TNPSC – MENSURATION

TNPSC – MENSURATION

MENSURATION:

This chapter is based finding on 2D & 3D figures and basically two-dimensional figures come under geometry. Only 3-dimensional figures are studied their area and volume.

Two Dimensional Figures:

=> Rectangle (opposite sides are equal and intersect each other at an angle of 90°)

Area = Length * Breadth

Perimeter = 2 (Length * Breadth)

Diagonal = √ [(Length)²+(Breadth)²]

=> Square (All the four sides are equal and the angle enclosed by adjacent sides is 90°)

Area = (Side)²

Perimeter = 4*side

Diagonal = √2*side

After this all the four sides figures —

Parallelogram

Rhombus

Trapezium & Scalene Quadrilateral should be given.

=> Triangles

(i) If a,b and c are the lengths of the first, second and third sides of a triangle respectively, then

S = (a+b+c)/2, where s=semi-perimeter and Area = √[s (s-a) (s-b) (s-c)]

(Heron’s formula),

Perimeter = a+b+c

(ii) For a right angled triangle,

Area = (1/2) base * height, and

Perimeter = a+b+c                            

(iii) For an equilateral triangle

Area = √ (3/4) * (side)² , and

Perimeter = 3*side

=> Circle

Area = π * (Radius)²

Circumference = 2π * Radius

Radius = Diameter/2

=> Area of the walls of a room

= 2 * height (length + breadth)

Height = Wall Area / 2 (length + breadth)

==> Parallelogram (Opposite sides are equal & parallel, but angel enclosed by adjacent sides is not a right angle)

Area = base * height

Height = length of perpendicular dropped from on opposite side to the base.

Perimeter = 2 (sum of opposite sides)

(ii) Rhombus (All the four sides are equal but the angle enclosed by adjacent sides is not a right angle)

Area = ½ * product of diagonals

Perimeter = 4 * side

(iii) Trapezium (One pair of opposite side is parallel)

Area = ½ * (Sum of parallel sides) * height

Perimeter = Sum of 4 sides, i.e. a+b+c+d.

(iv) Scalene Quadrilateral (All the four sides are unequal and non-parallel)

Area = ½ (DP+BQ) * AC

Perimeter = Sum of 4 sides, i.e. a+b+c+d.

Note:

Area is the portion enclosed by the figure.

Perimeter is the boundary sum of angles of a four sided figure is 360^o.

Three dimensional figures

=> Cuboid

If L, B and H are length, breadth and height of the cuboid, then

Volume = L*B*H

Surface area = 2 (L*B+B*H+H*L)

Diagonal = √ (L²+B²+H²) 

=> Cube

If a is each side of the cube, then

Volume = a*a*a=a³

Surface area = 2(a*a+ a*a+ a*a) = 6a²

Diagonal of cube = √ (a²+a²+a²) = √ (3a)

=> Cylinder

If radius of cylinder is r and height or length is h, then

Volume = πr²h

Latest surface area = 2πrh

Whole surface area = (2πrh + 2πr²)

=> Cone

If base-radius, vertical height and slanting height of a cone are r, h and l respectively, then

Volume = 1/3 (πr²h)

Lateral surface area = πrl

Total surface area = πrl+ πr²

Vertical height = l = √ (r²+h²)

Frustum of a cone:

A cone whose top is sliced off, is called frustum of a cone.

In the above figure, R is the radius of the base, r is the radius of the top, h is the vertical height and L is the slant height.

Volume = πh/3 (R²+r²+Rr)

Slant height = √ [(R-r)²+(h²)]

Curved surface area = π(R+r)L

Total surface area = π (RL+rL+r²+R²)

To find height of the original cone, following formula can be used –

                                                H = Rh/(R-r)

=>Sphere

If r is the radius of the sphere, then

Volume = 4/3 (πr³)

Surface area = 4πr²

=> Hemisphere

Volume = 2/3 (πr³)

Curved surface area = 2πr²

Total surface area = 2πr²+πr² = 3πr²

Some units related to volume:

1 litre = 1000 cm³

1 Hectometer³ = 1000000 meter³

1 Decameter³ = 1000 meter³

1 Meter³ = 1000000 cm³

1 Decimeter³ = 1000cm³

1 Millimetre³ = 1/1000 cm³

Some units to related to area:

1 Hectare = 10000 metre square

1 kilometre square = 1000000 metre square

1 Decametre square = 100 metre square

1 Decimetre square = 1/100 metre square

1 Centimetre square = 1/10000 metre square

1 Millimetre square = 1/1000000 metre square