A cone is a 3D geometric shape that tapers from a flat circular base to a single point called the apex. The formulas for the volume and surface area of a cone involve the radius of the base and the height or slant height. Word problems apply these formulas to calculate missing values like height, volume, or surface area when other values like radius or volume are given.

1. Start

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

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

12. The End
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