2. Whatare
Cones? • 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
3.
4. 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.
5. Nets
The 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
6. Formulas
1. VOLUME
r : radius
h : 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
7. 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.
8. Word Problems
1. 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 = 24p
S = ∏ r2 + ∏ rs
S = ∏ 32 + ∏ 3s
S = 3 ∏(3 + s)
Solving this equation for s we get
24 ∏= 3 ∏(3 + s)
8 = 3 + s
s = 5 cm Next
9. 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 - 9
h2 = 16
h = 4 cm
Now use r = 3 cm and h = 4 cm in
the formula for volume:
Answer: Height = 4 cm
Volume = 12∏ cm3
10. 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 in
Next
11. 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 = 169
s = 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
