Mensuration formulas of 2D and 3D shapes

1. Mensuration Formulas for RECTANGLE

• Area of Rectangle = Length × Breadth.
• Perimeter of a Rectangle = 2 × (Length + Breadth)
• Length of the Diagonal = √(Length2+ Breadth2)

2. Mensuration Formulas for SQUARE

• Area of a Square = Length × Length = (Length)2
• Perimeter of a square = 4 × Length
• Length of the Diagonal = √2 × Length

3. Mensuration Formulas for PARALLELOGRAM

• Area of a Parallelogram = Length × Height
• Perimeter of a Parallelogram = 2 × (Length + Breadth)

4. Mensuration Formulas for TRIANGLE

• Area of a triangle=(1/2)(Base × Height)=(1/2)(BC×AD)

For a triangle with sides measuring a, b and c, respectively:

• Perimeter = a + b + c
• s = semi perimeter = perimeter/2 =  (a+b+c)/2
• Area of Triangle, A=
  (This is also known as “Heron’s formula”)
  
• Area of isosceles triangle =
  (Where a = length of two equal side, b = length of base of isosceles triangle.)
  
• Area of an equilateral triangle =
  (Where, a is the side of an equilateral triangle)

5. Mensuration Formulas for TRAPEZIUM

• Area of a trapezium = (1/2) × (sum of parallel sides) × (distance between parallel sides)
  = (1/2) × (AB+DC) × AE
• Perimeter of a Trapezium = Sum of All Sides

6. Mensuration Formulas for RHOMBUS

• Area of a rhombus=(1/2)×Product of diagonals

• Perimeter of a rhombus = 4 × l
  (where l = length of a side)

7. Mensuration Formulas for CIRCLE and SEMICIRCLE

In the following formulae, r = radius and d = diameter of the circle

• Area of a circle = πr2= (πd2)/4
• Circumference of a circle = 2πr = πd
• Circumference of a semicircle = πr
• Perimeter of the semicircle = (πr+d)
• Area of semicircle =(πr2)/2
  
• Length of an arc = (2πrθ)/360, where θ is the central angle in degrees.
• Area of a sector = (1/2) × (length of arc) × r = (πr2θ)/360

Mensuration Formulas for 3D Figures

8. Mensuration Formulas for CUBOID

In the following formulae, l = length, b = breadth and h = height

• Total surface area of cuboid = 2 (lb + bh + lh)
• Length of diagonal of cuboid= √(l2+b2+h2)
• Volume of cuboid = l × b × h

9. Mensuration Formulas for CUBE

In the following formulae, a = side of a cube

• Volume of cube = a3
• Total surface area of cube = 6a2
• Length of Leading Diagonal of Cube = a√3

10. Mensuration Formulas for CONE

In the following formulae, r = radius of base, l = slant height of cone and h = height of the cone (perpendicular to base)

• Slant height of a cone = l =√(h2+r2 )
• Curved surface area of a cone = C = π × r × l
• Total surface area of a cone = π × r × (r + l)
• Volume of right circular cone =1/3 πr2h

11. Mensuration Formulas for CYLINDER

In the following formulae, r = radius of base, h = height of cylinder

• Curved surface area of a cylinder = 2πrh
• Total surface area of a cylinder = 2πr(r + h)
• Volume of a cylinder = πr2h

12. Mensuration Formulas for SPHERE

In the following formulae, r = radius of sphere, d = diameter of sphere

• Surface area of a sphere = 4πr2 = πd2
• Volume of a sphere = (4/3) πr3 = (1/6)πd3

13. Mensuration Formulas for HEMISPHERE

In the following formulae, r = radius of sphere

• Volume of a hemisphere =(2/3)πr3
• Curved surface area of a hemisphere = 2πr2
• Total surface area of a hemisphere = 3πr2

14. Mensuration Formulas for HOLLOW CYLINDER

Hollow cylinder made by cutting a smaller cylinder of same height and orientation out of a bigger cylinder.

• Volume of hollow cylinder = πh(R2– r2)
  (Where, R = radius of cylinder, r = radius of cavity, h = height of cylinder)

15. Mensuration Formulas for FRUSTUM OF A RIGHT CIRCULAR CONE

Frustum is created when a plane cuts a cone parallel to its base.
In the following formulae, R = radius of the base of the frustum, r = radius of the top of the frustum,
h = height of the frustum, l = slant height of the frustum

• If a cone is cut by a plane parallel to the base of the cone, the lower part is called the frustum of the cone.
• Slant height of the frustum =l=√(h2+(R-r)2)
• Curved surface area of frustum = π(R + r)l
• Total surface area of frustum = π(R + r)l + π(R2 + r2)
• Volume of the frustum=(1/3)πh(R2+r2+Rr)

16. Mensuration Formulas for PRISM

• Prism consists of two polygonal bases which are parallel to each other.
• These bases are joined by lateral faces, which are perpendicular to the polygonal bases.
• The number of lateral faces is equal to the number of sides in the polygonal base. Thus, the base of a prism could be of various shapes, namely, triangular, quadrangular, pentagonal etc.
• Volume of prism = Base area × height
• Lateral surface area of prism = perimeter of base × height
• Total surface area of prism = Lateral surface area + (2 × base area)

17. Mensuration Formula for PYRAMID

• Pyramid consists of a polygonal base and triangles at its sides. These triangles are called faces.
• The base could be of any shape, whereas the faces are generally isosceles triangles.
• All these triangular faces meet in a single point called the apex.
• Total surface area of pyramid = base area + (number of sides × ½ × slant height × base length)
• Volume of pyramid = (1/3) × base area × height