All formulas for area of plane figures

Area of a triangle, Heron's formula

area triangle Heron

 

a , b , c    -  sides of a triangle

s  - semiperimeter    

s F

 

Heron's formula for calculating the Area of a triangle if given all sides ( A ) :

area triangle Heron F

 

 

Area of an ellipse

area ellipse

R red  -  semi-major axis

rr red  -  semi-minor axis

pi1    3,14

 

Formula for calculating Area of an ellipse if given semi-major and semi-minor axes ( A ) :

area ellipse F

 

 

Area of an equilateral triangle

area equilateral triangle

 

h  - height  measured at right angles to the base

a  - equal sides

 

 

Calculate the Area of an equilateral triangle if given side or height ( A ) :

area equilateral triangle F1   area equilateral triangle F2

 

 

Area of a triangle - "side angle side" (SAS) method

area triangle 2sides angle

 

a , b , c blue    -  sides of a scalene triangle

alpha red1 , beta green , Y blue    -  angles

 

 

Calculate the Area of a triangle if given two sides and the included angle ( A ) :

area triangle 2sides angle F1

area triangle 2sides angle F2

area triangle 2sides angle F3

 

 

Area of a triangle from a side and two angles (AAS or ASA) method

area triangle 2sides angle

 

a , b , c blue    -  sides of a scalene triangle

alpha red1 , beta green , Y blue    -  angles

 

 

Calculate the Area of a triangle if given two angles and a side ( A ) :

area triangle side 2angles F1

area triangle side 2angles F2

area triangle side 2angles F3

 

 

Area of a rhombus 

1.  The area of a rhombus if you know diagonals and angle  

area rhombus

DD blue  -  larger diagonal

d blue  -  smaller diagonal

alpha red  -  acute angle

beta green  -  obtuse angle

 

Calculate the Area of a rhombus if given diagonals and angle ( A ) :

area rhombus F1

area rhombus Fd1  area rhombus Fd2

 

 

2.  The area of a rhombus if you know side and angle

area rhombus

 

a black  - side

alpha red , beta green -  angles

 

 

Calculate the Area of a rhombus when sides and angle are given ( A ) :

area rhombus Fa1   area rhombus Fa2

 

 

3.  The area of a rhombus if you know side and radius of the inscribed circle  

area rhombus radius1

 

a black  - side

h  - height

rr red  - radius  of the inscribed circle

 

Calculate the Area of a rhombus if given side and radius of the inscribed circle ( A ) :

area-rhombus-height-F    area-rhombus-radius-F

 

 

Area of a parallelogram 

1.  The area of a parallelogram if you know sides and angle

area parallelogram

 

a , b -  sides

alpha red , beta green -  angles

 

 

Formula for the area of a parallelogram when side and angle are given ( A ) :

area parallelogram F1      area parallelogram F2

 

 

2.  How to find the area of a parallelogram if you know side(base) and height

 

area parallelogram height

a , b -  sides - base

   Ha- height  measured at right angles to the side  

   Hb- height  measured at right angles to the side

 

Formula for the Area of a parallelogram if given side and height ( A ) :

area parallelogram height Fa      area parallelogram height Fb

 

 

3.  The area of a parallelogram if you know diagonals and angle between them  

area parallelogram diagonal

 

DD blue , d blue   -  diagonals

alpha red , beta green -  angles between the diagonals

 

 

Formula for the Area of a parallelogram with diagonals and angle between them ( A ) :

area parallelogram diagonal F1    area parallelogram diagonal F2

 

 

Area of a rectangle

area rectangle

 

a  - length  of a rectangle

b  - width  of a rectangle

 

 

Calculate the Area of a rectangle if given length and width ( A ) :

area rectangle F

 

 

Area of a square from a side or diagonal

area square

 

a  - side

d blue1  - diagonal

 

 

Calculate the Area of a square if given side (A) :

area square F

Calculate the Area of a square if given diagonal (A) :

area square F1

 

 

Area of a trapezoid 

1.  How to find the area of a trapezoid (called a trapezium in the UK) if you know bases, height and middle line of a trapezoid  

area of trapezoid m

 

a , b     -  bases

h  - height

m red - middle line

A-black  -  area of a trapezoid

 

Formula for area of a trapezoid if given bases, height and middle line. As well as you can use the calculator for calculating area. ( A ) :

area of trapezoid m formula1

area of trapezoid m formula2

 

 

2.  How to find the area of a trapezoid (trapezium UK) if you know diagonals and angle between them

area of trapezoid d

 

d1 blue , d2 blue -  diagonals

alpha red , beta green -  angles between the diagonals

A-black  -  area of a trapezoid

 

Formula for area of a trapezoid if given diagonals and angle between them. As well as you can use the calculator for calculating area ( A ) :

area of trapezoid d formula1  area of trapezoid d formula2

 

 

3.  How to find the area of a trapezoid (trapezium UK) if you know all sides  

area of trapezoid side

 

 

a , b , c , d black   -  sides

A-black  -  area of a trapezoid

 

Formula for area of a trapezoid if given all four sides. As well as you can use the calculator for calculating area ( A ) :

area of trapezoid side formula

 

 

Definition of a trapezoid (also known as trapezium)  

Trapezoid is a quadrilateral in which only one pair of opposite sides are parallel

 

Properties of a trapezoid (trapezium UK)  

1. The middle line of the trapezoid is parallel to the bases and is equal to their half-sum.

2. The bisector of any angle of the trapezoid cuts off on its base (or continuation) an interval equal to the leg.

3. The triangles , formed by segments of the diagonals and the bases of the trapezoid, are similar.

4. Triangles formed by segments of the diagonals and the leg of the trapezoid, have the same area.

5. A circle can be inscribed into a trapezoid if the sum of the bases of the trapezoid is equal to the sum of its lateral sides.

6. The segment connecting the midpoints of the diagonals is equal to the half-difference of the bases and lies on the midline.

7. The intersection point of the diagonals of the trapezoid, the point of intersection of the extensions of its lateral sides and the middle of the bases lie on one straight line.

8. If the sum of the angles at any base of the trapezoid is 90 °, then the segment connecting the midpoints of the bases is equal to their half-difference.

 

 

Area of an isosceles trapezoid 

1.  The area of an isosceles trapezoid if you know sides and angle  

area isosceles trapezoid side

a black  -  lower base

b black  -  upper base

c black  -  equal lateral sides

alpha green  - angle  at the lower base

 

Formula for calculating Area of a trapezoid if given all sides ( A ) :

area isosceles trapezoid side F

 

Formulas for calculating Area of a trapezoid if given sides and angle ( A ) :

area isosceles trapezoid side F1

area isosceles trapezoid side F2

 

 

2.  The area of an isosceles trapezoid in the case of a circle being inscribed in it  

area isosceles trapezoid radius

a , b     -  bases of an isosceles trapezoid

alpha brown , beta blue      -  angles of an isosceles trapezoid

R red  - radius  of the inscribed circle

D green  - diameter  of the inscribed circle

O black  - center  of the inscribed circle

H green  - height

Formulas  for calculating Area of a trapezoid if given radius, diameter or bases ( A ) :

area isosceles trapezoid radius F11   area isosceles trapezoid radius F12

area isosceles trapezoid radius F21   area isosceles trapezoid radius F22

area isosceles trapezoid radius F31   area isosceles trapezoid radius F32

area isosceles trapezoid radius F41   area isosceles trapezoid radius F42

 

For the circle inscribed into an isosceles trapezoid, the following is TRUE:

 area isosceles trapezoid radius F

 

 

3.  The area of an isosceles trapezoid in the case of a circle being inscribed in it and if you know middle line  

area isosceles trapezoid m

a , b     -  bases of an isosceles trapezoid

c black  -  equal lateral sides

R red  - radius  of the inscribed circle

O black  - center  of the inscribed circle

m green  - middle line

 

Formulas  for calculating Area of a trapezoid if given radius, sides or middle line ( A ) :

area isosceles trapezoid m F1

area isosceles trapezoid m F2

area isosceles trapezoid m F31   area isosceles trapezoid m F32

 

For the circle inscribed into an isosceles trapezoid, the following is TRUE:

 area isosceles trapezoid m F

 

 

4.  The area of a isosceles trapezoid if you know diagonals and angle between them

area isosceles trapezoid d

 

d blue1  -  diagonal of a trapezoid

alpha red , beta green -  angles between the diagonals

 

 

Formula for calculating Area of a isosceles trapezoid if given diagonals and angle between them ( A ) :

area isosceles trapezoid d F1   area isosceles trapezoid d F2

 

 

5.  The area of an isosceles trapezoid if you know lateral side, middle line and angle  

area isosceles trapezoid m1

 

c black  -  equal lateral sides

m red  - middle line

alpha green , beta blue - angles at the bases

 

Formula for calculating Area of a trapezoid if given lateral side, middle line and angle ( A ) :

area isosceles trapezoid m1 F1   area isosceles trapezoid m1 F2

 

 

6.  The area of an isosceles trapezoid if you know bases and height  

area isosceles trapezoid h

 

 

a , b     -  bases of an isosceles trapezoid

h- height  measured at right angles to the base

 

Formula for calculating Area of a trapezoid if given bases and height ( A ) :

area isosceles trapezoid h F

 

 

Area of regular polygon

area regular polygon

 

a  -  side

n black  -  number  of sides

 

 

Calculate the Area of a regular polygon if given side and number of sides   ( A ) :

area regular polygon F

 

 

Area of a circle

area circular disc

 

rr1 red -  radius

D green -  diameter

pi1  3,14

 

Calculate the Area of a circle if given radius or diameter ( A ) :

area circular disc F1

area circular disc F2

 

 

Area of a Sector of a Circle

area sector circular disc

rr1 red  -  radius

L  -  length of the arc    AB

alpha blue  -  angle of the sector AOB

pi1  3,14

 

Calculate the Area of sector of circle if given length of arc ( A ) :

area sector circular disc F

Calculate the Area of sector of circle if given central angle in degrees ( A ) :

area sector circular disc F1

 

 

Area of a circular segment

area segment circulat disc

 

R red  -  radius

alpha green-  angle of the segment

pi1  3,14

 

Calculate the Area of segment of circle if given radius and central angle in degrees ( A ) :

area segment circulat disc F

 

 

Area of an annulus

area annulus

 

R red -  radius of the outer circle

rr1 red -  radius of the inner circle

pi1  3,14

 

Calculate the Area of an annulus if given radii of the circles ( A ) :

area annulus Formula

 

 

Area of a sector of an annulus

area sector annulus

R red  -  radius of the outer circle

rr1 red  -  radius of the inner circle

alpha green  -  angle of the sector AOB

pi1    3,14

 

Calculate the Area of a sector of an annulus if given radii of the circles and central angle in degrees ( A ) :

area sector annulus F

 

 

Area of a right triangle

area right triangle

 

 

a , b   -  legs of a right triangle

 

 

 

Calculate the Area of a right triangle if given legs ( A ) :

area right triangle F

 

 

Area of a triangle with height

Area of a triangle

 

h  - height  measured at right angle to the base

a  - base

 

 

Find the Area of a triangle if given height and base ( A ) :

area triangle F

 

 

Area of an isosceles triangle

isosceles triangle

 

b blue  - base  of an isosceles triangle

h  - height  measured at right angles to the base

a  - equal sides

 

Calculate the Area of an isosceles triangle if given sides or height and base ( A ) :

area isosceles triangle F     area isosceles triangle F1