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7.3.5: Distinguishing Volume and Surface Area


Lesson

Let's work with surface area and volume in context.

Exercise (PageIndex{1}): THe Science Fair

Mai’s science teacher told her that when there is more ice touching the water in a glass, the ice melts faster. She wants to test this statement so she designs her science fair project to determine if crushed ice or ice cubes will melt faster in a drink.

She begins with two cups of warm water. In one cup, she puts a cube of ice. In a second cup, she puts crushed ice with the same volume as the cube. What is your hypothesis? Will the ice cube or crushed ice melt faster, or will they melt at the same rate? Explain your reasoning.

Exercise (PageIndex{2}): Revisiting the Box of Chocolates

The other day, you calculated the volume of this heart-shaped box of chocolates.

The depth of the box is 2 inches. How much cardboard is needed to create the box?

Exercise (PageIndex{3}): Card Sort: Surface Area or Volume

Your teacher will give you cards with different figures and questions on them.

  1. Sort the cards into two groups based on whether it would make more sense to think about the surface area or the volume of the figure when answering the question. Pause here so your teacher can review your work.
  2. Your teacher will assign you a card to examine more closely. What additional information would you need to be able to answer the question on your card?
  3. Estimate reasonable measurements for the figure on your card.
  4. Use your estimated measurements to calculate the answer to the question.

Are you ready for more?

A cake is shaped like a square prism. The top is 20 centimeters on each side, and the cake is 10 centimeters tall. It has frosting on the sides and on the top, and a single candle on the top at the exact center of the square. You have a knife and a 20-centimeter ruler.

  1. Find a way to cut the cake into 4 fair portions, so that all 4 portions have the same amount of cake and frosting.
  2. Find another way to cut the cake into 4 fair portions.
  3. Find a way to cut the cake into 5 fair portions.

Exercise (PageIndex{4}): A Wheelbarrow of Concrete

A wheelbarrow is being used to carry wet concrete. Here are its dimensions.

  1. What volume of concrete would it take to fill the tray?
  2. After dumping the wet concrete, you notice that a thin film is left on the inside of the tray. What is the area of the concrete coating the tray? (Remember, there is no top.)

Summary

Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area.

Here are some examples of quantities related to volume:

  • How much water a container can hold
  • How much material it took to build a solid object

Volume is measured in cubic units, like in3 or m3.

Here are some examples of quantities related to surface area:

  • How much fabric is needed to cover a surface
  • How much of an object needs to be painted

Surface area is measured in square units, like in2 or m2.

Glossary Entries

Definition: Base (of a prism or pyramid)

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

Definition: Cross Section

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

Definition: Prism

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

Definition: Pyramid

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

Definition: Surface Area

The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is (6cdot 9), or 54 cm2.

Definition: Volume

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.

Practice

Exercise (PageIndex{5})

Here is the base of a prism.

  1. If the height of the prism is 5 cm, what is its surface area? What is its volume?
  2. If the height of the prism is 10 cm, what is its surface area? What is its volume?
  3. When the height doubled, what was the percent increase for the surface area? For the volume?

Exercise (PageIndex{6})

Select all the situations where knowing the volume of an object would be more useful than knowing its surface area.

  1. Determining the amount of paint needed to paint a barn.
  2. Determining the monetary value of a piece of gold jewelry.
  3. Filling an aquarium with buckets of water.
  4. Deciding how much wrapping paper a gift will need.
  5. Packing a box with watermelons for shipping.
  6. Charging a company for ad space on your race car.
  7. Measuring the amount of gasoline left in the tank of a tractor.

Exercise (PageIndex{7})

Han draws a triangle with a (50^{circ}) angle, a (40^{circ}) angle, and a side of length 4 cm as shown. Can you draw a different triangle with the same conditions?

(From Unit 7.2.4)

Exercise (PageIndex{8})

Angle (H) is half as large as angle (J). Angle (J) is one fourth as large as angle (K). Angle (K) has measure 240 degrees. What is the measure of angle (H)?

(From Unit 7.1.3)

Exercise (PageIndex{9})

The Colorado state flag consists of three horizontal stripes of equal height. The side lengths of the flag are in the ratio (2:3). The diameter of the gold-colored disk is equal to the height of the center stripe. What percentage of the flag is gold?

(From Unit 4.2.4)


Surface-area-to-volume ratio

The surface-area-to-volume ratio, also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. In chemical reactions involving a solid material, the surface area to volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed.

For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a ball, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with acute-angled spikes will have very large surface area for a given volume.


Surface Area vs Volume

The difference between Surface Area and Volume is that the Surface Area measures the area occupied by the uppermost layer of a surface or put differently it is the area of all the shapes/planes that make up the figures/solids while Volume is the measure of carrying capacity of a figure/shape or the space enclosed within the figure.


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Contents

Gabriel's horn is formed by taking the graph of

with the domain x ≥ 1 and rotating it in three dimensions about the x -axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today, calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a , where a > 1 . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume V and the surface area A :

The value a can be as large as required, but it can be seen from the equation that the volume of the part of the horn between x = 1 and x = a will never exceed π however, it does gradually draw nearer to π as a increases. Mathematically, the volume approaches π as a approaches infinity. Using the limit notation of calculus:

The surface area formula above gives a lower bound for the area as 2 π times the natural logarithm of a . There is no upper bound for the natural logarithm of a , as a approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say,

When the properties of Gabriel's horn were discovered, the fact that the rotation of an infinitely large section of the xy -plane about the x -axis generates an object of finite volume was considered a paradox. While the section lying in the xy -plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the "weighted sum" of sections, is finite.

The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei. [1]

Painter's paradox Edit

The converse of Gabriel's horn—a surface of revolution that has a finite surface area but an infinite volume—cannot occur when revolving a continuous function on a closed set:

Theorem Edit

Let f : [1,∞) → [0,∞) be a continuously differentiable function. Write S for the solid of revolution of the graph y = f(x) about the x -axis. If the surface area of S is finite, then so is the volume.

Proof Edit

Since the lateral surface area A is finite, the limit superior:

lim t → ∞ sup x ≥ t f ( x ) 2 − f ( 1 ) 2 = lim sup t → ∞ ∫ 1 t ( f ( x ) 2 ) ′ d x ≤ ∫ 1 ∞ | ( f ( x ) 2 ) ′ | d x = ∫ 1 ∞ 2 f ( x ) | f ′ ( x ) | d x ≤ ∫ 1 ∞ 2 f ( x ) 1 + f ′ ( x ) 2 d x = A π < ∞ . lim _sup _f(x)^

Therefore, there exists a t0 such that the supremum sup< f(x) | xt0 > is finite. Hence,

M = sup< f(x) | x ≥ 1 > must be finite since f is a continuous function, which implies that f is bounded on the interval [1,∞) .

Therefore: if the area A is finite, then the volume V must also be finite.


Illustrative Mathematics Unit 6.1, Lesson 16: Distinguishing Between Surface Area and Volume

Learn more about the contrast between surface area and volume of three-dimensional shapes, and about the differences between one-, two-, and three-dimensional measurements and units. After trying the questions, click on the buttons to view answers and explanations in text or video.

Distinguishing Between Surface Area and Volume
Let’s contrast surface area and volume.

16.1 - Attributes and Their Measures

For each quantity, choose one or more appropriate units of measurement.

For the last two rows, think of a quantity that could be appropriately measured with the given units.

  1. Perimeter of a parking lot:
  2. Volume of a semi truck:
  3. Surface area of a refrigerator:
  4. Length of an eyelash:
  5. Area of a state:
  6. Volume of an ocean:
  7. ______________________________: miles
  8. ______________________________: cubic meters
  • millimeters (mm)
  • feet (ft)
  • meters (m)
  • square inches (sq in)
  • square feet (sq ft)
  • square miles (sq mi)
  • cubic kilometers (cu km)
  • cubic yards (cu yd)
  1. Perimeter of a parking lot: feet or meters
  2. Volume of a semi truck: cubic yards
  3. Surface area of a refrigerator: square inches
  4. Length of an eyelash: millimeters
  5. Area of a state: square miles
  6. Volume of an ocean: cubic kilometers
  7. Length of a highway: miles
  8. Volume of a swimming pool: cubic meters

Area is always measured in square units and volume is always measured in cubic units.

16.2 - Building with 8 Cubes

Open the applet. Drag on the red point on the cube to move it, and click on the point to switch between vertical and horizontal movement. The gray square will give you 16 cubes. Build 2 different shapes using 8 cubes for each.

For each shape, determine the following information and write it down:
Give a name or a label (e.g., Mae’s First Shape or Eric’s Steps).
Determine its volume.
Determine its surface area.

You may create prisms or non-prisms. You can find the surface area or volume of any shape you create with the applet.

When calculating or counting surface area of your shape, come up with a system to avoid omitting or double-counting faces.

The volume of the rectangular prism is 4 × 2 × 1 = 8 cubic units. This is the same result as counting the number of cubes used to build the prism.

In the rectangular prism, face A has an area of 4 × 2 = 8 square units. Face B has an area of 2 square units. Face C has an area of 4 square units.
Faces A, B, and C all have congruent opposing faces.
The surface area of the rectangular prism, including the bottom, is therefore 2(8) + 2(2) + 2(4) = 28 square units.

The volume of the square donut, from counting the number of cubes it occupies, is 8 cubic units.

In the rectangular prism, face D and E have the same area of 3 square units each.
Face F has an area of 8 square units.
Faces D, E, and F all have congruent opposing faces.
There are 4 faces in the donut hole, which have a total area of 4 square units.
The surface area of the square donut is therefore 4(3) + 2(8) + 4 = 32 square units.

The volumes of both shapes are the same, because volume measures the number of unit cubes that can be packed into a figure. Both shapes are built using the same number of cubes. Building shapes with different volumes would mean using fewer or more cubes.

Shapes with the same volume like these two can have different surface areas. Shapes with larger surface areas are more spread out and have more faces exposed. Shapes with smaller surface areas are more compact and have more of their faces hidden or shared with other cubes.

16.3 - Comparing Prisms Without Building Them

Three rectangular prisms each have a height of 1 cm.

Prism A has a base that is 1 cm by 11 cm.
Prism B has a base that is 2 cm by 7 cm.
Prism C has a base that is 3 cm by 5 cm.

1. Find the surface area and volume of each prism. Use the dot paper to draw the prisms, if needed.

2. Analyze the volumes and surface areas of the prisms. What do you notice? Write 1–2 observations about them.

1. A: Volume = 11 cubic cm
Surface area = 4(11) + 2(1) = 46 sq cm
B: Volume = 2 × 7 × 1 = 14 cubic cm
Surface area = 2(2 × 7) + 2(7) + 2(2) = 46 sq cm
C: Volume = 3 × 5 × 1 = 15 cubic cm
Surface area = 2(3 × 5) + 2(5) + 2(3) = 46 sq cm

2. The volumes of the prisms are all different, but the surface areas are the same.
Shapes with different volumes can have the same surface area.
Volume is described in terms of unit cubes and surface area in terms of the exposed faces of those unit cubes.

Can you find more examples of prisms that have the same surface areas but different volumes? How many can you find?

These are 3 examples of prisms that all have a surface area of 54 sq cm, but different volumes. These are not necessarily the only prisms which can be drawn for a surface area of 54 sq cm.

1. A: Volume = 3 × 3 × 3 = 27 cubic cm
Surface area = 6(3 × 3) = 54 sq cm
B: Volume = 3 × 6 × 1 = 18 cubic cm
Surface area = 2(3 × 6) + 2(6) + 2(3) = 54 sq cm
C: Volume = 13 × 1 × 1 = 13 cubic cm
Surface area = 4(13) + 2(1) = 54 sq cm

Length is a one-dimensional attribute of a geometric figure. We measure lengths using units like millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles.

Area is a two-dimensional attribute. We measure area in square units. For example, a square that is 1 centimeter on each side has an area of 1 square centimeter.

Volume is a three-dimensional attribute. We measure volume in cubic units. For example, a cube that is 1 kilometer on each side has a volume of 1 cubic kilometer.

Surface area and volume are different attributes of three-dimensional figures. Surface area is a two-dimensional measure, while volume is a three-dimensional measure.

Two figures can have the same volume but different surface areas. For example:

A rectangular prism with side lengths of 1 cm, 2 cm, and 2 cm has a volume of 4 cu cm and a surface area of 16 sq cm.
A rectangular prism with side lengths of 1 cm, 1 cm, and 4 cm has the same volume but a surface area of 18 sq cm.

Similarly, two figures can have the same surface area but different volumes.

A rectangular prism with side lengths of 1 cm, 1 cm, and 5 cm has a surface area of 22 sq cm and a volume of 5 cu cm.
A rectangular prism with side lengths of 1 cm, 2 cm, and 3 cm has the same surface area but a volume of 6 cu cm.

1. Match each quantity with an appropriate unit of measurement.

  1. The surface area of a tissue box
  2. The amount of soil in a planter box
  3. The area of a parking lot
  4. The length of a soccer field
  5. The volume of a fish tank
  1. Square meters
  2. Yards
  3. Cubic inches
  4. Cubic feet
  5. Square centimeters
  1. The surface area of a tissue box: Square centimeters
  2. The amount of soil in a planter box: Cubic inches
  3. The area of a parking lot: Square meters
  4. The length of a soccer field: Yards
  5. The volume of a fish tank: Cubic inches

2. Here is a figure built from snap cubes.

a. Find the volume of the figure in cubic units.
b. Find the surface area of the figure in square units.
c. True or false: If we double the number of cubes being stacked, both the volume and surface area will double. Explain or show how you know.

a. Volume of the figure = 1 unit × 1 unit × 4 units = 4 cubic units
b. Surface area of the figure = 4(4 units) + 2(1 unit) = 18 square units
c. False. The volume will double to 8 cubic units, but the new surface area will be 4(8 units) + 2(1 unit) = 34 square units, which does not equal 18 × 2.

3. Lin said, "Two figures with the same volume also have the same surface area."

a. Which two figures suggest that her statement is true?
b. Which two figures could show that her statement is not true?

A: Volume = 6 cubic units, surface area = 26 square units
B: Volume = 6 cubic units, surface area = 24 square units
C: Volume = 6 cubic units, surface area = 24 square units
D: Volume = 7 cubic units, surface area = 26 square units
E: Volume = 5 cubic units, surface area = 22 square units

B and C have the same volume and same surface area.
A and C have the same volume, but a different surface area, showing that Lin's statement is not true.

4. Draw a pentagon (five-sided polygon) that has an area of 32 square units. Label all relevant sides or segments with their measurements, and show that the area is 32 square units.

This polygon does not need to be a regular pentagon, as long as it has 5 sides. Try a composite of a square and a triangle, and try different areas for the square. The triangle should have the same base length as the square's sides.

Other compositions are possible.

5. a. Draw a net for this rectangular prism.
b. Find the surface area of the rectangular prism.

a.

b. The surface area is 2(5 × 10) + 2(2 × 10) + 2(2 × 5) = 160 sq cm.

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What is the Difference Between Volume and Surface Area?

Volume and surface area are two related concepts in the study of mathematics. They’re both important to understand, but equally important is understanding how they differ and what they mean. This is especially the case when it comes to computing the volume and surface areas of a prism or a cylinder.

If you think of wrapping a present in a box, you can get a good sense of how volume and surface area differ. First, you have to consider the size of the box, when you consider the size of the present. How much interior space does your box need to have so a present will fit? The measurement of the box’s capacity, how much it will hold, is its volume. Next you have to wrap the present. The amount of wrapping paper, which will cover the exterior of the box, is a very different calculation than the capacity of the box. You’ll need a separate measurement or some good guessing, to figure out the sum of the sides of all the surfaces or the surface area.

Volume of a square or rectangular box is pretty easy to compute. Simply multiply height times length times width to get the measurement. With a square it’s even easier, you merely cube one side’s length, since they all measure the same. If the side length is a, the formula is a x a x a or a 3 . When you are comparing volume and surface area, you’ll note a very different formula. You need to get the area of each face, and then add the areas of all faces together. With a square prism or cube, you’d essentially compute the area a x a or a 2 , multiplied by 6 (6a 2 ). When you’re working with a rectangular prism, you’ll have to the area the of 3 pairs of equal sides, which needed to be added together to determine surface area.

Work on volume and surface area are differ a little when you are trying to calculate the area of a cylinder. The formula for a volume of a cylinder is the area of one circular face multiplied times the height of the cylinder. It reads: πr 2 x h, or pi times the radius squared times height. Getting the surface area of the cylinder is a little trickier since the circular portion is essentially one continuous face. Computing surface area of a cylinder means computing the lateral area of this face.

Lateral area formula is the following πr2r or πd (pi times the radius doubled or pi times the diameter), multiplied to the height, πr2r x h. This is essentially the circumference of one circle times the height of the cylinder. To compute the entire formula you also need to add in the top and bottom circular faces’ areas. Since in a cylinder these are equal, the formula is 2 πr 2 . This calculation is then added to the lateral area to compute the whole surface area in the following expression:

πr2r x h + 2πr 2 = lateral area.

You can also view difference between volume and cylinder as a difference between what is inside and can be contained and the exterior of a three-dimensional object. These are valuable differences to understand in many applications, such as construction, engineering, or even present wrapping. When children complain that math is useless outside of math class, you might point out to them that knowing the difference between volume and surface area meant they got a very nicely wrapped gift for their birthday.

Tricia has a Literature degree from Sonoma State University and has been a frequent InfoBloom contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.

Tricia has a Literature degree from Sonoma State University and has been a frequent InfoBloom contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.


Lesson 15 Summary

Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area.

Here are some examples of quantities related to volume:

  • How much water a container can hold
  • How much material it took to build a solid object

Volume is measured in cubic units, like in 3 or m 3 .

Here are some examples of quantities related to surface area:

  • How much fabric is needed to cover a surface
  • How much of an object needs to be painted

Surface area is measured in square units, like in 2 or m 2 .




Volume is the quantification of the three-dimensional space a substance occupies. The SI unit for volume is the cubic meter, or m 3 . By convention, the volume of a container is typically its capacity, and how much fluid it is able to hold, rather than the amount of space that the actual container displaces. Volumes of many shapes can be calculated by using well-defined formulas. In some cases, more complicated shapes can be broken down into their simpler aggregate shapes, and the sum of their volumes used to determine total volume. The volumes of other even more complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary. Beyond this, shapes that cannot be described by known equations can be estimated using mathematical methods, such as the finite element method. Alternatively, if the density of a substance is known, and is uniform, the volume can be calculated using its weight. This calculator computes volumes for some of the most common simple shapes.

Sphere

A sphere is the three-dimensional counterpart of the two-dimensional circle. It is a perfectly round geometrical object that mathematically, is the set of points that are equidistant from a given point at its center, where the distance between the center and any point on the sphere is the radius r. Likely the most commonly known spherical object is a perfectly round ball. Within mathematics, there is a distinction between a ball and a sphere, where a ball comprises the space bounded by a sphere. Regardless of this distinction, a ball and a sphere share the same radius, center, and diameter, and the calculation of their volumes is the same. As with a circle, the longest line segment that connects two points of a sphere through its center is called the diameter, d. The equation for calculating the volume of a sphere is provided below:

EX: Claire wants to fill a perfectly spherical water balloon with radius 0.15 ft with vinegar to use in the water balloon fight against her arch-nemesis Hilda this coming weekend. The volume of vinegar necessary can be calculated using the equation provided below:

volume = 4/3 × &pi × 0.15 3 = 0.141 ft 3

A cone is a three-dimensional shape that tapers smoothly from its typically circular base to a common point called the apex (or vertex). Mathematically, a cone is formed similarly to a circle, by a set of line segments connected to a common center point, except that the center point is not included in the plane that contains the circle (or some other base). Only the case of a finite right circular cone is considered on this page. Cones comprised of half-lines, non-circular bases, etc. that extend infinitely will not be addressed. The equation for calculating the volume of a cone is as follows:

where r is radius and h is height of the cone

EX: Bea is determined to walk out of the ice cream store with her hard earned $5 well spent. While she has a preference for regular sugar cones, the waffle cones are indisputably larger. She determines that she has a 15% preference for regular sugar cones over waffle cones and needs to determine whether the potential volume of the waffle cone is &ge 15% more than that of the sugar cone. The volume of the waffle cone with a circular base with radius 1.5 in and height 5 in can be computed using the equation below:

volume = 1/3 × &pi × 1.5 2 × 5 = 11.781 in 3

Bea also calculates the volume of the sugar cone and finds that the difference is < 15%, and decides to purchase a sugar cone. Now all she has to do is use her angelic, childlike appeal to manipulate the staff into emptying the containers of ice cream into her cone.

A cube is the three-dimensional analog of a square, and is an object bounded by six square faces, three of which meet at each of its vertices, and all of which are perpendicular to their respective adjacent faces. The cube is a special case of many classifications of shapes in geometry including being a square parallelepiped, an equilateral cuboid, and a right rhombohedron. Below is the equation for calculating the volume of a cube:

volume = a 3
where a is edge length of the cube

EX: Bob, who was born in Wyoming (and has never left the state), recently visited his ancestral homeland of Nebraska. Overwhelmed by the magnificence of Nebraska and the environment unlike any other he had previously experienced, Bob knew that he had to bring some of Nebraska home with him. Bob has a cubic suitcase with edge lengths of 2 feet, and calculates the volume of soil that he can carry home with him as follows:

Cylinder

A cylinder in its simplest form is defined as the surface formed by points at a fixed distance from a given straight line axis. In common use however, "cylinder" refers to a right circular cylinder, where the bases of the cylinder are circles connected through their centers by an axis perpendicular to the planes of its bases, with given height h and radius r. The equation for calculating the volume of a cylinder is shown below:

volume = &pir 2 h
where r is radius and h is height of the tank

EX: Caelum wants to build a sandcastle in the living room of his house. Because he is a firm advocate of recycling, he has recovered three cylindrical barrels from an illegal dumping site and has cleaned the chemical waste from the barrels using dishwashing detergent and water. The barrels each have a radius of 3 ft and a height of 4 ft, and Caelum determines the volume of sand that each can hold using the equation below:

volume = &pi × 3 2 × 4 = 113.097 ft 3

He successfully builds a sandcastle in his house, and as an added bonus, manages to save electricity on nighttime lighting, since his sandcastle glows bright green in the dark.

Rectangular Tank

A rectangular tank is a generalized form of a cube, where the sides can have varied lengths. It is bounded by six faces, three of which meet at its vertices, and all of which are perpendicular to their respective adjacent faces. The equation for calculating the volume of a rectangle is shown below:

volume= length × width × height

EX: Darby likes cake. She goes to the gym for 4 hours a day, every day, to compensate for her love of cake. She plans to hike the Kalalau Trail in Kauai and though extremely fit, Darby worries about her ability to complete the trail due to her lack of cake. She decides to pack only the essentials and wants to stuff her perfectly rectangular pack of length, width, and height 4 ft, 3 ft and 2 ft respectively, with cake. The exact volume of cake she can fit into her pack is calculated below:

Capsule

A capsule is a three-dimensional geometric shape comprised of a cylinder and two hemispherical ends, where a hemisphere is half a sphere. It follows that the volume of a capsule can be calculated by combining the volume equations for a sphere and a right circular cylinder:

where r is radius and h is height of the cylindrical portion

EX: Given a capsule with a radius of 1.5 ft and a height of 3 ft, determine the volume of melted milk chocolate m&m's that Joe can carry in the time capsule he wants to bury for future generations on his journey of self-discovery through the Himalayas:

volume = &pi × 1.5 2 × 3 + 4/3 ×&pi ࡧ.5 3 = 35.343 ft 3

Spherical Cap

A spherical cap is a portion of a sphere that is separated from the rest of the sphere by a plane. If the plane passes through the center of the sphere, the spherical cap is referred to as a hemisphere. Other distinctions exist including a spherical segment, where a sphere is segmented with two parallel planes and two different radii where the planes pass through the sphere. The equation for calculating the volume of a spherical cap is derived from that of a spherical segment, where the second radius is 0. In reference to the spherical cap shown in the calculator:

Given two values, the calculator provided computes the third value and the volume. The equations for converting between the height and the radii are shown below:

EX: Jack really wants to beat his friend James in a game of golf to impress Jill, and rather than practicing, decides to sabotage James' golf ball. He cuts off a perfect spherical cap from the top of James' golf ball, and needs to calculate the volume of the material necessary to replace the spherical cap and skew the weight of James' golf ball. Given James' golf ball has a radius of 1.68 inches, and the height of the spherical cap that Jack cut off is 0.3 inches, the volume can be calculated as follows:

volume = 1/3 × &pi × 0.3 2 (3 × 1.68 - 0.3) = 0.447 in 3

Unfortunately for Jack, James happened to receive a new shipment of balls the day before their game, and all of Jack's efforts were in vain.

Conical Frustum

A conical frustum is the portion of a solid that remains when a cone is cut by two parallel planes. This calculator calculates the volume for a right circular cone specifically. Typical conical frustums found in everyday life include lampshades, buckets, and some drinking glasses. The volume of a right conical frustum is calculated using the following equation:

where r and R are the radii of the bases, h is the height of the frustum

EX: Bea has successfully acquired some ice cream in a sugar cone, and has just eaten it in a way that leaves the ice cream packed within the cone, and the ice cream surface level and parallel to the plane of the cone's opening. She is about to start eating her cone and the remaining ice cream when her brother grabs her cone and bites off a section of the bottom of her cone that is perfectly parallel to the previously sole opening. Bea is now left with a right conical frustum leaking ice cream, and has to calculate the volume of ice cream she must quickly consume given a frustum height of 4 inches, with radii 1.5 inches and 0.2 inches:

volume=1/3 × &pi × 4(0.2 2 + 0.2 × 1.5 + 1.5 2 ) = 10.849 in 3

Ellipsoid

An ellipsoid is the three-dimensional counterpart of an ellipse, and is a surface that can be described as the deformation of a sphere through scaling of directional elements. The center of an ellipsoid is the point at which three pairwise perpendicular axes of symmetry intersect, and the line segments delimiting these axes of symmetry are called the principle axes. If all three have different lengths, the ellipsoid is commonly described as tri-axial. The equation for calculating the volume of an ellipsoid is as follows:

where a, b, and c are the lengths of the axes

EX: Xabat only likes eating meat, but his mother insists that he consumes too much, and only allows him to eat as much meat as he can fit within an ellipsoid shaped bun. As such, Xabat hollows out the bun to maximize the volume of meat that he can fit in his sandwich. Given that his bun has axis lengths of 1.5 inches, 2 inches, and 5 inches, Xabat calculates the volume of meat he can fit in each hollowed bun as follows:

volume = 4/3 × &pi × 1.5 × 2 × 5 = 62.832 in 3

Square Pyramid

A pyramid in geometry is a three-dimensional solid formed by connecting a polygonal base to a point called its apex, where a polygon is a shape in a plane bounded by a finite number of straight line segments. There are many possible polygonal bases for a pyramid, but a square pyramid is a pyramid in which the base is a square. Another distinction involving pyramids involves the location of the apex. Right pyramids have an apex that is directly above the centroid of its base. Regardless of where the apex of the pyramid is, as long as its height is measured as the perpendicular distance from the plane containing the base to its apex, the volume of the pyramid can be written as:

EX: Wan is fascinated by ancient Egypt and particularly enjoys anything related to the pyramids. Being the eldest of his siblings Too, Tree and Fore, he is able to easily corral and deploy them at his will. Taking advantage of this, Wan decides to re-enact ancient Egyptian times and have his siblings act as workers building him a pyramid of mud with edge length 5 feet and height 12 feet, the volume of which can be calculated using the equation for a square pyramid:

volume = 1/3 × 5 2 × 12 = 100 ft 3

Tube Pyramid

A tube, often also referred to as a pipe, is a hollow cylinder that is often used to transfer fluids or gas. Calculating the volume of a tube essentially involves the same formula as a cylinder (volume=pr 2 h), except that in this case the diameter is used rather than the radius, and length is used rather than height. The formula therefore involves measuring the diameters of the inner and outer cylinder, as shown in the figure above, calculating each of their volumes, and subtracting the volume of the inner cylinder from that of the outer one. Considering the use of length and diameter mentioned above, the formula for calculating the volume of a tube is shown below:

where d1 is outer diameter, d2 is inner diameter, and l is length of the tube

EX: Beulah is dedicated to environmental conservation. Her construction company uses only the most environmentally friendly of materials. She also prides herself on meeting customer needs. One of her customers has a vacation home built in the woods, across a creek. He wants easier access to his house, and requests that Beulah build him a road, while ensuring that the creek can flow freely so as not to disrupt his favorite fishing spot. She decides that the pesky beaver dams would be a good point to build a pipe through the creek. The volume of patented low-impact concrete required to build a pipe of outer diameter 3 feet, inner diameter 2.5 feet, and length of 10 feet, can be calculated as follows:


Surface area of a triangular prism

The surface area formula for a triangular prism is 2 * (height x base / 2) + length x width1 + length x width2 + length x base, as seen in the figure below:

A triangular prism is a stack of triangles, so the usualy triangle solving rules apply when calculating the area of the bases.


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