Namrata Das Exams Prep Master
Exams Prep Master
Surface area and volume are performed for any three-dimensional geometrical shape. The surface area of any given object means the area or region occupied by the surface of the object. The surface area can be classified broadly into three areas which are Curved Surface Area (CSA), Lateral Surface Area (LSA) and Total surface area (TSA). Let us have a closer look at the topic and discuss some important related to it.
Check also: Surface Areas and Volumes Revision Notes
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Key Terms: Shapes, cuboid, cylinder, cones, sphere, Total surface area (TSA), Lateral Surface Area (LSA), Curved Surface Area (CSA).
Surface Area and Volume
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Some of the major shapes to which the students of class 9 are introduced are cuboid, cylinder, cone and sphere. We will discuss each one of them one by one.
Cuboid
The shape of a cuboid has rectangular faces which are six in number and are placed right angled to each other. The formula of the total surface area of a cuboid is equal to the sum total of the area of all the six rectangles.
Cuboid
If we have to take out the lateral surface area of a cuboid, it is the area of all the sides except the area of the top and the bottom surfaces. LSA = 2h(l+b).
Also read: Area of Hollow Cylinder
Cube
A cube is similar to the cuboid which has all the six surfaces, that is the length, breadth and height are equal. The number of edges and vertices are 12 and 8 respectively. The total surface area of a cube is 6s2. This is because here length= breadth= height (let us assume it s cm)
Total surface area = 2*3 s2
Cube
Lateral surface area of the cube is the area of the four sides. Therefore it is 4s2
Right Circular Cylinder
Right circular cylinder is a surface in which two curved surfaces are present in order to connect the two parallel circular bases. The two bases will be exactly over one another. The formula of the total surface area of a right circular cylinder is as follows:
TSA = 2π × r × h + 2 × πr2
⇒ TSA = 2πr(h + r)
Here r is the radius of the surface and h is the height of the cylinder. If we have to find the curved surface area then it will be 2 rh
Right Circular Cylinder
Read more: Surface Area of a Cylinder Formula
Right Circular Cone
If we first take out the relation between the slant height (l) and height of the cone then by using the pythagoras theorem it will be l2=h2+r2. The curved surface area of the right circular cone will be the total sum of the small triangles formed. The final formula for the CSA will be rl
The total surface area of the right circular cone will be the total of CSA and area of the base. As mentioned above theCSA will be rl and the area of the base will be r²
Therefore the TSA of the right circular cone will be r(l+r).
Right Circular Cone
Sphere
A three dimensional closed solid figure is a sphere. In a sphere the distance of the surface from a common point which is the center is equal. In a sphere the total surface area and the curved surface area are equal which is 4r2
The video below explains this:
Surface Area and Volume Detailed Video Explanation:
Volume of the Surfaces
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Now that we have taken a look at the total surface area and the curved or lateral surface area of the three dimensional structure. Now let us look at the volume of these surfaces.
- The formula for the cuboid is lbh.
- The formula for volume of a cube is base area multiplied by height. As all the surfaces of the cube are equal. The formula generated is a³
- The formula for volume of a right circular cylinder is the base area multiplied by height. The formula formed will be r2h
- The formula for volume of a right circular cone is ar2h.
Read more: Volume and Capacity
Things to Remember
- Cube is similar to the cuboid with all the sides equal.
- The TSA and CSA of a sphere is equal that is 4r2
- TSA of the cube is 6a2 and that of the LSA is 4a2
- Total surface area is the area that include the base(s) and the curved part, and is the total of the area covered by the surface of the object.
- Curved surface area, also referred to as lateral surface area for shapes such as a cylinder, is the area of only the curved part of the shape excluding its base(s).
Sample Questions
Ques: If the height and the radius of the right circular cone is 18 and 3 respectively. Find the volume of the cone. (1 mark)
Ans: Volume of the cone = ar2h.
a* 3.14 * 9*18
169.56 cm³
Ques: The curved surface area of a cone is 12320 sq. cm, if the radius of its base is 56 cm, find its height. (2 marks)
Ans: Here, radius of base of a cone (r) = 56 cm
And, curved surface area = 12320 cm2
πrl = 12320
l = 12320/ πr
= 12320×7/22×56
= 70 cm
Again, we have
r2 + h2 = l2
h2 = l2 – r2 = 702 – 562
= 4900 – 3136 = 1764
h = √1764 = 42 cm
Hence, the height of the cone is 42 cm.
Ques: Find the capacity in liters of a conical vessel having height 8 cm and slant height 10 cm. (2 marks)
Ans: Height of conical vessel (h) = 8 cm
Slant height of conical vessel (l) = 10 cm
∴ r2 + h2 = l2
⇒ r2 + 82 = 102
⇒ r2 = 100 – 64 = 36
⇒ r = 6 cm
Now, volume of conical vessel = 13πr2h = 13 × 227 × 6 × 8 = 301.71 cm3 = 0.30171 litre
Ques: Two cubes of edge 6 cm are joined to form a cuboid. Find the total surface area of the cuboid. (2 marks)
Ans. When two cubes are joined end to end, then
Length of the cuboid = 6 + 6 = 12 cm
Breadth of the cuboid = 6 cm
Height of the cuboid = 6 cm
Total surface area of the cuboid = 2 (lb + bh + hi)
= 2(12 x6 + 6×6 + 6×12)
= 2(72 + 36 + 72) = 2(180) = 360 cm2
Ques: The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and ceiling at the rate of Rs 7.50 per m2. (3 marks)
Ans. Length (l) of room = 5m
Breadth (b) of room = 4m
Height (h) of room = 3m
It can be observed that four walls and the ceiling of the room are to be white washed.
Total area to be white washed = Area of walls + Area of ceiling of room
= 2lh+2bh+lb
= [2×5×3+2×4×3+5×4]
= (30+24+20)
= 74
Area = 74 m2
Also,
Cost of white wash per m2 area = Rs.7.50 (Given)
Cost of white washing 74 m2 area = Rs. (74×7.50)
= Rs. 555
Ques: The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder. (Assume π =22/7 ) (2 marks)
Ans. Height of cylinder, h = 14cm
Let the diameter of the cylinder be d
Curved surface area of cylinder = 88 cm2
We know that, formula to find the Curved surface area of a cylinder is 2πrh.
So 2πrh =88 cm2 (r is the radius of the base of the cylinder)
2×(22/7)×r×14 = 88 cm2
2r = 2 cm
d =2 cm
Therefore, the diameter of the base of the cylinder is 2 cm
Ques: Praveen wanted to make a temporary shelter for her car, by making a box – like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5m, with base dimensions 4m×3m? (3 marks)
Ans. Let l, b and h be the length, breadth and height of the shelter.
Given:
l = 4m
b = 3m
h = 2.5m
Tarpaulin will be required for the top and four wall sides of the shelter.
Using formula, Area of tarpaulin required = 2(lh+bh)+lb
On putting the values of l, b and h, we get
= [2(4×2.5+3×2.5)+4×3] m2
= [2(10+7.5)+12]m2
= 47m2
Therefore, 47 m2 tarpaulin will be required.
Ques: Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area (Assume π=22/7) (2 marks)
Ans. Radius of the base of cone = diameter/ 2 = (10.5/2)cm = 5.25cm
Slant height of cone, say l = 10 cm
CSA of cone is = πrl
= (22/7)×5.25×10 = 165 cm2
Therefore, the curved surface area of the cone is 165 cm2.
Ques: Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. (Assume π = 22/7) (2 marks)
Ans. Radius of cone, r = 24/2 m = 12m
Slant height, l = 21 m
Formula: Total Surface area of the cone = πr(l+r)
Total Surface area of the cone = (22/7)×12×(21+12) m2
= 1244.57m2
Ques: The slant height and base diameter of conical tomb are 25m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs. 210 per 100 m2. (Assume π = 22/7) (2 marks)
Ans. Slant height of conical tomb, l = 25m
Base radius, r = diameter/2 = 14/2 m = 7m
CSA of conical tomb = πrl
= (22/7)×7×25 = 550
CSA of conical tomb= 550m2
Cost of white-washing 550 m2 area, which is Rs (210×550)/100
= Rs. 1155
Therefore, the cost will be Rs. 1155 while white-washing tomb.
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