In this chapter we will introduce the concept of cone and show the formula to calculate volume and surface area of cone.
What is cone ?
It’s a three dimensional figure formed when infinite line segment from circular base intersect at common point called vertex.
The shape of the cone is given below;
The above cone has base circle of radius r and all the points from the circle join the point A through straight line.
Properties of cone
(a) cone has one circular base
(b) It has one vertex
(c) The vertex A is just above the center of base circle.
(d) Ice-cream cone is one common example of cone in practical life.
Parts of cone
The shape of cone has three important elements;
(a) Radius of base (r)
It is the radius of circular base of the cone
(b) Height of cone (h)
The distance from center of base to apex point (A) is the height of cone
(c) Slant height (l)
The distance from apex to any point on circumference of base circle is called slant height.
I hope you understood the basic concept, let us now understand the formulas.
Volume of cone
The total space captured by the figure is called volume of cone.
Volume = \mathtt{\frac{1}{3} \pi r^{2} h}
We get the volume in cubic units ( \mathtt{cm^{3} ,\ m^{3} ,\ etc.} )
Curved surface area of cone
The area of the slant surface is called curved surface area of cone. It is also called lateral surface area.
Curved surface area = 𝜋. r. l
Curved surface area is expressed in square units ( \mathtt{cm^{2} ,\ m^{2} ,\ etc} )
Total surface area
Cone is made of lateral surface and circular base. So, total surface area is given as;
Total surface area = circular base + Lateral surface
Total surface area = 𝜋.r.(l + r)
The value of total surface is expressed in square units ( \mathtt{cm^{2} ,\ m^{2} ,\ etc} )
Questions on volume and surface area of cone
Question 01
Find the volume of cone if radius of base is 14 cm and height is 30 cm.
Solution
Radius (r) = 14 cm
Height (h) = 30 cm
Volume of cube = \mathtt{\frac{1}{3} \pi r^{2} h}
Putting the values in formula;
\mathtt{Volume\ =\ \frac{1}{3} \times \frac{22}{7} \times 14\times 14\times 30\ }\\\ \\ \mathtt{Volume\ =\ 6160\ cm^{3}}
Hence, the volume of cone is 6160 cu. cm
Question 02
Consider the cone of radius 6 cm and height 8 cm. Find the curved surface area of cone.
Solution
In the given question;
r = 6 cm
h = 8 cm
Consider triangle AOB. Using Pythagoras theorem;
\mathtt{l^{2} =h^{2} +\ r^{2}}\\\ \\ \mathtt{l^{2} =\ 8^{2} +6^{2}}\\\ \\ \mathtt{l^{2} =\ 64\ +\ 36\ =\ 100}\\\ \\ \mathtt{l\ =\ 10\ cm}
So the slant height is 10 cm.
Now calculating the curved surface area.
Curved surface = 𝜋.r. l
Curved surface = (22/7) x 6 x 10 = 188.6 sq cm
Hence, the curved surface of cone is 188.6 sq. cm.
Question 03
The curved surface area of cone with radius 5 cm is given as 220 sq. cm. Calculate the slant height of the given cone.
Solution
Radius (r) = 5 cm
Curved surface area = 220 sq. cm
Curved surface area = 𝜋.r. l
220 = (22/7) . (5) . l
\mathtt{l=\ \frac{220\times 7}{22\times 5} =14\ cm}
Hence, the slant height of given cone is 14 cm .
Question 04
Find the area of base of the cone if height and slant height is 21 and 28 cm respectively.
Solution
Height (h) = 21 cm
Slant height = 28 cm
Taking triangle AOB and applying Pythagoras theorem.
\mathtt{l^{2} =h^{2} +\ r^{2}}\\\ \\ \mathtt{28^{2} =\ 21^{2} +r^{2}}\\\ \\ \mathtt{r^{2} =784-441}\\\ \\ \mathtt{r^{2} \ =\ 343\ cm}
We know that base of cone is circular in nature.
Area = \mathtt{\pi r^{2}}
Putting the values;
\mathtt{Area\ =\frac{22}{7} \times 343\ =\ 1078\ cm^{2}}
Hence, area of circular base is 1078 sq. cm