Revision Notes on Volume and Surface area of solids
Mensuration
The part of mathematics in which we measure area, perimeter, and volume of the plane and solid figures is called mensuration.
Area
The surface covered by the border line of the figure is the area of the plain shape.
Unit of the area is square if the length unit.
Perimeter
The length of the boundary of the plane shape is called perimeter.
The unit of the perimeter is same as the length unit.
Volume
The amount of space that something contains or fill by a solid figure is called Volume.
The unit of volume is a cubic unit.
Solid Shapes
The 3-dimensional shapes which occupy some spaces are called solid shapes.
Example- Cube, Cuboid, Cylinder, Cone, Sphere etc.
Surface Area
Sum of the areas of all the faces of solid shape is called total surface area.
The unit of surface area is a square unit.
Lateral or Curved Surface Area
If we leave the top and bottom faces of the solid shape then the area of the rest of the figure is the lateral surface of the shape.
The unit of lateral surface area is a square unit.
Surface Area and Volume of Cube, Cuboid and Cylinder
Example 1
There is a shoe box whose length, breadth and height is 9 cm, 3 cm and 4 cm respectively. Find the surface area and volume of the shoe box.
Solution:
Given,
length = 9 cm
Breadth = 3 cm
Height = 4 cm
Area of cuboid = 2(lb + bh + lh)
= 2(9 × 3 + 3×4 + 9 × 4)
= 2(27 + 12 + 36)
= 2(75)
= 150 cm2
Volume of cuboid = lbh
= 9 × 3 × 4
= 108 cm3
Example 2
If there is a cold drink can whose height is 7 cm and the radius of its round top is 3 cm then wh
at will be the lateral surface area and volume of that cylinder? (π = 3.14)
Solution:
Given,
radius = 3 cm
Height = 7 cm
Lateral surface area of cylinder = 2πrh
= 2 × 3.14 × 3 × 7
= 131.88 cm2
Volume of cylinder = πr2h
= 3.14 × 3 × 3 × 7
= 197.82 cm3
Example 3
If there is a box of cube shape with the length of 4 cm then what will be the capacity of this box. Also, find the surface area of the box if it is open from the top.
Solution:
Given, side = 4 cm
Capacity or volume of the box = s3
= 43 = 64 cm3
The total surface area of the box = 6s2
But, if the box is open from the top then the surface area will be total surface area minus the area of one face of the cube.
Surface Area = Total Surface Area – Area of one face
= 6s2 – s2
= 5s2 = 5 × 42
= 80 cm2
Volume and Capacity
Volume and capacity are one and the same thing.
Volume is the amount of space occupied by a shape.
Capacity is the quantity that a container can hold.
Capacity can be measured in form of liters.
We can see the relation between liter and cm3 as,
1 L = 1000 mL
1 mL = 1 cm3,
1 L = 1000 cm3.
Thus, 1 m3 = 1000000 cm3 = 1000 L.
Example
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
Solution
Given,
Length = 20 cm
Breadth = 8 cm
Height = 30 cm
Total surface area of the cereal box = 2(lb + bh + lh)
= 2(20 × 8 + 8 × 30 + 20 × 30)
= 2(160 + 240 + 600)
= 2(1000) = 2000 cm2.
Surface Area and Volume of a Right Circular Cylinder
If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.
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Curved surface area of a Right circular cylinder = 2πrh |
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Total surface area of a Right circular cylinder = 2πr2 + 2πrh = 2πr(r + h) |
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The volume of a Right circular cylinder = πr2 |
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Where, r = radius, h = height |
Surface Area and Volume of a Hollow Right Circular Cylinder
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
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Curved surface area of a Right circular cylinder = 2πh (R + r) |
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Total surface area of a Right circular cylinder = 2πh (R + r) + 2π (R2 – r2) |
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R = outer radius, r = inner radius |
Example
Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)
Solution
Given, h = 22 cm
R = 7 cm
r = 6 cm (thickness of the wall is 1 cm)
Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2(π) (22) (7+6) + 2(π)(72 – 62)
= 572 π + 26 π = 598 π
= 1878.67 cm2
Surface Area and Volume of a Right Circular Cone
If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.
Surface Area and Volume of a Sphere |
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
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Curved or Lateral surface area of a Sphere = 4πr2 |
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Total surface area of a Sphere = 4πr2 |
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Volume of a Sphere = (4/3) πr3 |
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Where, R = radius |
Surface Area and Volume of a Hemisphere
If we cut the sphere in two parts then is said to be a hemisphere.
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Curved or Lateral surface area of a Sphere = 2πr2 |
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Total surface area of a Sphere = 3πr2 |
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Volume of a Sphere = (2/3) πr3 |
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Where, r = radius |
Example
If we have a metal piece of cone shape with volume 523.33 cm3 and we mould it in a sphere then what will be the surface area of that sphere?
Solution
Given, volume of cone = 523.33 cm3
Volume of cone = Volume of Sphere
Volume of sphere = 100 π cm3
125 = r3
r = 5
Surface area of a sphere = 4πr2
= 314.28 cm2.