Have you ever eaten ice cream? One of the shapes is cone. Besides that, other cone things are traffic cone, cone snail, party hat, megaphone, etc..
Table of Contents
Definition of Cone
Cone is a three-dimensional shape that has a flat base, a curved surface, and a point in the top (apex). The flat base is circular shape so that it has a radius. The height is distance between the point in the top with the base.
Properties of Cone
There are properties of cone:
- It has one flat base
- It has one curved surface
- There is a point in the top that is called apex
- It has no edge
Types of cone
There are two type of cone.
1. Right Cone
Right cone is a cone with the height (axis) is perpendicular with the center of the base. The base is circular. It is called “right” because of the angle between height and the base is right angle. Most of the cone in geometry is right cone.
2. Oblique Cone
Oblique cone is a cone that the height (axis) is not perpendicular with the center of the base. The vertex is not above of center of the base exactly. The height is perpendicular to another side of the base. It is also called as slanted cone.
Surface Area of Cone
Surface area of a cone means total area of curved and flat (base) surface.
Surface area = curved surface + base
= prs + pr2
= pr(s+r)
Note:
s means slant height. It is distance between vertex and the outer line of base.
Volume of Cone
Volume of cone means total capacity in the cone. It can be determined by multiplying one-third of base area and the height.
Volume = (1/3)pr2h
Examples
1. Determine the surface area and volume of a cone if it has r = 10 cm and the height = 15cm.
r = 10
h = 15
- s = √(r2+h2)
= √(102+152)
= √(100+225)
= √(325)
= 5√13
- Surface area = pr(s+r)
= p(10)(5√13+10)
= (50√13+100)p cm2
- Volume = (1/3)pr2h
= (1/3)p(10)2(15)
= (1/3)p (1500)
= 500p cm3
2. There is a cone that has surface area 125p cm2 and the r = 5 cm. Determine the volume.
Surface area = 125p
r = 5
- Determine s
Surface area = pr(s+r)
125p = pr(s+r)
125 = 5(s+5)
25 = s+5
s = 20
- Determine height (h)
s = 20
s = √(r2+h2)
20 = √(52+h2)
400 = 25+ h2
375 = h2
h = 5√15
Then
- Volume = (1/3)pr2h
= (1/3)p(5)2(5√15)
= (1/3)p.25. (5√15)
= (1/3).125√15p
= (125/3)(√15)p cm3
3. There is a party hat that diameter to the head is 24 cm. The owner wants to make it not too tall so that the height is just 16 cm. If it is made from a special paper that the price is $2 in each m2 and there will be 25 participants, determine the total price.
diameter = 24 → r = 12 cm
height = 16 cm
before calculating the total price, first step is determining total paper that is needed.
Paper is only in curved surface, so
Total paper = 25 x prs
= 25 x (3.14) x 12 x √(122+162)
= 25 x (3.14) x 12 x 20
= 18,840 cm2
= 188.4 m2
Price of paper = $3 in each 1 m2
So
Total price = 188.4 m2 x $2 = $.376.8