Table of contents  
Introduction  
Surface Area of Cones  
Surface Area of Sphere  
Surface Area of HemiSphere  
Volume of a Cone  
Volume of a Sphere  
Volume of a HemiSphere 
Ever wondered how we measure the wrapping needed for a gift or the space inside a box? That's where surface area and volume come into play. Surface area is like the total paper to cover a 3D object, while volume is the space it occupies.
Wrapping Paper
It's how we quantify the 'outside' and 'inside' of shapes. So, how do we calculate these for different objects? Let's explore the answers by unraveling the mystery of surface area and delving into the 'roominess' of volume!
Cones are threedimensional geometric shapes characterized by a circular base that tapers to a single point called the apex or vertex. Picture an ice cream cone – it's a classic example.
The curved surface of a cone forms a lateral surface, and the distance from the apex to the base is the height. Cones appear in various contexts, from ice cream treats to traffic cones, and their properties make them intriguing in geometry and realworld applications. Calculating their surface area and volume involves understanding their unique geometric features.
To understand the formation of a right circular cone, perform the following activity:
The curved surface area (CSA) of a cone is calculated using the formula:
Curved Surface Area=πrl
Here, r is the base radius, l is the slant height, and π is a constant.
Note: The relationship l^{2}=r^{2}+h^{2} can be derived from the cone's crosssection, where h is the cone's height.
If the base is closed, a circular piece of paper with an area of πr^{2} is required, adding to the total surface area.
Example: Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
Solution: The slant height of cone (l) = 60 cm
The radius of the base of the cone (r) = 21 cm
Now,
Curved surface area of the right circular cone = πrl = 22/7 x 21 x 60 = 3960 cm^{2}
Therefore the curved surface area of the right circular cone is 3960 cm^{2}
Total Surface Area = CSA+πr^{2}=πrl+πr^{2}=πr(l+r)
Examples: Find the curved surface area of a right circular cone with a slant height of 10 cm and a base radius of 7 cm.
Solution: CSA=π×7×10=220cm^{2}
Spheres are perfectly round threedimensional shapes, resembling balls or globes. Every point on the surface of a sphere is equidistant from its center. Think of a basketball or the Earth, which is nearly a sphere. Spheres have no edges or vertices.
The surface area of a sphere is calculated as:
Surface Area=4πr^{2}
Here, r is the sphere's radius.
Example: Find the surface area of a tennis ball of radius 14 cm.
Clearly, the tennis ball is in the form of a sphere.
Here, the radius of the sphere is 14 cm.
We know that,
Surface Area of the sphere =4πr^{2}.
Therefore, the Surface Area of a tennis ball =4πr^{2}.
= 4 × 22/7 × (14 cm)^{2}.
= 4 × 22/7 × 14cm × 14 cm.
= 4 × 22 × 2cm × 14 cm.
= 2464 cm^{2}.
Hence, the Surface area of the tennis ball is 2464 cm^{2}.
Note: A sphere doesn't have separate curved and total surface areas because its entire surface is curved. In other words, there are no flat or planar sections on a sphere. The term "surface area" for a sphere typically refers to the total surface area, which includes both the curved surface area and the area of the sphere's base (which is also curved).
A hemisphere, being half of a sphere, has a surface area equal to 3/2 times the surface area of a sphere.
Surface Area of Hemisphere= 3/2×4πr^{2}=3πr^{2}
Therefore,
Curved Surface Area of a Hemisphere = 2πr^{2}
Where r is the radius of the sphere of which the hemisphere is a part.
Now taking the two faces of a hemisphere, its total surface area is 2πr^{2} + πr^{2}.
So, the Total surface area of a hemisphere = 3 πr^{2}.
Example: Find the curved surface area and the total surface area of the half slice of watermelon of radius 7 cm.
Clearly, half a slice of watermelon is in the form of a hemisphere.
Here, the radius of the hemisphere is 7 cm.
We know that,
The curved surface area of the hemisphere =2πr^{2}.
Therefore, the Curved surface Area of half a slice of watermelon =2πr^{2}.
= 2 × 22/7 × (7 cm)^{2}.
= 2 × 22/7 × 7cm × 7 cm.
= 2 × 22 × 1cm × 7 cm.
= 308 cm^{2}.
Hence, Curved Surface area of half a slice of watermelon is 308 cm^{2}.
Also, we know that
The total surface area of hemisphere = 3πr^{2}.
So, the total surface area of half a slice of watermelon = 3πr^{2}.
= 3 ×227× (7 cm)^{2}.
= 3 × 22/7 × 7cm × 7 cm
= 3 × 22 × 1cm × 7 cm.
= 462 cm^{2}.
Hence, the total surface area of half slice of watermelon = 462 cm^{2}.
Volume refers to the amount of space occupied by a threedimensional object. It is a measure of how much "stuff" or substance an object can hold. The concept of volume is often applied to various geometric shapes, such as cubes, spheres, cylinders, and cones.
The volume (V) of a cone is calculated using the formula:
V= 1/3πr^{2}h
Here, r is the base radius, and h is the height of the cone.
Given the height (h) and slant height (l) of a cone as 21 cm and 28 cm respectively, find the volume.
Solution: Slant height (l) = 28cm; Height of cone (h) = 21cm ; Let radius of cone = r cm
we know that,
The volume (V) of a sphere is given by the formula:
V= 4/3 πr^{3}
Here, r is the radius of the sphere.
Find the volume of a sphere with a radius of 11.2 cm.
Solution:
The volume of a hemisphere, being half of a sphere .
The volume (V) of a hemisphere (half of a sphere) can be calculated using the formula:
V= 2/3πr^{3}
where:
Example: A particular dome of a building is in the shape of a hemisphere. From inside, it was whitewashed at the price of Rs. 4989.60. If the cost of whitewashing is Rs 20 per square metre, find the
(i) The required inside surface area of the dome
(ii) The volume of the air inside the dome
Solution: (i) The cost of whitewashing the dome from inside = Rs 4989.60
Cost of whitewashing 1m² area = Rs 20
The curved surface area of the inner side of the dome = 498.96/2 m² = 249.48 m²
(ii) Let the required inner radius of the hemispherical dome be r.
Curved Surface Area of the inner side of dome = 249.48 m² (from (i))
Formula to find CSA of a hemisphere = 2πr²
2πr = 249.48
2×(22/7)×r² = 249.48
r² = (249.48×7)/(2×22)
r² = 39.69
r = 6.3
So, the radius is 6.3 m
The volume of air inside the given dome = Volume of hemispherical dome
Using the formula, the volume of the hemisphere = 2/3 πr³
= (2/3)×(22/7)×6.3×6.3×6.3
= 523.908
= 523.9(approx.)
The required volume of air inside the dome is 523.9 m³.
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NCERT Solutions: Surface Areas & Volumes Doc  20 pages 
Short Answer Type Questions: Surface Areas & Volumes Doc  4 pages 
NCERT Exemplar Solutions: Surface Area & Volume Doc  18 pages 
1. What is the formula for finding the surface area of a cone? 
2. How do you calculate the volume of a sphere? 
3. What is the surface area of a hemisphere? 
4. How can you find the volume of a cone? 
5. What is the surface area of a sphere? 
NCERT Solutions: Surface Areas & Volumes Doc  20 pages 
Short Answer Type Questions: Surface Areas & Volumes Doc  4 pages 
NCERT Exemplar Solutions: Surface Area & Volume Doc  18 pages 

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