Important Mensuration Formulas-Area & Perimeter

Area and perimeter

TRIANGLE

1. Equilateral Triangle

Equilateral Triangle

Area =\,\frac{\sqrt{3}}{4} \,a^{2}

Height (h) =\,\frac{\sqrt{3}}{2} \,a

Perimeter = 3a

2. Isosceles Triangle

Area =\,\frac{b}{4} \sqrt{4 a^{2}- b^{2}}

h = \frac{1}{2}\sqrt{4 a^{2}- b^{2}}

Perimeter= 2a+b

3. Scalene triangle

Scalene Triangle

Area = \sqrt{S(S-a)(S-b)(S-c)} = \frac{1}{2}\times\,C\times\,h    Where S = \frac{a+b+c}{2}

Perimeter = a + b + c

4. Right angled Triangle

Right angles triangle
Area = \frac{1}{2}\times\,b\times\,h

Perimeter = p + b + h

 h^{2} = b^{2} + p^{2}

Note: Side of the maximum size square inscribed in a right-angle triangle.

right angled triangle

a = \frac{P\times\,B}{P+B}

QUADRILATERAL

1. Parallelogram

parallelogram

Area = Base \times Height = b  \times  h

Perimeter = 2(a + b)

2. Trapezium

Trapezium

Area =\frac{1}{2} ( Sum of Parallel Sides \times  Height) = \frac{1}{2} (a + b) h

Perimeter = a + b+ c+ d

3. Rhombus

Rhombus

Area = \frac{1}{2} \times d1 \times d2

a = \frac{1}{2} \sqrt{ (d_{1})^{2}+ (d_{2})^{2}}

Perimeter = 4a

4a ^{2} =  (d_{1})^{2}+ (d_{2})^{2}

4. Rectangle

Rectangle

Area = l \times b

Perimeter = 2 (l + b)

d = \sqrt{ l^{2}+ b^{2}}

5. Square

Area =  a^{2} = \frac{ d^{2}}{2}

Perimeter = 4a

Diagonal (d) = a\sqrt{2}

REGULAR POLYGON

  • Each Exterior angle = \frac{360 ^{o}}{n}
  • Each Interior angle = 180 ^{o} - Exterior \, angle
  • Number of Diagonals= \left \{ \frac{n(n-1)}{2}-n \right \} or, \frac{n\,(n-3)}{2}
  • Sum of Exterior angle =  360^{o}
  • Sum of Interior angle = (n-2)\times 180^{0}
  • Sum of interior and Exterior = 180^{0}

CIRCLE

Circle

Area = \pi\,r ^{2}

Circumference = 2\pi\, r

Diameter= 2r

Length of arc (l) = \frac{\pi\, r\,\theta}{180^{o}}

Area of Sector = \frac{\pi\,r^{2}\,\theta}{360^{o}}

Perimeter of Sector =  \pi\,r\,\frac{\theta}{180^{0}} + 2r

For Circular Ring

Area = \pi (R^{2}-r^{2})

SEGMENT

segment

Area = area of sector OACB – area of ∆OAB = \pi\,r^{2}\,\frac{\theta}{360^{0}}-\frac{1}{{2}}\,r^{2}\sin\theta

Perimeter = length of ARC ACB + Chord length AB = (2πr) \frac{\theta}{{360^{o}}}+2\p\,sin\frac{\theta}2}

AB = 2r sin\frac{\theta}2}

IMPORTANT POINTS TO BE REMEMBER

  • If the length and breadth of a rectangle are increased by a% and b% respectably, then area will be increased by \left ( a+b+\frac{ab}{100} \right )%
  • If all the increasing sides of any two dimensional figure are changed by a%, then the area will be changed by \left ( 2a+\frac{a ^{2}}{100} \right )%

In case of circle, radius (or diameter) is increased in place of sides.

  • If all the measuring sides of any two-dimensional figure are changed by a% , then its perimeter also changes by a%
  • If area of a square is a square unit, then the area of the circle formed by the same perimeter is given by  \frac{4a}{\pi} square unit.
  • The area of the largest triangle inscribed in a semi circle of radius r =  r^{2}
  • Area of square inscribed in a circle of radius r =  2r^{2}

 

 

 

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