Area Formulas for Geometric Figures

Area of a triangle formulas

1. Area of a triangle when we know the base and the height

   The area of a triangle is equal to half of base times height.

   A =	1	a · h
   	2
2. Area of a triangle when we know the lengths of all three of its sides
   

   Heron's formula

   A = √s(s - a)(s - b)(s - c)
3. Area of a triangle when we know two sides and the included angle

   The area of a triangle is equal to half of a product of two sides and sine of the angle between this sides.

   A =	1	a · b · sin γ
   	2

   A =	1	a · c · sin β
   	2

   A =	1	b · c · sin α
   	2

4. Area of a triangle when we know three sides and circumradius

   A =	a · b · с
   	4R

5. Area of a triangle when we know semiperimeter and in-radius

   The area of a triangle is equal to semiperimeter times in-radius.

   A = s · r

   where A - the area of a triangle,
   a, b, c - the length of sides BC,AC,AB accordingly,
   h - the height, the length of the altitude,
   α, β, γ - the angles,
   r - the length of the in-radius,
   R - the length of the circumradius,
   

   s =	a + b + c	- the semiperimeter, or half of the triangle's perimeter.
   	2

Area of a square formulas

1. Area of a square when we know the length of a side

   The area of a square is equal to squared side.

   A = a²
2. Area of a square when we know the length of diagonal

   The area of a square is half of squared diagonal.

   A =	1	d2
   	2

   where A - the area of a square,
   a - the lenth of side,
   d - the length of diagonal.

Area of a rectangle formula

Area of a parallelogram formulas

1. Area of a parallelogram when we know the side and the height

   The area of a parallelogram is equal to side times height.

   A = a · h
2. Area of a parallelogram when we know two sides and the included angle

   The area of a parallelogram is equal to a product of two sides and sine of the angle between this sides.

   A = a · b · sin α
where A - the area of a parallelogram,
a, b - lengths of sides,
h - the height, the length of the altitude,
α - the measure of the angle between sides.

Area of a rhombus formulas

1. Area of a rhombus when we know the side and the height

   The area of a rhombus is equal to side times height.

   A = a · h
2. Area of a rhombus when we know length of a side and the included angle

   The area of a rhombus is equal to a product of the squared side and sine of the angle between sides of a rhombus.

   A = a² · sin α
3. Area of a rhombus when we know length of its diagonals

   The area of a rhombus is equal to half a product of it diagonals lengths.

   A =	1	d1 · d2
   	2

   where A - the area of a rhombus,
   a - the length of a side,
   h - the height, the length of the altitude,
   α - the measure of the angle between sides of a rhombus,
   d₁, d₂ - the length of diagonals.

Area of a trapezium formulas

1. Heron's formula for a trapezium

   A =	a + b	√(s-a)(s-b)(s-a-c)(s-a-d)
   	|a - b|

2. Area of a trapezium when we know length of 2 bases and the height

   The area of a trapezium is equal to product of the height and half-sum of two bases

   A =	1	(a + b) · h
   	2

   where A - the area of a trapezium,
   a, b - the length of the 2 bases,i.e., the parallel sides,
   c, d - length of the legs (the lateral sides),
   

   s =	a + b + c + d	- the semiperimeter, or half of the trapezium's perimeter.
   	2

Area of a quadrangle formulas

1. Area of a quadrangle when we know length of its diagonals and angle between diagonals

   The area of a quadrangle is equal to product of its diagonals and and sine of the angle between diagonals

   A =	1	d1 d2 sin α
   	2

   
   where A - the area of a quadrangle,
   d₁, d₂ - the length of diagonals,
   α - the angle between diagonals.
   
   
2. Area of a quadrangle when we know length of its perimeter and in-radius

   The area of a quadrangle is equal to perimeter timesin-radius

   A = s · r
3. 
   Area of a quadrangle when we know length of its sides and value of opposite corners

   A = √(s - a)(s - b)(s - c)(s - d) - abcd cos²θ
   where A -the area of a quadrangle,
   a, b, c, d - the length of sides,
   

   s =	a + b + c + d	- semiperimeter of quadrangle,
   	2

   θ =	α + β	- half the sum of two opposite angles of a quadrilateral.
   	2

   
   

Area of a circle formulas

1. Area of a circle when we know its radius

   The area of a circle is equal to a product of squared radius and pi.

   A = π r²
2. Area of a circle when we know its diameter

   The area of a circle is equal to a quarter product of squared diameter and pi.

   A =	1	π d2
   	4

   where A - the area of a circle,
   r - the length of the radius,
   d - the length of the diameter.

Area of a ellipse formulas