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Perimeter and Area Formulas for two dimensional geometrical figures

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This page provided formulas of Perimeter and Area for two-dimensional geometrical figures like square, rectangle, Rhombus, parallelogram, trapezium or trapezoid, triangle, right angle triangle, ellipse, circle, sector of a circle, segment of a circle etc

Formulas for Two Dimensional Geometrical Figures

Square

Area of the square, Perimeter of the square, Length of diagonal and Perimeter of the square- sugarprocesstech

Side for square = a  &  Length of diagonal = d

Area of the square = A = side2  = a2  = (1/2) d2

Perimeter of the square = P = 4 × side = 4 a

 

\text{Length of diagonal} = d = a \sqrt{2} = \sqrt{2 \times A} = \frac{P}{4} \sqrt{2}

Rectangle

Area of the rectangle, Perimeter of the rectangle, Length of diagonal and Perimeter of the rectangle- sugarprocesstechPerimeter of the rectangle = 2 × (length + width) = 2 (a + b)

Area of the rectangle = length × width = ab

Length of diagonal  ( d )   = √ (a2 + b2)

Rhombus

Formulas for Area of the Rhombus, Perimeter of the Rhombus - sugarprocesstechFor Rhombus all sides are equal = a, Vertical Height = d &  h1, h2 are the diagonals  also  AB || DC , AD || BC

Perimeter of rhombus = 4a

Area of the Rhombus = ad = (1/2) d1 d2

 

Parallelogram

Formulas for Area of the parallelogram, Perimeter of the parallelogram - sugarprocesstech

Area of  parallelogram = base × height = a x h

Perimeter of  parallelogram = 2 × (side1 + side2) = 2 ( a + b)

Trapezium (Trapezoid)

Formulas for Area of the Trapezium or Trapezoid, Perimeter of the Trapezium or Trapezoid - sugarprocesstech

In trapezium  two sides are in parallel,

 

 

\text{Area of Trapezium or Trapezoid} = \frac{1}{2} (a + b) h

\text{Perimeter of Trapezium or Trapezoid} = a + b + h \left[ \frac{1}{\sin \alpha} + \frac{1}{\sin \beta} \right]

Triangle

Formulas for Area of the Triangle, Perimeter of the Triangle - sugarprocesstechFormulas for triangle

Area of the triangle = (1/2) x Base x Height  

\text{Area of the triangle} = \left( \frac{1}{2} \right) \times \text{Base} \times \text{Height} = \frac{1}{2} \times a \times h

\text{Area of the triangle} = \sqrt{S(S-a)(S-b)(S-c)}

Where Semi perimeter = S = \frac{a+b+c}{2}
Radius of incircle triangle =  r = \frac{\text{Area}}{S}  

Formulas for equilateral triangle

Perimeter of the equilateral triangle = 3 x Side = 3a

Area of the equilateral triangle  = \text{Area} = \frac{\sqrt{3}}{4} \times (\text{Side})^2 = 0.433 \times (\text{Side})^2

S = \frac{ \text{Side}^3 }{ 4 \times \text{Area of Triangle} } = \frac{a}{\sqrt{3}}

\text{Area} = \pi \times \left( \frac{a}{\sqrt{3}} \right)^2

\text{Radius} = \frac{ \text{Area of Triangle} }{ \text{Semi perimeter} (S) } = \frac{a}{2\sqrt{3}} = \frac{a}{2 \sqrt{3}}

Radius of circumference circle of an equilateral triangle = S = \frac{\text{Side}^3}{4 \times \text{Area of Triangle}} = \frac{a}{\sqrt{3}}

Area of circumference circle of an equilateral triangle =  \pi \times \left( \frac{a}{\sqrt{3}} \right)^2

The radius of incircle of an equilateral triangle = = \frac{\text{Area of Triangle}}{\text{Semi perimeter (S)}} = \frac{a}{2\sqrt{3}}

 

Formulas for Right Angle Triangle & Formula for the length of a triangle when given by one side and angle

Formulas for Area of the right angle Triangle, Perimeter of the right angle Triangle - How to find length of a triangle given one side and angle-sugarprocesstech

In the above figure, ABC is a right angle triangle x = length of adjacent side, y = length of hypotenuse & h = height of the triangle.

Pythagoras’ Law –  x^2 + h^2 = y^2

Area of the right angle triangle = \frac{1}{2} x h

Sin β = h / y  ⇒  h  = y  Sin β

Cos β = x / y  ⇒  x  = y  Cos β

Tan β = h / x ⇒  h  = x Tan β

Cosine Law

c 2 = a 2 + b 2 – 2 ab Cos γ

b2 = a2 + c2 – 2 ac Cos β

a2 = b2 + c2 – 2 bc Cos α

Sine Law    

\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}
frac{a}{sin alpha} = frac{b}{sin beta} = frac{c}{sin gamma}

Area of a Triangle

\frac{b c \sin \alpha}{2} = \frac{a c \sin \beta}{2} = \frac{a b \sin \gamma}{2}

Circle

Circle Formulas, Area of a circle , Circumference of a circle, Arc and sector of a circle, Segment of and perimeter of segment in a circle

Formulas for Area and circumference of the circle

Area of a circle (A ) =  π x (Radius) 2   =  π r 2 = ( π/4 ) D2 = 0.7854 x D2

Diameter of a circle (D) = \sqrt{\frac{\text{Area of a circle}}{0.7854}}

Circumference of a circle ( C ) = 2 π x Radius =  2 π r = π D

Area of circle =( 1/2) x Circumference x radius = (1/2) x C x  r

Formula for Arc and sector of a circle

\text{Arc length of circle} \ (l) = \frac{\theta}{360} \times 2\pi r = \frac{\theta \pi r}{180}

\text{Area of the sector (minor)} = \frac{\theta}{360} \times \pi r^2

If the angle θ is in radians mean   one radian = 180/π, then

\text{The area of the sector} = \frac{\theta}{2} \times r^2

\text{Sector angle of a circle} \ \theta = \frac{180 \times l}{\pi r}

Segment of circle and perimeter of segment

\text{Area of the segment} = \left( \frac{\theta}{360} \right) \pi r^2 - \left( \frac{1}{2} \right) \sin \theta \, r^2

\text{The perimeter of the segment} = \left( \frac{\theta \pi r}{180} \right) - 2r \sin \left( \frac{\theta}{2} \right)

Arc  Length of the circle segment   =  l  = 0.01745 x r x θ

\text{The chord length of the circle} = 2r \sin \left( \frac{\theta}{2} \right) = 2 \sqrt{h(2r - h)}

Area of the circular ring

Area of a circular ring  = = (π/4)  ( D 2 – d 2)   = 0.7854 (D 2 – d 2)

Area of a circular ring  = π (R 2 – r 2 )

Ellipse

Formulas for Area of the Ellipse, Perimeter of the Ellipse - sugarprocesstech

D = Diameter of higher half-axis & d = Diameter of lower half-axis

Area of an Ellipse = \frac{\pi}{4} \times D \times d     = 0.7854 x D x d

 

D = 2 x radius = 2 m , d = 2 x radius = 2 n

\text{Perimeter of an Ellipse (Formula - 1)} \approx 2\pi \sqrt{\frac{m^2 + n^2}{2}}

\text{Perimeter of an Ellipse (Formula - 2)} \approx \pi \left[ 3(m + n) - \sqrt{(3m + n)(m + 3n)} \right]

Both formulas gives approximate values but the formula- 2 gives better approch which is derived by Indian mathematician Ramanujan

Perimeter and Area of 2D geometrical figures of square, rectangle, Rhombus, parallelogram, trapezium or trapezoid, triangle, right angle triangle, ellipse, circle, sector of a circle, segment of a circle etc

 

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