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The figures which we can draw on a piece of paper or which lie on a plane are called Plane Figures. Example - Circle, Square, Rectangle etc.
The 3D shapes which occupy some space are called Solid Figures. Example - Cube, Cuboid, Sphere etc.
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
The area of all the faces of the solid shape is the total surface area of that figure. The unit of surface area is a square unit.
The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.
Cube is a solid shape having 6 equal square faces.
Lateral surface area of a cube
4l2
Total surface area of a cube
6l2
The volume of a cube
l3
Diagonal
√3 l, l = edge of the cube
What is the capacity of a cubical vessel having each side of 8 cm?
Given side = 8 cm
So, Volume of the cubical vessel = l3
= (8)3 = 256 cm3.
Cuboid is a solid shape having 6 rectangular faces at a right angle.
Lateral surface area of a cuboid
2h(l + b)
Total surface area of a cuboid
2(lb + bh + lh)
Volume of a cuboid
lbh
l = length, b = breadth, h = height
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
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.
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.
Curved surface area of a Right circular cylinder
2πrh
Total surface area of a Right circular cylinder
2πr2 + 2πrh = 2πr(r + h)
The volume of a Right circular cylinder
πr2
r = radius, h = height
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
2πh (R + r)
2πh (R + r) + 2π(R2 - r2)
R = outer radius, r = inner radius
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)
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
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.
Curved surface area of a Right Circular Cone
Total surface area of a Right Circular Cone
πr2 + πrl = πr(r + l)
The volume of Right Circular Cone
(1/3) πr2h
r = radius, h = height, l = slant height
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere
4πr2
Total surface area of a Sphere
Volume of a Sphere
(4/3) πr3
R = radius
If we cut the sphere in two parts then is said to be a hemisphere.
2πr2
3πr2
(2/3) πr3
r = radius
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?
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.
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