Start

Wha

t are

Co

nes?

• A cone is an n-dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex.

More

If one end of a line is

rotated about a

second fixed line while

keeping the line's

other end fixed, then a

cone is formed.

The point about which the

line is rotated is called the

vertex and the base of the

cone is a circle.

A cone is said to be right when the vertex is directly above the centre of the base.When the vertex of a cone is not vertically above the center of the base, it is called an oblique cone.

NetsThe net of a cone consists of the following two parts:•a circle that gives the base; and •a sector that gives the curved surface

Examples of Cones

Formulas1. VOLUME

r : radiush : height (the perpendicular distance from the base to the apex).

Example: Calculate the volume of a cone if the height is 12 cm and the radius is 7 cm. Solution: Volume

2. SURFACE AREA

Surface area of cone = Area of sector + area of circle

Solution: Area = πr(r + s) =

= 1,257.14 cm2

Example: A cone has a circular base of radius 10 cm and a slant height of 30 cm. Calculate the surface area.

Word Problems1. The radius of a right cone is 3 cm and its surface area is 24∏ cm2. Find the height and volume of this cone.

Solution:Start with the equation for surface area since the radius is given as 3 cm and the surface area as 24∏.S = 24pS = ∏ r2 + ∏ rs S = ∏ 32 + ∏ 3s S = 3 ∏(3 + s)

Solving this equation for s we get 24 ∏= 3 ∏(3 + s)8 = 3 + ss = 5 cm Next

To calculate the volume we need to find the values of h. Since h, r, and s form a right triangle, we can use the Pythagorean Theorem to calculate the value of h.

h2 + r2 = s2

h2 + 32 = 52

h2 = 25 - 9h2 = 16h = 4 cm

Now use r = 3 cm and h = 4 cm in the formula for volume:

Answer: Height = 4 cm Volume = 12∏ cm3

2. The radius of a cone is 5 inches and the volume is 100∏ cubic inches. Find the slant height and surface area of this cone.

Solution:Using the formula for the volume of a cone and the fact that r = 5 in:

Solve the equation for h:

h = 12 inNext

Use r = 5 and h = 12 in the Pythagorean Theorem to find the value for the slant height s.

h2 + r2 = s2

122 + 52= s2

144 + 25 = s2

s2 = 169s = 13 inches

Use r = 5 and s = 13 in the formula for surface area:S = ∏ r 2 + ∏ rs S = ∏ 52 + ∏ (5)(13)S = ∏ (25 + 65)S = 90 ∏ square inches

Answer: Slant height = 13 inches Surface Area = 90 ∏ square inches

The End

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