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Surface Area and Volume of Cones

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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? After completing this Concept, you'll be able to find the surface area and volume of a cone.

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Cones CK-12

Guidance

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 .

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 .

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=\frac{1}{3} \pi r^2 h .

Example A

What is the surface area of the cone?

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

l^2 &= 9^2+21^2\\&= 81+441\\l &= \sqrt{522} \approx 22.85

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

Example B

Find the volume of the cone.

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

5^2+h^2 &=15^2\\h &= \sqrt{200}=10\sqrt{2}\\V &= \frac{1}{3}(5^2)\left(10\sqrt{2}\right) \pi \approx 370.24 \ units^3

Example C

Find the volume of the cone.

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

V=\frac{1}{3} \pi (3^2)(6)=18 \pi \approx 56.55 \ units^3

Cones CK-12

Guided Practice

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

2. The volume of a cone is 484 \pi \ cm^3 and the height is 12 cm. What is the radius?

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

Answers:

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

36 \pi &= \pi 4^2+\pi 4l\\36 &= 16+4l \qquad When \ each \ term \ has \ a \ \pi, \ they \ cancel \ out.\\20 &= 4l\\5 &= l

2. Plug what you know to the volume formula.

484 \pi &= \frac{1}{3} \pi r^2 (12)\\121 &= r^2\\11 \ cm &= r

3. First we need to find the radius. Use the Pythagorean Theorem.

r^2 +40^2 &=41^2 \\ r^2 &= 81 \\ r&=9

Now use the formulas to find surface area and volume. Use the \pi button on your calculator to help approximate your answer at the end.

SA&= \pi r^2 + \pi r l\\ SA &= 81 \pi + 369 \pi \\ SA &= 450 \pi \\ SA &=1413.72

Now for volume:

V &= \frac{1}{3} \pi r^2 h \\ V &= \frac{1}{3} \pi (9^2)(40)\\ V&= 1080 \pi \\ V&=3392.92

Practice

Use the cone to fill in the blanks.

  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.

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

  1. If the lateral surface area of a cone is 30 \pi \ cm^2 and the radius is 5 cm, what is the slant height?
  2. If the surface area of a cone is 105 \pi \ cm^2 and the slant height is 8 cm, what is the radius?
  3. If the volume of a cone is 30 \pi \ cm^3 and the radius is 5 cm, what is the height?
  4. If the volume of a cone is 105 \pi \ cm^3 and the height is 35 cm, what is the radius?

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