Surface Area And Volume

Solid:
A physical body which occupies some space is called a solid. It has three dimensions in space called length, breadth and height.
Example: a brick, a table, a ball ,an antenna etc.

Volume:
The space occupied by a solid body is called its volume. Cubic centimeters (cm3) and cubic meters (m3) are the common units of volume.
Cuboid or Rectangular Parallelepiped: It is a solid with six rectangular faces.
If length, breadth and height of a cuboid are l, b and h respectively then,

Areas
square = a^2

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b1 + b2)

circle = pi r^2

ellipse = pi r1 r2

triangle = (1/2) b h

equilateral triangle = (1/4)sqrt(3) a^2

triangle given SAS = (1/2) a b sin C

triangle given a,b,c = [sqrt][s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)

regular polygon = (1/2) n sin(360°/n) S^2
when n = # of sides and S = length from center to a corner

Volumes
cube = a^3

rectangular prism = a b c

irregular prism = b h

cylinder = b h = [pi] r^2 h

pyramid = (1/3) b h

cone = (1/3) b h = 1/3 [pi] r^2 h

sphere = (4/3) [pi] r3

ellipsoid = (4/3) pi r1 r2 r3

Surface Areas
cube = 6 a^2

prism:
(lateral area) = perimeter(b) L
(total area) = perimeter(b) L + 2b

sphere = 4 [pi] r^2

Cuboid

The surface area of a rectangular solid is equal to the sum of the areas of all the faces.
Total surface area = 2(lb + bh + hl )
Curved surface area = 2h(l + b)
Volume = lbh, Where l= length, b= breadth, h= height

Curved surface area of cuboid or surface
The Curved surface area of cuboid is the sum of the areas of its six faces. The surface area is the total area of all the faces of a 3D shape. CSA of a cuboid is calculated using the given Curved Surface Area (CSA) Of Cuboid Formula. First add both length and breadth, then multiply by 2 with Height.

Body diagonal of a cuboid = Length of the longest rod that can be kept inside a rectangular room is =
?L^2 + B^2 + H^2

Formulas for Cube :
For a cube of side n*n*n painted on all sides which is uniformly cut into smaller cubes of dimension 1*1*1,
Number of cubes with 0 side painted= (n-2) ^3
Number of cubes with 1 sides painted =6(n – 2) ^2
Number of cubes with 2 sides painted= 12(n-2)
Number of cubes with 3 sidess painted= 8(always)
For a cuboid of dimension a*b*c painted on all sides which is cut into smaller cubes of dimension 1*1*1,
Number of cubes with 0 side painted= (a-2) (b-2) (c-2)
Number of cubes with 1 sides painted =2[(a-2) (b-2) + (b-2)(c-2) + (a-2)(c-2) ]
Number of cubes with 2 sides painted= 4(a+b+c -6)
Number of cubes with 3 sidess painted= 8

Perimeter formula:
Square 4 × side
Rectangle 2 × (length + width)
Parallelogram 2 × (side1 + side2)
Triangle side1 + side2 + side3
Regular n-polygon n × side
Trapezoid height × (base1 + base2) / 2
Trapezoid base1 + base2 + height × [csc(theta1) + csc(theta2)]
Circle 2 × pi × radius
Ellipse 4 × radius1 × E(k,pi/2)
E(k,pi/2) is the Complete Elliptic Integral of the Second Kind
k = (1/radius1) × sqrt(radius1^2 - radius2^2)

Area formula:
Square side^2
Rectangle length × width
Parallelogram base × height
Triangle base × height / 2
Regular n-polygon (1/4) × n × side^2 × cot(pi/n)
Trapezoid height × (base1 + base2) / 2
Circle pi × radius^2
Ellipse pi × radius1 × radius^2
Cube (surface) 6 × side^2
Sphere (surface) 4 × pi × radius^2
Cylinder (surface of side) perimeter of circle × height
2 × pi × radius × height
Cylinder (whole surface) Areas of top and bottom circles + Area of the side
2(pi × radius^2) + 2 × pi × radius × height
Cone (surface) pi × radius × side
Torus (surface) pi^2 × (radius2^2 - radius1^2)

Volume formula:
Cube side^3
Rectangular Prism side1 × side2 × side3
Sphere (4/3) × pi × radius3
Ellipsoid (4/3) × pi × radius1 × radius2 × radius3
Cylinder pi × radius^2 × height
Cone (1/3) × pi × radius^2 × height
Pyramid (1/3) × (base area) × height
Torus (1/4) × pi^2 × (r1 + r2) × (r1 - r2)^2

Hemisphere:
A plane passing through the centre of a sphere divides the sphere into two equal parts. Each part is called a hemisphere.
Let the radius of the hemisphere be r, then
(i) Volume of the hemisphere = 2/3 ?r3 cu. units
(ii) Curved surface area = 2?r2 sq. units
(iii) Total surface area = 3?r2 sq. units

Spherical Shell:
The difference of two solid concentric spheres is called a spherical shell.
Let the outer radius and inner radius of a spherical shell are R and r respectively, then (i) Volume of spherical shell = 4/3? (R3 - r3) cub. Units
(ii) External surface area = 4?R2 sq. units
(iii) Internal surface area = 4?r2sq. units

Surace Area Of Combination Of Solids:
I. Total surface area of the solid given in figure = Curved surface area of one hemisphere
+ Curved surface area of right circular cylinder
+ Curved surface area of other hemisphere
= 2?r2 + 2?rh + 2?r2
= 4?r2 + 2?rh
= 2?r (2r + h)
II. Total surface area of the solid [like toy / top (lattu)] (given in figure)
= Curved surface area of hemisphere + curved surface area of cone
= 2?r2 + ?r l
= ?r (2r + l)