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Thursday, 25 September 2014

Mensuration


Mensuration Formulas
Mensuration is the branch of mathematics which deals with the study of Geometric shapes, their area, volume and related parameters.
Some important mensuration formulas are:

1. Area of rectangle (A) = length(l) × Breath(b)
A = l \times b


2. Perimeter of a rectangle (P) = 2 × (Length(l) + Breath(b))
P = 2 \times(l + b)

3. Area of a square (A) = Length (l) × Length (l)
A = l \times l


4. Perimeter of a square (P) = 4 × Length (l)
P = 4 \times l

5. Area of a parallelogram(A) = Length(l) × Height(h)
A = l \times h
Parallelogram

6. Perimeter of a parallelogram (P) = 2 × (length(l) + Breadth(b))
P = 2 \times (l + b)

7. Area of a triangle (A) = (Base(b) × Height(b)) / 2
A = \frac{1}{2} \times b \times h
Triangle
And for a triangle with sides measuring “a” , “b” and “c” , Perimeter = a+b+c
and s = semi perimeter = perimeter / 2 = (a+b+c)/2
And also: Area of triangle = A = \sqrt{s(s-a)(s-b)(s-c)}
This formulas is also knows as “Heron’s formula”.

8. Area of triangle(A) = \frac{1}{2} a \times b \times \angle C = \frac{1}{2} b \times c \times \angle A = \frac{1}{2} a \times c \times \angle B
Where A, B and C are the vertex and angle A , B , C are respective angles of triangles and  a , b , c are the respective opposite sides of the angles as shown in figure below:
area of triangle - mensuration
area of triangle - mensuration

9. Area of isosceles triangle = \frac{b}{4}\sqrt{4a^2 - b^2}
Where a = length of two equal side , b= length of base of isosceles triangle.

10. Area of trapezium (A) = \frac{1}{2} (a+b) \times h
Where “a” and “b” are the length of parallel sides and “h” is the perpendicular distance between “a” and “b” .
Trapezium

11. Perimeter of a trapezium (P) = sum of all sides

12. Area of rhombus (A) =  Product of diagonals / 2

13. Perimeter of a rhombus (P) = 4 × l
where l = length of a side

14. Area of quadrilateral (A) = 1/2 × Diagonal × (Sum of offsets)
quadrilateral

15.  Area of a Kite (A) = 1/2 × product of it’s diagonals

16. Perimeter of a Kite (A) = 2 × Sum on non-adjacent sides

17.  Area of a Circle (A) = \pi r^2 = \frac{\pi d^2}{4}
Where r = radius of the circle and d = diameter of the circle.

18. Circumference of a Circle = 2 \pi r = \pi d
r= radius of circle
d= diameter of circle

19. Total surface area of cuboid = 2 (lb + bh + lh)
where l= length , b=breadth , h=height

20. Total surface area of cuboid = 6 l^2
where l= length

21. length of diagonal of cuboid = \sqrt{l^2+b^2+h^2}

22. length of diagonal of cube = \sqrt{3 l}

23. Volume of cuboid = l × b × h

24. Volume of cube = l × l × l

25. Area of base of a cone = \pi r^2

26.  Curved surface area of a cone = C = \pi \times r \times l
Where r = radius of base , l = slanting height of cone

27. Total surface area of a cone = \pi r (r+l)

28. Volume of right circular cone = \frac{1}{3} \pi r^2 h
Where r = radius of base of cone , h= height of the cone (perpendicular to base)

29. Surface area of triangular prism = (P × height) + (2 × area of triangle)
Where p = perimeter of base

30. Surface area of polygonal prism = (Perimeter of base × height ) + (Area of polygonal base × 2)

31. Lateral surface area of prism = Perimeter of base × height

32. Volume of  Triangular prism = Area of the triangular base × height

33. Curved surface area of  a cylinder = 2 \pi r h
Where r = radius of base, h = height of cylinder

34. Total surface area of a cylinder = 2 \pi r(r + h)

35. Volume of a cylinder = \pi r^2 h

36. Surface area of sphere = 4 \pi r^2 = \pi d^2
where r= radius of sphere, d= diameter of sphere

37. Volume of a sphere = \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3

38. Volume of hollow cylinder = \pi r h(R^2-r^2)
where , R = radius of cylinder , r= radius of hollow , h = height of cylinder

39. Right Square Pyramid:
If a = length of base , b= length of equal side  ; of the isosceles triangle forming the slanting face , as shown in figure:
net diagram of right square pyramid
net diagram of right square pyramid
39.a Surface area of a right square pyramid = a \sqrt{4b^2 - a^2}
39.b Volume of a right square pyramid = \frac{1}{2} \times base \, \, area \times height

40. Square Pyramid:
40.a. Johnson Pyramid:
net diagram of johnson pyramid
net diagram of johnson pyramid
Volume = (1+ \sqrt{3})\times a^2
Total Surface Area: \frac{\sqrt{2}}{6} \times a^3
40.b. Normal Square pyramid:
If a = length of square base and h = height of the pyramid then:
Volume = V=\frac{1}{3}a^2h
Total Surface Area = a^2+a\sqrt{a^2+(2h)^2}

41. Area of a regular hexagon = \frac{3\sqrt{3}a^2}{2}

42. area of equilateral triangle = \frac{\sqrt{3}}{4} a^2

43. Curved surface area of a Frustums = \pi h (r_1 + r_2)

44. Total surface area of a Frustums = \pi (r_1^2 + h(r_1+r_2) + r_2^2)

45. Curved surface area of a Hemisphere = 2 \pi r^2

46. Total surface area of a Hemisphere = 3 \pi r^2

47. Volume of a Hemisphere =  \frac{2}{3} \pi r^3 = \frac{1}{12} \pi d^3

48. Area of sector of a circle = \frac{\theta r^2 \pi}{360}
where \theta = measure of angle of the sector , r= radius of the sector
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