is its surface area?
A right rectangular prism has dimensions of 12.4 x 2.3 x 33. What is its surface area?
There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 12.4 x 2.3, a face that is 2.3 x 33 and a face that is 33 x 12.4. To think this through, imagine that the front face is 12.4 x 2.3, the left side is 2.3 x 33, and the top is 33 x 12.4. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).
Therefore, we know we have the following areas for the faces of our prism:
2 * 12.4 * 2.3 = 57.04
2 * 2.3 * 33 = 151.8
2 * 12.4 * 33 = 818.4
Add these to get the total surface area:
57.04 + 151.8 + 818.4 = 1027.24
There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 12.4 x 2.3, a face that is 2.3 x 33 and a face that is 33 x 12.4. To think this through, imagine that the front face is 12.4 x 2.3, the left side is 2.3 x 33, and the top is 33 x 12.4. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).
Therefore, we know we have the following areas for the faces of our prism:
2 * 12.4 * 2.3 = 57.04
2 * 2.3 * 33 = 151.8
2 * 12.4 * 33 = 818.4
Add these to get the total surface area:
57.04 + 151.8 + 818.4 = 1027.24
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What is the surface area of a cube with a volume of 1728 in3?
What is the surface area of a cube with a volume of 1728 in3?
This problem is relatively simple. We know that the volume of a cube is equal to s3, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s3 = 1728. Taking the cubed root, we get s = 12.
Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in2. Therefore, the total surface area is 6 * 144 = 864 in2.
This problem is relatively simple. We know that the volume of a cube is equal to s3, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s3 = 1728. Taking the cubed root, we get s = 12.
Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in2. Therefore, the total surface area is 6 * 144 = 864 in2.
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The dimensions of a right rectangular prism are such that the second dimension is twice the length of the first and the third is twice the length of the second. If the volume of the prism is 216 cubic units, what is its surface area?
The dimensions of a right rectangular prism are such that the second dimension is twice the length of the first and the third is twice the length of the second. If the volume of the prism is 216 cubic units, what is its surface area?
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 2 * Dim1 = 2x
Dim3: 2 * Dim2 = 2 * 2x = 4x
More directly stated, therefore, our dimensions are: x, 2x, and 4x. Therefore, the volume is x * 2x * 4x = 216, which simplifies to 8x3 = 216 or x3 = 27. Solving for x, we find x = 3. Therefore, our dimensions are:
x = 3
2x = 6
4x = 12
Or: 3 x 6 x 12
Now, to find the surface area, we must consider that this means that our prism has sides of the following dimensions: 3 x 6, 6 x 12, and 3 x 12. Since each side has a "matching" side opposite it, we know that we have the following values for the areas of the faces:
2 * 3 * 6 = 36
2 * 6 * 12 = 144
2 * 3 * 12 = 72
The total surface area therefore equals: 36 + 144 + 72 = 252 square units.
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 2 * Dim1 = 2x
Dim3: 2 * Dim2 = 2 * 2x = 4x
More directly stated, therefore, our dimensions are: x, 2x, and 4x. Therefore, the volume is x * 2x * 4x = 216, which simplifies to 8x3 = 216 or x3 = 27. Solving for x, we find x = 3. Therefore, our dimensions are:
x = 3
2x = 6
4x = 12
Or: 3 x 6 x 12
Now, to find the surface area, we must consider that this means that our prism has sides of the following dimensions: 3 x 6, 6 x 12, and 3 x 12. Since each side has a "matching" side opposite it, we know that we have the following values for the areas of the faces:
2 * 3 * 6 = 36
2 * 6 * 12 = 144
2 * 3 * 12 = 72
The total surface area therefore equals: 36 + 144 + 72 = 252 square units.
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A right circular cylinder has a height of 41 in. and a lateral area (excluding top and bottom) 512.5π in2. What is the area of its bases?
A right circular cylinder has a height of 41 in. and a lateral area (excluding top and bottom) 512.5π in2. What is the area of its bases?
The lateral area (not including its bases) is equal to the circumference of the base times the height of the cylinder. Think of it like a label that is wrapped around a soup can. Therefore, we can write this area as:
A = h * π * d or A = h * π * 2r = 2πrh
Now, substituting in our values, we get:
512.5π = 2 * 41*rπ; 512.5π = 82rπ
Solve for r by dividing both sides by 82π:
6.25 = r
From here, we can calculate the area of a base:
A = 6.252π = 39.0625π
NOTE: The question asks for the area of the bases. Therefore, the answer is 2 * 39.0625π or 78.125π in2.
The lateral area (not including its bases) is equal to the circumference of the base times the height of the cylinder. Think of it like a label that is wrapped around a soup can. Therefore, we can write this area as:
A = h * π * d or A = h * π * 2r = 2πrh
Now, substituting in our values, we get:
512.5π = 2 * 41*rπ; 512.5π = 82rπ
Solve for r by dividing both sides by 82π:
6.25 = r
From here, we can calculate the area of a base:
A = 6.252π = 39.0625π
NOTE: The question asks for the area of the bases. Therefore, the answer is 2 * 39.0625π or 78.125π in2.
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The area of a given object is 30,096 in2. What is the area of this object in ft2?
The area of a given object is 30,096 in2. What is the area of this object in ft2?
Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely divide the initial value (30,096) by 12, as though you were converting from inches to feet.
Begin by thinking this through as follows. In the case of a single dimension, we know that:
1 ft = 12 in or 1 in = (1/12) ft
Now, think the case of a square with dimensions 1 ft x 1 ft. This square has the following dimensions in inches: 12 in x 12 in. The area is therefore 12 * 12 = 144 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:
1 ft2 = 144 in2 or 1 in2 = (1/144) ft2
Based on this, we can convert our value 30,096 in2 thus: 30,096/144 = 209 ft2.
Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely divide the initial value (30,096) by 12, as though you were converting from inches to feet.
Begin by thinking this through as follows. In the case of a single dimension, we know that:
1 ft = 12 in or 1 in = (1/12) ft
Now, think the case of a square with dimensions 1 ft x 1 ft. This square has the following dimensions in inches: 12 in x 12 in. The area is therefore 12 * 12 = 144 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:
1 ft2 = 144 in2 or 1 in2 = (1/144) ft2
Based on this, we can convert our value 30,096 in2 thus: 30,096/144 = 209 ft2.
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The area of a given object is 24 yd2. What is the area of this object in in2?
The area of a given object is 24 yd2. What is the area of this object in in2?
Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely multiply the initial value (24) by 36, as though you were converting from yards to inches.
Begin by thinking this through as follows. In the case of a single dimension, we know that:
1 yd = 36 in
Now, think the case of a square with dimensions 1 yd x 1 yd. This square has the following dimensions in inches: 36 in x 36 in. The area is therefore 36 * 36 = 1296 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:
1 yd2 = 1296 in2
Based on this, we can convert our value 24 yd2 thus: 24 * 1296 = 31,104 in2.
Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely multiply the initial value (24) by 36, as though you were converting from yards to inches.
Begin by thinking this through as follows. In the case of a single dimension, we know that:
1 yd = 36 in
Now, think the case of a square with dimensions 1 yd x 1 yd. This square has the following dimensions in inches: 36 in x 36 in. The area is therefore 36 * 36 = 1296 in2. This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:
1 yd2 = 1296 in2
Based on this, we can convert our value 24 yd2 thus: 24 * 1296 = 31,104 in2.
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The number of square units in the surface area of a right circular cylinder is equal to the number of cubic units in its volume. If r and h represent the length in units of the cylinder's radius and height, respectively, which of the following is equivalent to r in terms of h?
The number of square units in the surface area of a right circular cylinder is equal to the number of cubic units in its volume. If r and h represent the length in units of the cylinder's radius and height, respectively, which of the following is equivalent to r in terms of h?
We need to find expressions for the surface area and the volume of a cylinder. The surface area of the cylinder consists of the sum of the surface areas of the two bases plus the lateral surface area.
surface area of cylinder = surface area of bases + lateral surface area
The bases of the cylinder will be two circles with radius r. Thus, the area of each will be _πr_2, and their combined surface area will be 2_πr_2.
The lateral surface area of the cylinder is equal to the circumference of the circular base multiplied by the height. The circumferece of a circle is 2_πr_, and the height is h, so the lateral area is 2_πrh_.
surface area of cylinder = 2_πr_2 + 2_πrh_
Next, we need to find an expression for the volume. The volume of a cylinder is equal to the product of the height and the area of one of the bases. The area of the base is _πr_2, and the height is h, so the volume of the cylinder is πr_2_h.
volume = πr_2_h
Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.
2_πr_2 + 2_πrh_ = πr_2_h
First, let's factor out 2_πr_ from the left side.
2_πr_(r + h) = πr_2_h
We can divide both sides by π.
2_r_(r + h) = r_2_h
We can also divide both sides by r, because the radius cannot equal zero.
2(r + h) = rh
Let's now distribute the 2 on the left side.
2_r_ + 2_h_ = rh
Subtract 2_r_ from both sides to get all the r's on one side.
2_h_ = rh – 2_r_
r_h – 2_r = 2_h_
Factor out an r from the left side.
r(h – 2) = 2_h_
Divide both sides by h – 2
r = 2_h_/(h – 2)
The answer is r = 2_h_/(h – 2).
We need to find expressions for the surface area and the volume of a cylinder. The surface area of the cylinder consists of the sum of the surface areas of the two bases plus the lateral surface area.
surface area of cylinder = surface area of bases + lateral surface area
The bases of the cylinder will be two circles with radius r. Thus, the area of each will be _πr_2, and their combined surface area will be 2_πr_2.
The lateral surface area of the cylinder is equal to the circumference of the circular base multiplied by the height. The circumferece of a circle is 2_πr_, and the height is h, so the lateral area is 2_πrh_.
surface area of cylinder = 2_πr_2 + 2_πrh_
Next, we need to find an expression for the volume. The volume of a cylinder is equal to the product of the height and the area of one of the bases. The area of the base is _πr_2, and the height is h, so the volume of the cylinder is πr_2_h.
volume = πr_2_h
Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.
2_πr_2 + 2_πrh_ = πr_2_h
First, let's factor out 2_πr_ from the left side.
2_πr_(r + h) = πr_2_h
We can divide both sides by π.
2_r_(r + h) = r_2_h
We can also divide both sides by r, because the radius cannot equal zero.
2(r + h) = rh
Let's now distribute the 2 on the left side.
2_r_ + 2_h_ = rh
Subtract 2_r_ from both sides to get all the r's on one side.
2_h_ = rh – 2_r_
r_h – 2_r = 2_h_
Factor out an r from the left side.
r(h – 2) = 2_h_
Divide both sides by h – 2
r = 2_h_/(h – 2)
The answer is r = 2_h_/(h – 2).
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The area of a circle with radius 4 divided by the surface area of a sphere with radius 2 is equal to:
The area of a circle with radius 4 divided by the surface area of a sphere with radius 2 is equal to:
The surface area of a sphere is 4_πr_2. The area of a circle is _πr_2. 16/16 is equal to 1.
The surface area of a sphere is 4_πr_2. The area of a circle is _πr_2. 16/16 is equal to 1.
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What is the ratio of the surface area of a cube to the surface area of a sphere inscribed within it?
What is the ratio of the surface area of a cube to the surface area of a sphere inscribed within it?
Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:
A = 4_πr_2
Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2_r_. This means that the side length of the cube is also 2_r_.
The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:
surface area of cube = 6_s_2
The formula for surface area of a cube comes from the fact that each face of the cube has an area of _s_2, and there are 6 faces total on a cube.
Since we already determined that the side length of the cube is the same as 2_r_, we can replace s with 2_r_.
surface area of cube = 6(2_r_)2 = 6(2_r_)(2_r_) = 24_r_2.
We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.
ratio = (24_r_2)/(4_πr_2)
The _r_2 term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.
ratio = (24_r_2)/(4_πr_2) = 6/π
The answer is 6/π.
Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:
A = 4_πr_2
Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2_r_. This means that the side length of the cube is also 2_r_.
The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:
surface area of cube = 6_s_2
The formula for surface area of a cube comes from the fact that each face of the cube has an area of _s_2, and there are 6 faces total on a cube.
Since we already determined that the side length of the cube is the same as 2_r_, we can replace s with 2_r_.
surface area of cube = 6(2_r_)2 = 6(2_r_)(2_r_) = 24_r_2.
We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.
ratio = (24_r_2)/(4_πr_2)
The _r_2 term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.
ratio = (24_r_2)/(4_πr_2) = 6/π
The answer is 6/π.
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You have a cube with sides of 4.5 inches. What is the surface area of the cube?
You have a cube with sides of 4.5 inches. What is the surface area of the cube?
The area of one side of the cube is:
A cube has 6 sides, so the total surface area of the cube is
The area of one side of the cube is:
A cube has 6 sides, so the total surface area of the cube is
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What is the surface area of a cylinder with a radius of 
and a height of 
?
What is the surface area of a cylinder with a radius of 
When you're calculating the surface area of a cylinder, note that the cylinder will have two circles, one for the top and one for the bottom, and one rectangle that wraps around the "side" of the cylinder (it's helpful to picture peeling the label off a can of soup - it's curved when it's on the can, but really it's a rectangle that has been wrapped around). You know the area of the circle formula; for the rectangle, note that the height is given to you but the width of the rectangle is one you have to intuit: it's the circumference of the circle, because the entire distance around the circle from one point around and back again is the horizontal distance that the area must cover.
Therefore the surface area of a cylinder = 
When you're calculating the surface area of a cylinder, note that the cylinder will have two circles, one for the top and one for the bottom, and one rectangle that wraps around the "side" of the cylinder (it's helpful to picture peeling the label off a can of soup - it's curved when it's on the can, but really it's a rectangle that has been wrapped around). You know the area of the circle formula; for the rectangle, note that the height is given to you but the width of the rectangle is one you have to intuit: it's the circumference of the circle, because the entire distance around the circle from one point around and back again is the horizontal distance that the area must cover.
Therefore the surface area of a cylinder =
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If the volume of a cube is 216 cubic units, then what is its surface area in square units?
If the volume of a cube is 216 cubic units, then what is its surface area in square units?
The volume of a cube is given by the formula V = 
, where V is the volume, and s is the length of each side. We can set V to 216 and then solve for s.
In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.
![\sqrt[3]{216}= 216^{1/3}=6=s](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/17880/gif.latex)
This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to 
. The formula for surface area comes from the fact that each face of the cube has an area of 
, and there are 6 faces in a cube.
surface area = 
The surface area of the square is 216 square units.
The answer is 216.
The volume of a cube is given by the formula V =
In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.
This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to
surface area =
The surface area of the square is 216 square units.
The answer is 216.
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