Revision Notes on Surface Areas and Volumes

Plane figure

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.

Solid figures

The 3D shapes which occupy some space are called Solid Figures. Example - Cube, Cuboid, Sphere etc.

Volume

Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.

Surface Area

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.

Lateral or Curved Surface Area

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.

Surface Area and Volume of a Cube

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

Example

What is the capacity of a cubical vessel having each side of 8 cm?

Solution

Given side = 8 cm

So, Volume of the cubical vessel = l³

= (8)³ = 256 cm³.

Surface Area and volume of a Cuboid

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
Diagonal	l = length, b = breadth, h = height

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.

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

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.

Curved surface area of a Right circular cylinder	2πh (R + r)
Total surface area of a Right circular cylinder	2πh (R + r) + 2π(R2 - r2)
	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.

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

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.

Curved or Lateral surface area of a Sphere	4πr2
Total surface area of a Sphere	4πr2
Volume of a Sphere	(4/3) πr3
	R = radius

Surface Area and Volume of a Hemisphere

If we cut the sphere in two parts then is said to be a hemisphere.

Curved or Lateral surface area of a Sphere	2πr2
Total surface area of a Sphere	3πr2
Volume of a Sphere	(2/3) πr3
	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.