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9.22: Surface Area and Volume of Cones

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    • Surface area equals the area of the circle plus the area of the outside of the cone

    Cones

    A cone is a solid with a circular base and sides that taper up towards a vertex. A cone is generated from rotating a right triangle, around one leg. A cone has a slant height.

    f-d_64f0dd36cb1db69f45c39ad2c00ae13f1079cc50620ae210364bde02+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    Surface Area

    Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit. For the surface area of a cone we need the sum of the area of the base and the area of the sides.

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

    f-d_2a3d3bd36dc8e45a69c188d011f56846043c307247cf1e8562d23e1e+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Area of the base: \(\pi r^{2}\)

    Area of the sides: \(\pi rl\)

    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.

    Volume of a Cone: \(V=\dfrac{1}{3} \pi r^{2} h\).

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    Figure \(\PageIndex{3}\)

    What if you were given a three-dimensional solid figure with a circular base and sides that taper up towards a vertex? How could you determine how much two-dimensional and three-dimensional space that figure occupies?

    Example \(\PageIndex{1}\)

    The surface area of a cone is \(36 \pi\) and the radius is 4 units. What is the slant height?

    Solution

    Plug what you know into the formula for the surface area of a cone and solve for \(l\).

    \(\begin{aligned} 36 \pi&= \pi 4^{2}+ \pi 4l \\ 36&=16+4l \qquad \text{ When each term has a } \pi \text{, they cancel out.} \\ 20&=4l \\ 5&=l\end{aligned}\)

    Example \(\PageIndex{2}\)

    The volume of a cone is \(484 \pi \text{ cm}^{3}\) and the height is 12 cm. What is the radius?

    f-d_42db0cd9bd34d190fcb5d32674faf7b396e9b35ed67f5ade28877a59+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    Plug what you know to the volume formula.

    \(\begin{aligned} 484 \pi&=\dfrac{1}{3} \pi r^{2}(12) \\ 121&=r^{2} \\ 11 \text{ cm}&=r\end{aligned}\)

    Example \(\PageIndex{3}\)

    What is the surface area of the cone?

    f-d_e44535defe66a5eeefe5f006e5ec89bc87e22754c09d4a3b5f5f9e8d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    First, we need to find the slant height. Use the Pythagorean Theorem.

    \(\begin{aligned} l^{2}&=9^{2}+21^{2} \\ &=81+441 \\ l&=\sqrt{522}\cong 22.85\end{aligned]\)

    The total surface area, then, is \(SA= \pi 9^{2}+ \pi(9)(22.85)\cong 900.54 \text{ units}^{2}\).

    Example \(\PageIndex{4}\)

    Find the volume of the cone.

    f-d_21b9e7ca872ebc0b64fcaa56f7c54594ba40d31e6f233349d33c19e7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    First, we need the height. Use the Pythagorean Theorem.

    \(\begin{aligned} 5^{2}+h^{2}&=15^{2} \\ h&=\sqrt{200}=10\sqrt{2} \\ V&=13(5^{2})(10\sqrt{2}) \pi \cong 370.24 \text{ units}^{3}\end{aligned}\)

    Example \(\PageIndex{5}\)

    Find the volume of the cone.

    f-d_9fde4657865d9740b8f4a2e90ebe63a1e412592e6efac6b8da2c6357+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    We can use the same volume formula. Find the radius.

    \(V=\dfrac{1}{3} \pi(3^{2})(6)=18 \pi \cong 56.55 \text{ units}^{3}\)

    Review

    Use the cone to fill in the blanks.

    f-d_d2e715278eb8137359d31e23c88f1183106a56bbc67bb45fb720e2a5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)
    1. v is the ___________.
    2. The height of the cone is ______.
    3. x is a __________ and it is the ___________ of the cone.
    4. w is the _____________ ____________.

    Sketch the following solid and answer the question. Your drawing should be to scale, but not one-to-one. Leave your answer in simplest radical form.

    1. Draw a right cone with a radius of 5 cm and a height of 15 cm. What is the slant height?

    Find the slant height, \(l\), of one lateral face in the cone. Round your answer to the nearest hundredth.

    1. f-d_0b03632f554d6f3d4d25aa3fba610caf860cc9c3eb87bda1d402fa42+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)

    Find the surface area and volume of the right cones. Round your answers to 2 decimal places.

    1. f-d_6fff7abfc702b4587badf6c6b33bea823224b0048c4a16d5745d5ca6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    2. f-d_a42b4f4eeb20c7561eedc2d6ae8dfd634e0fc191658f8ad4a50542bb+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    3. If the lateral surface area of a cone is \(30 \pi \text{ cm}^{2}\) and the radius is 5 cm, what is the slant height?
    4. If the surface area of a cone is \(105 \pi \text{ cm}^{2}\) and the slant height is 8 cm, what is the radius?
    5. If the volume of a cone is\( 30 \pi \text{ cm}^{3}\) and the radius is 5 cm, what is the height?
    6. If the volume of a cone is \(105 \pi \text{ cm}^{3}\) and the height is 35 cm, what is the radius?

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 11.6.

    Vocabulary

    Term Definition
    cone is a solid with a circular base and sides that taper up towards a vertex. A cone has a slant height.
    Slant Height The slant height is the height of a lateral face of a pyramid.

    Additional Resources

    Interactive Element

    Video: Surface Area of Cone - Overview

    Activities: Cones Discussion Questions

    Study Aids: Pyramids and Cones Study Guide

    Practice: Surface Area and Volume of Cones