What if you were given a threedimensional solid figure with a circular base and sides that taper up towards a vertex? How could you determine how much twodimensional and threedimensional 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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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 twodimensional 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:
Area of the base:
Area of the sides:
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:
Example A
What is the surface area of the cone?
First, we need to find the slant height. Use the Pythagorean Theorem.
The total surface area, then, is
Example B
Find the volume of the cone.
First, we need the height. Use the Pythagorean Theorem.
Example C
Find the volume of the cone.
We can use the same volume formula. Find the radius.
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Guided Practice
1. The surface area of a cone is
2. The volume of a cone is
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
2. Plug what you know to the volume formula.
3. First we need to find the radius. Use the Pythagorean Theorem.
Now use the formulas to find surface area and volume. Use the
Now for volume:
Explore More
Use the cone to fill in the blanks.

v is the ___________.  The height of the cone is ______.

x is a __________ and it is the ___________ of the cone. 
w is the _____________ ____________.
Sketch the following solid and answer the question. Your drawing should be to scale, but not onetoone. Leave your answer in simplest radical form.
 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,
Find the surface area and volume of the right cones. Round your answers to 2 decimal places.
 If the lateral surface area of a cone is
30π cm2 and the radius is 5 cm, what is the slant height?  If the surface area of a cone is
105π cm2 and the slant height is 8 cm, what is the radius?  If the volume of a cone is
30π cm3 and the radius is 5 cm, what is the height?  If the volume of a cone is
105π cm3 and the height is 35 cm, what is the radius?