# Area Formulas for Geometric Figures

## Area of a triangle formulas

**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 **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)**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 **Area of a triangle when we know three sides and circumradius**A = a · b · с 4R - where A - the area of a triangle,
**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

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

**Area of a square when we know the length of a side**The area of a square is equal to squared side.

A = a^{2}- where A - the area of a square,
**Area of a square when we know the length of diagonal**The area of a square is half of squared diagonal.

A = 1 d ^{2}2

a - the lenth of side,

d - the length of diagonal.

## Area of a rectangle formula

a, b - the length of sides(the height and width of rectangle).

## Area of a parallelogram formulas

**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**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

**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**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 = a^{2}· sin α- where A - the area of a rhombus,
**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 d _{1}· d_{2}2

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,

d_{1}, d_{2}- the length of diagonals.

## Area of a trapezium formulas

**Heron's formula for a trapezium**A = a + b √(s-a)(s-b)(s-a-c)(s-a-d) |a - b| - where A - the area of a trapezium,
**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

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

**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 d _{1}d_{2}sin α2

where A - the area of a quadrangle,

d_{1}, d_{2}- the length of diagonals,

α - the angle between diagonals.**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-
where A -the area of a quadrangle,
**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 cos^{2}θ

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

**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 r^{2}- where A - the area of a circle,
**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 d ^{2}4

r - the length of the radius,

d - the length of the diameter.

## Area of a ellipse formulas

a - the length of the major semiaxis,

b - the length of the minor semiaxis,

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