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9.18: Surface Area and Volume of Cylinders

  • Page ID
    6288
  • Surface area and volume of solids with congruent circular bases in parallel planes.

    Cylinders

    A cylinder is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed.

    A cylinder has a radius and a height.

    f-d_a2c7a0bf6a3fa75986f82c601dbcd425a3530ab78b5e6652c4ed1ae3+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    A cylinder can also be oblique (slanted) like the one below.

    f-d_1ebbd11b220c15d868602f79457baa066a620854a9723d2b4fce84fe+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Surface Area

    Surface area is the sum of the area of the faces of a solid. The basic unit of area is the square unit.

    Surface Area of a Right Cylinder: \(SA=2 \pi r^{2}+2 \pi r h\).

    f-d_8f3e273b79a92aee8275bbcf16eda9c65de8fc91402dbab9f3443aac+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    \(\underbrace{2 \pi r^{2}}_\text{area of both circles}+ \underbrace{2 \pi r h}_\text{length of rectangle}\)

    Volume

    To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit. For cylinders, volume is the area of the circular base times the height.

    Volume of a Cylinder: \(V= \pi r^{2}h\)

    f-d_9a2b061d4c74592e09a621a0d0458d2f24fd54720aaa6955acaae1e2+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    If an oblique cylinder has the same base area and height as another cylinder, then it will have the same volume. This is due to Cavalieri’s Principle, which states that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

    What if you were given a solid three-dimensional figure with congruent enclosed circular bases that are in parallel planes? How could you determine how much two-dimensional and three-dimensional space that figure occupies?

    Example \(\PageIndex{1}\)

    If the volume of a cylinder is \(484 \pi in^{3}\) and the height is 4 in, what is the radius?

    Solution

    Solve for \(r\).

    \(\begin{aligned} 484 \pi&= \pi r^{2}(4) \\ 121&=r^{2} \\ 11\text{ in}&=r\end{aligned}\)

    Example \(\PageIndex{2}\)

    The circumference of the base of a cylinder is \(80 \pi\) cm and the height is 36 cm. Find the total surface area.

    Solution

    We need to solve for the radius, using the circumference.

    \(\begin{aligned} 2 \pi r&=80 \pi \\ r&=40\end{aligned}\)

    Now, we can find the surface area.

    \(\begin{aligned} SA&=2 \pi(40)^{2}+(80 \pi)(36) \\ &=3200 \pi+2880 \pi \\ &=6080 \pi \text{ units}^{2}\end{aligned}\)

    Example \(\PageIndex{3}\)

    Find the surface area of the cylinder.

    f-d_3def65531a18514d5d965f09a23eff3d816e036b9ee8f20e873c3b4b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    \(r=4\) and \(h=12\).

    \(\begin{aligned} SA&=2 \pi(4)^{2}+2 \pi(4)(12) \\ &=32 \pi+96 \pi \\ &=128 \pi \text{ units}^{2}\end{aligned} \)

    Example \(\PageIndex{4}\)

    The circumference of the base of a cylinder is \(16 \pi\) and the height is 21. Find the surface area of the cylinder.

    Solution

    We need to solve for the radius, using the circumference.

    \(\begin{aligned} 2 \pi r&=16 \pi \\ r&=8\end{aligned}\)

    Now, we can find the surface area.

    \(\begin{aligned} SA&=2 \pi(8)2+(16 \pi)(21) \\ &=128 \pi+336 \pi \\ &=464 \pi \text{ units}^{2}\end{aligned}\)

    Example \(\PageIndex{5}\)

    Find the volume of the cylinder.

    f-d_65a6be18e8dcb8cc04805e2b77c8d1c5d1326f9c1ec73608cadc8175+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    If the diameter is 16, then the radius is 8.

    \(\begin{aligned} V&= \pi 8^{2}(21) \\ &=1344 \pi \text{ units}^{3}\end{aligned}\)

    Review

    1. Two cylinders have the same surface area. Do they have the same volume? How do you know?
    2. A cylinder has \(r=h\) and the radius is 4 cm. What is the volume?
    3. A cylinder has a volume of \(486 \pi \text{ ft}^{3}\). If the height is 6 ft., what is the diameter?
    1. A right cylinder has a 7 cm radius and a height of 18 cm. Find the volume.

    Find the volume of the following solids. Round your answers to the nearest hundredth.

    1. f-d_401c01cf9d8bf691ea7b647480a501fc98b5cd4e2d39d0f15d478e18+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. f-d_d02df2bbb9aee8bbedcda5dd3bc99a9903a53aaa80c3148a9faf4427+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)

    Find the value of x, given the volume.

    1. \(V=6144 \pi \text{ units}^{3}\)
      f-d_cb8e18cfbe72b3035018290a3aa222dc2c01471d24274037fd20a8b6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    2. The area of the base of a cylinder is \(49 \pi \text{ in}^{2}\) and the height is 6 in. Find the volume.
    3. The circumference of the base of a cylinder is \(34 \pi \text{ cm}\) and the height is 20 cm. Find the total surface area.
    4. The lateral surface area of a cylinder is \(30 \pi \text{ m}^{2}\) and the height is 5 m. What is the radius?

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 11.4.

    Vocabulary

    Term Definition
    cylinder A solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed. A cylinder has a radius and a height and can also be oblique (slanted).
    Surface Area Surface area is the total area of all of the surfaces of a three-dimensional object.
    Volume Volume is the amount of space inside the bounds of a three-dimensional object.
    Cavalieri's Principle States that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.
    Oblique Cylinder An oblique cylinder is a cylinder with bases that are not directly above one another.

    Additional Resources

    Interactive Element

    Video: Cylinders Principles - Basic

    Activities: Cylinders Discussion Questions

    Study Aids: Prisms and Cylinders Study Guide

    Practice: Surface Area and Volume of Cylinders

    Real World: Drill Teams