Area and volume of solid figure, math Assignment Help

Geometry Mathematics - Area and volume of solid figure, math

Area and volume of solid figure 

Solids: Bodies occupying space are called solids. A solid has three dimensions viz., length" breadth and thickness (or depth or height). 

Volume: The space occupied by a solid body is called its volume. Volume is measured in cubic units. Its S.I. unit is m3. i.e., the volume of a space region formed by a cube of m. 

Cuboid/rectangular parallelopiped:

It is solid with six rectangular faces.

(i)      Volume of cuboid = l × b × h

Where: l = length, b = breadth, h = height.

(ii)     Surface area of cuboid = 2 (lb + bh + hl)

(iii)    Surface area of 4 walls = 2(bh + hl) = 2(l + b) × h

(iv)    Surface area of cuboid, in which top face is open

          lb + 2 (bh + hl)

          (v)     Diagonal of faces of cuboid


(vi)    Diagonal of cuboid = 1208_sp2.jpg

          (vii)   Height of cuboid = volume/base area

          (viii)   Area of base = volume/height


Volume of material = Ext. volume - Int. volume.

Closed cuboid: If l, b, h are external dimension of closed cuboid of thickness 'a' then internal dimensions are : l - 2a, b-2a, h-2a.

Open cuboid: If the cuboid is an open cuboid then internal dimensions area l- 2a, b - 2a, h - a.

Cube: It is rectangular solid in which every face is square i.e.,

          l = b = h = 'a' (say)


(i)      Volume of cube = (edge)3 = a3

(ii)     Surface area of cube = 2 (aa + aa + aa)

= 2(a2 + a2 + a2)

= 2(3a2) = 6a2 = 6 (edge)2

(iii)    Diagonal of face of the cube = 2400_root1.jpg × a

(iv)    Diagonal of cube = 2263_root2.jpg × a

(v)     Edge of cube = (Volume)1/2


Right circular cylinder: If a rectangle is revolved about its one side as its axis, the solid formed is called a Right circular cylinder.



(1)     Volume of cylinder = (Area of base) × height

= (pr2) × h

= pr2 h m3 or cm3

(2)     Curved surface = (Perimeter of base) × height

= (2 × pr) × h

= 2prh m2 or cm2

(3)     Total surface area = Area of circular ends + Curved surface area

= 2pr2 + 2prh

= 2pr (r + h) m2 or cm2

where r = radius of the circular base of cylinder

h = height of cylinder.


Hollow cylinder: Solids like iron pipes, rubber tubes etc. are in shape of hollow cylinder.

A solid bounded by two co-axial cylinders of the same height, is called a hollow cylinder.

Cylinder with external and Internal radii: Cylinder of height h and with external and internal radii Rand r respectively, we have.

(1)     Volume of the material = pR2 h - pr2h

= ph (R2 - r)2

(2)     Curved surface = 2pRh + 2prh

= 2ph (R + r)

(3)     Total surface area = 2ph (R + r)

+ 2p (R2 - r2)

= 2p (R + r) (h + R - r).


Right circular cone: If a right angled triangle is revolved about one of the sides containing a right angle, the solid thus formed is called a light circular cone.


Slant height: Slant height of a right circular cone is the distance of its vertex from any point on the circumference of the base. OL is the slant height of the cone.

Semi vertical angle: It is the angle between the height and the slant height and is usually denoted by a.


(1)     Volume of Cone = 1/3 (Area or base) × height  

= 1/3 (pier2) × h = 1/3 pier2h cubic units.

(2)     Curved Surface Area = pierl sq. units.

(3)     Total Surface Area = (pierL + pier2) sq. units.


Frustum of a cone: If a cone is cut by a plane parallel to the base of the cone, then the portion between the plane and base is called the frustum of the cone.


Lateral surface area of frustum of a right circular cone = p(R + r) sq. units, where R, r be the radii of base and top of the frustum of a cone, h is the height of the frustum and 2 = h2 + (R -r)2. 

Volume of a frutrum of a cone = 626_1111111.jpg 

Total surface area of frustum of right circular cone

= Area of base + Area of top + Lateral surface area

= pR2 + pr2 + p(R + r) = pie [R2 + r2 + l(R + r)] sq. units.

Where l = slant height, r = radius of circular base, h = height.

Where l2 = h2 + r2


Sphere: If a circle is revolved about its diameter, the solid thus formed is called a sphere.      


          If r = radius of spare


(1)     Volume of sphere = 4/3 pier3 cubic units.

(2)     Surface area of sphere = 4 piel2



(1)     Volume of hemisphere =  460_pie.jpg

(2)     Curved area: Surface area of hemisphere = 2pr2

(3)     Surface (total surface area) of solid hemisphere = 2pr2 + pr2 = 3pr2

A plane through the centre of the sphere cuts it into two equal parts. Each part is called hemisphere.

(4)     Spherical shell: Volume of a spherical shell 2198_pie2.jpg


Prism: A prism is a solid whose side faces are parallelograms and whose bases are equal and parallel rectilinear figures.


A prism is called a right prism, if the axis is perpendicular to the base.

Surface area of a right prism = Perimeter of the base × height sq. units.

Total surface area of a right prism = Lateral area + 2 (area of one box) sq. units.

Volume of right prism = Area of the base × height cu. units. 


Pyramid: A polyhedron whose one face is a polygon and the other faces are triangles having a common vertex.

Volume of a right pyramid = 1/2 (area of the base) × height sq. units.

Surface area of a right pyramid = 1/2 (Perimeter of the base × slant height) sq. units.

Total surface area of a right pyramid

= Surface area + area of the base sq. 


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