Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math: FAQ

   Quadrilateral Formulas   

_____________________________________________
Geometric Formulas: Contents || Ask Dr. Math || Dr. Math FAQ || Search Dr. Math
_____________________________________________

general || square || rectangle || parallelogram || rhombus || trapezoid || kite || cyclic quadrilateral

also see Defining Geometric Figures

  Quadrilateral
 

  A polygon (plane figure) with 4 angles
and 4 sides.

Sides: a, b, c, d
Angles: A, B, C, D
Around the quadrilateral are a, A, b, B, c, C, d, D, and back to a, in that order

Altitudes: ha , etc.
Diagonals: p = BD, q = AC, intersect at O
Angle between diagonals: theta

Perimeter: P
Semiperimeter: s
Area: K

Radius of circumscribed circle: R
Radius of inscribed circle: r

 

 
To read about quadrilaterals, visit The Geometry Center. 




  General

 
 

   
P = a + b + c + d.
s = P/2 = (a+b+c+d)/2
 
A + B + C + D = 2 Pi radians = 360o
 
K = pq sin(theta)/2
K = (b2+d2-a2-c2)tan(theta)/4
K = sqrt[4p2q2-(b2+d2-a2-c2)2]/4
 
K = sqrt[(s-a)(s-b)(s-c)(s-d)-abcd
     cos2([A+C]/2)]
     (Bretschneider's Formula)



  Square


 

 
 
A quadrilateral with four right angles and all four sides of equal length.

a = b = c = d
A = B = C = D = Pi/2 radians = 90o
theta = Pi/2 radians = 90o

    ha = a
p = q = a sqrt(2)
P = 4a
s = 2a
K = a2
R = a sqrt(2)/2
r = a/2
   

 
JavaSketchpad exploration: Square




  Rectangle

 
 
A quadrilateral with adjacent sides perpendicular
(all four angles are therefore right angles).

a = c, b = d.
A = B = C = D = Pi/2 radians = 90o

     
ha = b
hb = a
p = q = sqrt(a2+b2)
theta = 2 arctan(a/b)
P = 2(a+b)
s = a + b
K = ab
R = p/2 = sqrt(a2+b2)/2
r = minimum(a,b)/2
   

 
JavaSketchpad exploration:
Rectangle




  Parallelogram  
   
A quadrilateral with opposite sides parallel.

a = c, b = d
A = C, B = D
 



 

    A + B = Pi radians = 180o
ha = b sin(A) = b cos(B-Pi/2)
hb = a sin(A) = a cos(B-Pi/2)
 
p = sqrt[a2+b2-2ab cos(A)]
q = sqrt[a2+b2-2ab cos(B)]
p2+q2 = 2(a2+b2)
theta = arccos([a2-b2]/pq)
P = 2*(a+b)
 
s = a + b
K = ab sin(A) = ab sin(B) = bhb
   = pq sin(theta)/2
   

 
JavaSketchpad exploration: Parallelogram




  Rhombus

   
A parallelogram with all sides equal.

a = b = c = d
A = C, B = D
theta = Pi/2 radians = 90o
 

    A + B = Pi radians = 180o
ha = a sin(A) = a cos(B-Pi/2)
ha = hb
 
p = a sqrt[2-2 cos(A)]
q = a sqrt[2-2 cos(B)]
 
p2+q2 = 4a2
P = 4a
s = 2a
K = a2sin(A) = a2sin(B)
     = aha = pq/2
   

 
JavaSketchpad exploration: Rhombus




  Trapezoid (American)
  Trapezium (British)
*

   
a parallel to c, m = (a+c)/2
A + B = C + D = Pi radians = 180o
 
        P = a + b + c + d

    K = ham = ha(a+c)/2

  If a = c, the trapezoid is actually a parallelogram, so b = d, and the height and area cannot be determined from a, b, c, and d alone. If a and c are not equal, then

        ha2 = (a+b-c+d)(-a+b+c+d)(a-b-c+d)(a+b-c-d)/[4(a-c)2].

  If ha2 < 0, no trapezoid having those side lengths exists.



*From The Words of Mathematics by Steven Schwartzman (1994, Mathematical Association of America):

trapezoid (noun); trapezoidal (adjective); trapezium, plural trapezia (noun): The Greek word trapeza "table" was composed of tetra "four" and the Indo-European root ped- "foot." A Greek table must have had four feet (= legs). The suffix -oid (q.v.) means "looking like," so that a trapezoid is a figure that looks like a table (at least in somebody's imagination). Some Americans define a trapezoid as a quadrilateral with at least one pair of parallel sides. Under that definition, a parallelogram is a special kind of trapezoid. For other Americans, however, a trapezoid is a quadrilateral with one and only one pair of parallel sides, in which case a parallelogram is not a trapezoid. The situation is further confused by the fact that in Europe a trapezoid is defined as a quadrilateral with no sides equal. Even more confusing is the existence of the similar word trapezium, which in American usage means "a quadrilateral with no sides equal," but which in European usage is a synonym of what Americans call a trapezoid. Apparently to cut down on the confusion, trapezium is not used in American textbooks. The trapeze used in a circus is also related, since a trapeze has or must once have had four "sides": two ropes, the bar at the bottom, and a support bar at the top.

   

JavaSketchpad exploration: Trapezoid




  Kite

   
A quadrilateral with two pairs of distinct
adjacent sides equal in length.

a = b, c = d
theta = Pi/2 radians = 90o
  

    OB = OD = p/2, OA = h, OC = q - h
 
h = sqrt(a2-p2/4)
q = sqrt(a2-p2/4) + sqrt(c2-p2/4)
 
P = 2(a+c)
K = pq/2
   

 
JavaSketchpad exploration: Kite





  Cyclic Quadrilateral

   
A quadrilateral all of whose vertices lie on a circle.

Points A, B, C, and D lie on a circle of radius R.
A + C = B + D = Pi radians = 180o
  

    K = sqrt[(s-a)(s-b)(s-c)(s-d)]
     (Brahmagupta's Formula)
K = sqrt[(ab+cd)(ac+bd)(ad+bc)]/4*R
 
p = sqrt[(ac+bd)(ad+bc)/(ab+cd)]
q = sqrt[(ab+cd)(ac+bd)/(ad+bc)]
 
R = sqrt[(ab+cd)(ac+bd)(ad+bc) /
     (s-a)(s-b)(s-c)(s-d)]/4
 
theta = arcsin[2K/(ac+bd)]




  Cyclic-Inscriptable

   
A quadrilateral within which a circle can be
inscribed, tangent to all four sides.

Points A, B, C, and D lie on a circle of radius R.
Sides a, b, c, and d are tangent to a circle of radius r.
m = distance between the centers of the two circles.

A + C = B + D = Pi radians = 180o
a + c = b + d

   
K = sqrt[abcd]

r = sqrt[abcd]/s
R = sqrt[(ab+cd)(ac+bd)(ad+bc)/abcd]/4
1/(R+m)2 + 1/(R-m)2 = 1/r2

Back to Contents

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. Math ®
© 1994-2014 Drexel University. All rights reserved.
http://mathforum.org/
The Math Forum is a research and educational enterprise of the Drexel University School of Education.The Math Forum is a research and educational enterprise of the Drexel University School of Education.