maths.free › Arithmetic › 9. Math Models and Geometry › Solve Geometry Applications: Volume and Surface Area

Solve Geometry Applications: Volume and Surface Area

Find volume and surface area of rectangular solids

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See ). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in has length \(4\) units, width \(2\) units, and height \(3\) units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Altogether there are \(24\) cubic units. Notice that \(24\) is the \(\text{length}\ \times \ \text{width}\ \times \ \text{height}\text{.}\)

The volume, \(V,\) of any rectangular solid is the product of the length, width, and height.

\[V=LWH\]

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, \(B,\) is equal to \(\text{length}\times \text{width}\text{.}\)

\[B=L\cdot W\]

We can substitute \(B\) for \(L\cdot W\) in the volume formula to get another form of the volume formula.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the \(4\ \times \ 2\ \times \ 3\) rectangular solid we started with. See .

\[\begin{array}{lllllll}{A}_{\text{front}}=L\times W & & & {A}_{\text{side}}=L\times W & & & {A}_{\text{top}}=L\times W \\ {A}_{\text{front}}=4\cdot 3 & & & {A}_{\text{side}}=2\cdot 3 & & & {A}_{\text{top}}=4\cdot 2 \\ {A}_{\text{front}}=12 & & & {A}_{\text{side}}=6 & & & {A}_{\text{top}}=8\end{array}\]\[\begin{array}{l}S=(\text{front}+\text{back})\text{+}(\text{left side}+\text{right side})+(\text{top}+\text{bottom}) \\ S=(2\cdot \text{front})+(\text{2}\cdot \text{left side})+(\text{2}\cdot \text{top}) \\ S=2\cdot 12+2\cdot 6+2\cdot 8 \\ S=24+12+16 \\ S=52\ \text{sq. units}\end{array}\]\[S=2LH+2LW+2WH\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate \(\pi\) with \(3.14.\)

Example

Try it.

A sphere has a radius \(6\) inches. Find its ⓐ volume and ⓑ surface area.

Solution

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and label
it with the given information.
ⓐ
Step 2. Identify what you are looking for.the volume of the sphere
Step 3. Name. Choose a variable to represent it.let V = volume
Step 4. Translate.
Write the appropriate formula.
\(V=\frac{4}{3}\pi {r}^{3}\)
Step 5. Solve.\(V\approx \frac{4}{3}(3.14){6}^{3}\)
\(V\approx 904.32\ \text{cubic inches}\)
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question.The volume is approximately 904.32 cubic inches.
ⓑ
Step 2. Identify what you are looking for.the surface area of the sphere
Step 3. Name. Choose a variable to represent it.let S = surface area
Step 4. Translate.
Write the appropriate formula.
\(S=4\pi {r}^{2}\)
Step 5. Solve.\(S\approx 4(3.14){6}^{2}\)
\(S\approx 452.16\)
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question.The surface area is approximately 452.16 square inches.

Condensed — the full section is in OpenStax Prealgebra 2e.

Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height \(h\) of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, \(h\) , will be perpendicular to the bases.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, \(V=Bh\) , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, \(B\) , is the area of the rectangular base, length × width. For a cylinder, the area of the base, \(B,\) is the area of its circular base, \(\pi {r}^{2}.\) compares how the formula \(V=Bh\) is used for rectangular solids and cylinders.

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See .

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length \(L\) of the rectangular label. The height of the cylinder is the width \(W\) of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius \(r\) and height \(h,\) is

\[S=2\pi {r}^{2}+2\pi rh\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See .

Earlier in this section, we saw that the volume of a cylinder is \(V=\text{\pi }{r}^{2}h.\) We can think of a cone as part of a cylinder. shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

Since the base of a cone is a circle, we can substitute the formula of area of a circle, \(\text{\pi }{r}^{2}\) , for \(B\) to get the formula for volume of a cone.

In this book, we will only find the volume of a cone, and not its surface area.

Example

Try it.

Find the volume of a cone with height \(6\) inches and radius of its base \(2\) inches.

Solution
Step 1. Read the problem. Draw the figure and label it
with the given information.
Step 2. Identify what you are looking for.the volume of the cone
Step 3. Name. Choose a variable to represent it.let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for \(\pi\))
\(V=\frac{1}{3}\ \pi \ {r}^{2}\ h\)
\(V\approx \frac{1}{3}\ 3.14\ {(2)}^{2}\ (6)\)
Step 5. Solve.\(V\approx 25.12\)
Step 6. Check: We leave it to you to check your
calculations.
Step 7. Answer the question.The volume is approximately 25.12 cubic inches.

Condensed — the full section is in OpenStax Prealgebra 2e.

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

Key Concepts

  • Volume and Surface Area of a Rectangular Solid
    • \(V=LWH\)
    • \(S=2LH+2LW+2WH\)
  • Volume and Surface Area of a Cube
    • \(V={s}^{3}\)
    • \(S=6{s}^{2}\)
  • Volume and Surface Area of a Sphere
    • \(V=\frac{4}{3}\pi {r}^{3}\)
    • \(S=4\pi {r}^{2}\)
  • Volume and Surface Area of a Cylinder
    • \(V=\pi {r}^{2}h\)
    • \(S=2\pi {r}^{2}+2\pi rh\)
  • Volume of a Cone
    • For a cone with radius \(r\) and height \(h\):
      Volume: \(V=\frac{1}{3}\pi {r}^{2}h\)

Solve Geometry Applications: Volume and Surface Area

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.

Try it.

length \(2\) meters, width \(1.5\) meters, height \(3\) meters

Solution

  1. ⓐ 9 cu. m
  2. ⓑ 27 sq. m

Try it.

length \(5\) feet, width \(8\) feet, height \(2.5\) feet

Try it.

length \(3.5\) yards, width \(2.1\) yards, height \(2.4\) yards

Solution

  1. ⓐ 17.64 cu. yd.
  2. ⓑ 41.58 sq. yd.

Try it.

length \(8.8\) centimeters, width \(6.5\) centimeters, height \(4.2\) centimeters

In the following exercises, solve.

Try it.

Moving van A rectangular moving van has length \(16\) feet, width \(8\) feet, and height \(8\) feet. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 1,024 cu. ft
  2. ⓑ 640 sq. ft

Try it.

Gift box A rectangular gift box has length \(26\) inches, width \(16\) inches, and height \(4\) inches. Find its ⓐ volume and ⓑ surface area.

Try it.

Carton A rectangular carton has length \(21.3\) cm, width \(24.2\) cm, and height \(6.5\) cm. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 3,350.49 cu. cm
  2. ⓑ 1,622.42 sq. cm

Try it.

Shipping container A rectangular shipping container has length \(22.8\) feet, width \(8.5\) feet, and height \(8.2\) feet. Find its ⓐ volume and ⓑ surface area.

In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.

Try it.

\(5\) centimeters

Solution

  1. ⓐ 125 cu. cm
  2. ⓑ 150 sq. cm

Try it.

\(6\) inches

Try it.

\(10.4\) feet

Solution

  1. ⓐ 1124.864 cu. ft.
  2. ⓑ 648.96 sq. ft

Try it.

\(12.5\) meters

In the following exercises, solve.

Try it.

Science center Each side of the cube at the Discovery Science Center in Santa Ana is \(64\) feet long. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 262,144 cu. ft
  2. ⓑ 24,576 sq. ft

Try it.

Museum A cube-shaped museum has sides \(45\) meters long. Find its ⓐ volume and ⓑ surface area.

Try it.

Base of statue The base of a statue is a cube with sides \(2.8\) meters long. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 21.952 cu. m
  2. ⓑ 47.04 sq. m

Try it.

Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its ⓐ volume and ⓑ surface area.

Find the Volume and Surface Area of Spheres

In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

Try it.

\(3\) centimeters

Solution

  1. ⓐ 113.04 cu. cm
  2. ⓑ 113.04 sq. cm

Try it.

\(9\) inches

Try it.

\(7.5\) feet

Solution

  1. ⓐ 1,766.25 cu. ft
  2. ⓑ 706.5 sq. ft

Try it.

\(2.1\) yards

In the following exercises, solve. Round answers to the nearest hundredth.

Try it.

Exercise ball An exercise ball has a radius of \(15\) inches. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 14,130 cu. in.
  2. ⓑ 2,826 sq. in.

Try it.

Balloon ride The Great Park Balloon is a big orange sphere with a radius of \(36\) feet . Find its ⓐ volume and ⓑ surface area.

Try it.

Golf ball A golf ball has a radius of \(4.5\) centimeters. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 381.51 cu. cm
  2. ⓑ 254.34 sq. cm

Try it.

Baseball A baseball has a radius of \(2.9\) inches. Find its ⓐ volume and ⓑ surface area.

Find the Volume and Surface Area of a Cylinder

In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

Try it.

radius \(3\) feet, height \(9\) feet

Solution

  1. ⓐ 254.34 cu. ft
  2. ⓑ 226.08 sq. ft

Try it.

radius \(5\) centimeters, height \(15\) centimeters

Try it.

radius \(1.5\) meters, height \(4.2\) meters

Solution

  1. ⓐ 29.673 cu. m
  2. ⓑ 53.694 sq. m

Try it.

radius \(1.3\) yards, height \(2.8\) yards

In the following exercises, solve. Round answers to the nearest hundredth.

Try it.

Coffee can A can of coffee has a radius of \(5\) cm and a height of \(13\) cm. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 1,020.5 cu. cm
  2. ⓑ 565.2 sq. cm

Try it.

Snack pack A snack pack of cookies is shaped like a cylinder with radius \(4\) cm and height \(3\) cm. Find its ⓐ volume and ⓑ surface area.

Try it.

Barber shop pole A cylindrical barber shop pole has a diameter of \(6\) inches and height of \(24\) inches. Find its ⓐ volume and ⓑ surface area.

Solution

  1. ⓐ 678.24 cu. in.
  2. ⓑ 508.68 sq. in.

Try it.

Architecture A cylindrical column has a diameter of \(8\) feet and a height of \(28\) feet. Find its ⓐ volume and ⓑ surface area.

Find the Volume of Cones

In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

Try it.

height \(9\) feet and radius \(2\) feet

Solution

37.68 cu. ft

Try it.

height \(8\) inches and radius \(6\) inches

Try it.

height \(12.4\) centimeters and radius \(5\) cm

Solution

324.47 cu. cm

Try it.

height \(15.2\) meters and radius \(4\) meters

In the following exercises, solve. Round answers to the nearest hundredth.

Try it.

Teepee What is the volume of a cone-shaped teepee tent that is \(10\) feet tall and \(10\) feet across at the base?

Solution

261.67 cu. ft

Try it.

Popcorn cup What is the volume of a cone-shaped popcorn cup that is \(8\) inches tall and \(6\) inches across at the base?

Try it.

Silo What is the volume of a cone-shaped silo that is \(50\) feet tall and \(70\) feet across at the base?

Solution

64,108.33 cu. ft

Try it.

Sand pile What is the volume of a cone-shaped pile of sand that is \(12\) meters tall and \(30\) meters across at the base?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Evaluate \({x}^{3}\) when \(x=5.\)
    If you missed this problem, review .

    Vastuse avalikustamine

    \(125\)

  2. Evaluate \({2}^{x}\) when \(x=5.\)
    If you missed this problem, review .

    Vastuse avalikustamine

    \(32\)

  3. Find the area of a circle with radius \(\frac{7}{2}.\)
    If you missed this problem, review .

    Vastuse avalikustamine

    \(\frac{77}{2}\)

  4. For a rectangular solid with length \(14\) cm, height \(17\) cm, and width \(9\) cm, find the ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the rectangular solid
    Step 3. Name. Choose a variable to represent it.Let \(V\)= volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    \(V=LWH\)
    \(V=14⋅9⋅17\)
    Step 5. Solve the equation.\(V=2,142\)
    Step 6. Check
    We leave it to you to check your calculations.
    Step 7. Answer the question.The volume is \(\text{2,142}\) cubic centimeters.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the solid
    Step 3. Name. Choose a variable to represent it.Let \(S\)= surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    \(S=2LH+2LW+2WH\)
    \(S=2(14⋅17)+2(14⋅9)+2(9⋅17)\)
    Step 5. Solve the equation.\(S=1,034\)
    Step 6. Check: Double-check with a calculator.
    Step 7. Answer the question.The surface area is 1,034 square centimeters.

  5. Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length \(8\) feet, width \(9\) feet, and height \(11\) feet.

    Vastuse avalikustamine

    1. ⓐ 792 cu. ft
    2. ⓑ 518 sq. ft

  6. Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length \(15\) feet, width \(12\) feet, and height \(8\) feet.

    Vastuse avalikustamine

    1. ⓐ 1,440 cu. ft
    2. ⓑ 792 sq. ft

  7. A rectangular crate has a length of \(30\) inches, width of \(25\) inches, and height of \(20\) inches. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the crate
    Step 3. Name. Choose a variable to represent it.let \(V\)= volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    \(V=LWH\)
    \(V=30⋅25⋅20\)
    Step 5. Solve the equation.\(V=15,000\)
    Step 6. Check: Double check your math.
    Step 7. Answer the question.The volume is 15,000 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the crate
    Step 3. Name. Choose a variable to represent it.let \(S\)= surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    \(S=2LH+2LW+2WH\)
    \(S=2(30⋅20)+2(30⋅25)+2(25⋅20)\)
    Step 5. Solve the equation.\(S=3,700\)
    Step 6. Check: Check it yourself!
    Step 7. Answer the question.The surface area is 3,700 square inches.

  8. A rectangular box has length \(9\) feet, width \(4\) feet, and height \(6\) feet. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 216 cu. ft
    2. ⓑ 228 sq. ft

  9. A rectangular suitcase has length \(22\) inches, width \(14\) inches, and height \(9\) inches. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 2,772 cu. in.
    2. ⓑ 1,264 sq. in.

  10. A cube is \(2.5\) inches on each side. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the cube
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    \(V={s}^{3}\)
    Step 5. Solve. Substitute and solve.\(V={(2.5)}^{3}\)
    \(V=15.625\)
    Step 6. Check: Check your work.
    Step 7. Answer the question.The volume is 15.625 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the cube
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    \(S=6{s}^{2}\)
    Step 5. Solve. Substitute and solve.\(S=6⋅{(2.5)}^{2}\)
    \(S=37.5\)
    Step 6. Check: The check is left to you.
    Step 7. Answer the question.The surface area is 37.5 square inches.

  11. For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube.

    Vastuse avalikustamine

    1. ⓐ 91.125 cu. m
    2. ⓑ 121.5 sq. m

  12. For a cube with side 7.3 yards, find the ⓐ volume and ⓑ surface area of the cube.

    Vastuse avalikustamine

    1. ⓐ 389.017 cu. yd.
    2. ⓑ 319.74 sq. yd.

  13. A notepad cube measures \(2\) inches on each side. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the cube
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    \(V={s}^{3}\)
    Step 5. Solve the equation.\(V={2}^{3}\)
    \(V=8\)
    Step 6. Check: Check that you did the calculations
    correctly.
    Step 7. Answer the question.The volume is 8 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the cube
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    \(S=6{s}^{2}\)
    Step 5. Solve the equation.\(S=6⋅{2}^{2}\)
    \(S=24\)
    Step 6. Check: The check is left to you.
    Step 7. Answer the question.The surface area is 24 square inches.

  14. A packing box is a cube measuring \(4\) feet on each side. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 64 cu. ft
    2. ⓑ 96 sq. ft

  15. A wall is made up of cube-shaped bricks. Each cube is \(16\) inches on each side. Find the ⓐ volume and ⓑ surface area of each cube.

    Vastuse avalikustamine

    1. ⓐ 4,096 cu. in.
    2. ⓑ 1536 sq. in.

  16. A sphere has a radius \(6\) inches. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and label
    it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the sphere
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    \(V=\frac{4}{3}\pi {r}^{3}\)
    Step 5. Solve.\(V\approx \frac{4}{3}(3.14){6}^{3}\)
    \(V\approx 904.32\ \text{cubic inches}\)
    Step 6. Check: Double-check your math on a calculator.
    Step 7. Answer the question.The volume is approximately 904.32 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the sphere
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    \(S=4\pi {r}^{2}\)
    Step 5. Solve.\(S\approx 4(3.14){6}^{2}\)
    \(S\approx 452.16\)
    Step 6. Check: Double-check your math on a calculator
    Step 7. Answer the question.The surface area is approximately 452.16 square inches.

  17. Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters.

    Vastuse avalikustamine

    1. ⓐ 113.04 cu. cm
    2. ⓑ 113.04 sq. cm

  18. Find the ⓐ volume and ⓑ surface area of each sphere with a radius of \(1\) foot

    Vastuse avalikustamine

    1. ⓐ 4.19 cu. ft
    2. ⓑ 12.56 sq. ft

  19. A globe of Earth is in the shape of a sphere with radius \(14\) inches. Find its ⓐ volume and ⓑ surface area. Round the answer to the nearest hundredth.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw a figure with the
    given information and label it.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the sphere
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(V=\frac{4}{3}\pi {r}^{3}\)
    \(V\approx \frac{4}{3}(3.14){14}^{3}\)
    Step 5. Solve.\(V\approx 11,488.21\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The volume is approximately 11,488.21 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the sphere
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(S=4\pi {r}^{2}\)
    \(S\approx 4(3.14){14}^{2}\)
    Step 5. Solve.\(S\approx 2461.76\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The surface area is approximately 2461.76 square inches.

  20. A beach ball is in the shape of a sphere with radius of \(9\) inches. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 3052.08 cu. in.
    2. ⓑ 1017.36 sq. in.

  21. A Roman statue depicts Atlas holding a globe with radius of \(1.5\) feet. Find the ⓐ volume and ⓑ surface area of the globe.

    Vastuse avalikustamine
    1. ⓐ 14.13 cu. ft
    2. ⓑ 28.26 sq. ft

  22. A cylinder has height \(5\) inches and radius \(3\) inches. Find the ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw the figure and label
    it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the cylinder
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(V=\pi {r}^{2}h\)
    \(V\approx (3.14){3}^{2}⋅5\)
    Step 5. Solve.\(V\approx 141.3\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The volume is approximately 141.3 cubic inches.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the cylinder
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(S=2\pi {r}^{2}+2\pi rh\)
    \(S\approx 2(3.14){3}^{2}+2(3.14)(3)5\)
    Step 5. Solve.\(S\approx 150.72\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The surface area is approximately 150.72 square inches.

  23. Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm.

    Vastuse avalikustamine

    1. ⓐ 351.68 cu. cm
    2. ⓑ 276.32 sq. cm

  24. Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft.

    Vastuse avalikustamine

    1. ⓐ 100.48 cu. ft
    2. ⓑ 125.6 sq. ft

  25. Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is \(4\) centimeters and the height is \(13\) centimeters. Assume the can is shaped exactly like a cylinder.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    ⓐ
    Step 2. Identify what you are looking for.the volume of the cylinder
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(V=\pi {r}^{2}h\)
    \(V\approx (3.14){4}^{2}⋅13\)
    Step 5. Solve.\(V\approx 653.12\)
    Step 6. Check: We leave it to you to check.
    Step 7. Answer the question.The volume is approximately 653.12 cubic centimeters.
    ⓑ
    Step 2. Identify what you are looking for.the surface area of the cylinder
    Step 3. Name. Choose a variable to represent it.let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(S=2\pi {r}^{2}+2\pi rh\)
    \(S\approx 2(3.14){4}^{2}+2(3.14)(4)13\)
    Step 5. Solve.\(S\approx 427.04\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The surface area is approximately 427.04 square centimeters.

  26. Find the ⓐ volume and ⓑ surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.

    Vastuse avalikustamine

    1. ⓐ 3,818.24 cu. cm
    2. ⓑ 1,356.48 sq. cm

  27. Find the ⓐ volume and ⓑ surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

    Vastuse avalikustamine

    1. ⓐ 91.5624 cu. ft
    2. ⓑ 113.6052 sq. ft

  28. Find the volume of a cone with height \(6\) inches and radius of its base \(2\) inches.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw the figure and label it
    with the given information.
    Step 2. Identify what you are looking for.the volume of the cone
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for \(\pi\))
    \(V=\frac{1}{3}\ \pi \ {r}^{2}\ h\)
    \(V\approx \frac{1}{3}\ 3.14\ {(2)}^{2}\ (6)\)
    Step 5. Solve.\(V\approx 25.12\)
    Step 6. Check: We leave it to you to check your
    calculations.
    Step 7. Answer the question.The volume is approximately 25.12 cubic inches.

  29. Find the volume of a cone with height \(7\) inches and radius \(3\) inches

    Vastuse avalikustamine

    65.94 cu. in.

  30. Find the volume of a cone with height \(9\) centimeters and radius \(5\) centimeters

    Vastuse avalikustamine

    235.5 cu. cm

  31. Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is \(8\) inches tall and \(5\) inches in diameter? Round the answer to the nearest hundredth.

    Vastuse avalikustamine
    Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
    Step 2. Identify what you are looking for.the volume of the cone
    Step 3. Name. Choose a variable to represent it.let V = volume
    Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for \(\pi\), and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)
    \(V=\frac{1}{3}\ \pi \ {r}^{2}\ h\)
    \(V\approx \frac{1}{3}\ 3.14\ {(2.5)}^{2}\ (8)\)
    Step 5. Solve.\(V\approx 52.33\)
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question.The volume of the wrap is approximately 52.33 cubic inches.

  32. How many cubic inches of candy will fit in a cone-shaped piñata that is \(18\) inches long and \(12\) inches across its base? Round the answer to the nearest hundredth.

    Vastuse avalikustamine

    678.24 cu. in.

  33. What is the volume of a cone-shaped party hat that is \(10\) inches tall and \(7\) inches across at the base? Round the answer to the nearest hundredth.

    Vastuse avalikustamine

    128.2 cu. in.

  34. length \(2\) meters, width \(1.5\) meters, height \(3\) meters

    Vastuse avalikustamine

    1. ⓐ 9 cu. m
    2. ⓑ 27 sq. m

  35. length \(5\) feet, width \(8\) feet, height \(2.5\) feet

  36. length \(3.5\) yards, width \(2.1\) yards, height \(2.4\) yards

    Vastuse avalikustamine

    1. ⓐ 17.64 cu. yd.
    2. ⓑ 41.58 sq. yd.

  37. length \(8.8\) centimeters, width \(6.5\) centimeters, height \(4.2\) centimeters

  38. Moving van A rectangular moving van has length \(16\) feet, width \(8\) feet, and height \(8\) feet. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 1,024 cu. ft
    2. ⓑ 640 sq. ft

  39. Gift box A rectangular gift box has length \(26\) inches, width \(16\) inches, and height \(4\) inches. Find its ⓐ volume and ⓑ surface area.

  40. Carton A rectangular carton has length \(21.3\) cm, width \(24.2\) cm, and height \(6.5\) cm. Find its ⓐ volume and ⓑ surface area.

    Vastuse avalikustamine

    1. ⓐ 3,350.49 cu. cm
    2. ⓑ 1,622.42 sq. cm

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\approx
approximately equal
Equal to the precision shown, not exactly.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Solve Geometry Applications: Volume and Surface Area

  1. Find volume and surface area of rectangular solids
  2. Find volume and surface area of spheres
  3. Find volume and surface area of cylinders
  4. Find volume of cones
  5. For a cone with radius

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Proovi ise.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Rohkem Arithmetic

Order of operationsFractionsPercentagesPowers and roots