Quadrilateral Definitions and Examples

Quadrilateral Definitions, Formulas, & Examples

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    Quadrilateral Definitions and Examples

    A quadrilateral is a geometric shape with four sides and four angles. The four sides can be of any length, and the angles can be of any size. There are many different types of quadrilaterals, each with their own properties and definitions. In this blog post, we will explore some of the most common quadrilaterals, their definitions, and examples. From squares and rectangles to trapezoids and parallelograms, we will cover all the basics you need to know about quadrilaterals.

    What is a Quadrilateral?

    A quadrilateral is a two-dimensional shape with four sides and four angles. The four sides can be of any length, and the four angles can be of any size. The most well-known quadrilateral is the rectangle, which has four equal-length sides and four right angles. Other examples of quadrilaterals include squares, parallelograms, rhombuses, and trapezoids.

    Quadrilateral Definition

    A quadrilateral is a geometric figure with four sides and four angles. The sides can be of any length, but all four angles must be 90 degrees for the figure to be considered a true quadrilateral.

    There are many different types of quadrilaterals, each with their own defining characteristics. Some common examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and kites.

    To learn more about the different types of quadrilaterals and see some examples of each, check out the blog post “Quadrilateral Definitions and Examples”.

    Types of Quadrilaterals

    There are many different types of quadrilaterals, each with its own unique properties and definitions. Here are some of the most common types of quadrilaterals:

    -Rectangles: A rectangle is a quadrilateral with four right angles. All sides are equal in length, making it a parallelogram.

    -Squares: A square is a rectangle with all four sides having equal length. This makes it both a parallelogram and a rhombus.

    -Trapezoids: A trapezoid is a quadrilateral with only two sides parallel to each other.

    -Kites: A kite is a quadrilateral with exactly two sets of adjacent sides that are equal in length.

    -Parallelograms: A parallelogram is a quadrilateral with both pairs of opposite sides being parallel to each other.

    Properties of Quadrilaterals

    A quadrilateral is a geometric shape with four sides and four angles. Quadrilaterals are classified by their side lengths and angle measurements.

    There are several types of quadrilaterals, including squares, rectangles, parallelograms, trapezoids, and rhombuses. Each type has its own unique properties that can be used to identify it.

    Squares have all sides of equal length and all angles measuring 90 degrees. Rectangles have two pairs of parallel sides and all angles measuring 90 degrees. Parallelograms have two pairs of parallel sides but the angle measurements can vary. Trapezoids have one pair of parallel sides and the other two sides are not parallel. Rhombuses have two pairs of adjacent sides that are equal in length but the angle measurements can vary.

    Knowing the properties of quadrilaterals can be helpful in many situations, such as when trying to solve a geometry problem or when identifying a shape in the world around us.

    Square

    A square is a four-sided figure with all sides equal in length. A square has four right angles (90°) and all of its sides are of equal length.

    squares also have diagonals that bisect each other at 90° angles. The word “square” can also refer to the resulting figure after someone folds a sheet of paper into quarters and cuts off the excess corners.

    Rectangle

    A rectangle is a four-sided polygon with two pairs of parallel sides. In other words, the opposite sides of a rectangle are always equal in length. Rectangles are often referred to as oblongs, although that term is ambiguous, since it can also refer to other quadrilaterals with unequal sides.

    There are several special types of rectangles:

    Square: A square is a rectangle with all four sides of equal length.

    Rhombus: A rhombus is a parallelogram with all four sides of equal length.

    Kite: A kite is a quadrilateral with two pairs of adjacent sides that are perpendicular to each other.

    Parallelogram

    A parallelogram is a four-sided polygon with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the angles between the parallel sides are also equal.

    One way to think of a parallelogram is as a rectangle with two adjacent sides that are not parallel. That means that all rectangles are parallelograms, but not all parallelograms are rectangles (for example, a rhombus is a type of parallelogram that is not a rectangle).

    Trapezium

    A trapezium is a four-sided polygon with two non-parallel sides, or more generally any quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases, and the other two sides are called the legs. The point where the legs meet is called the vertex.

    Rhombus

    A rhombus is a four-sided figure with two sets of parallel sides. All the sides have equal length, and the angle between any two adjacent sides is 90 degrees. The word “rhombus” comes from the Greek word for “rotund”, which means “round”.

    A square is a special type of rhombus in which all four angles are right angles (90 degrees).

    Kite

    A kite is a geometric shape with four sides and two diagonals. Kites typically have two pairs of congruent sides, meaning the length of each side is equal to the length of the corresponding side on the other side of the kite. The diagonals of a kite intersect at 90-degree angles.

    Kites are named for their similarity to the airborne toy of the same name. Kites are often used as a symbol for freedom or power.

    The most famous kite is probably the one flown by George Washington during the Battle of Trenton, which helped turn the tide of the war in favor of the Americans.

    Area of Quadrilaterals

    A quadrilateral is a four-sided closed figure. The sum of the angles in a quadrilateral is always 360 degrees. There are many types of quadrilaterals, each with its own defining characteristics.

    The area of a quadrilateral is the amount of space enclosed by the figure. To find the area of a quadrilateral, we can use one of several formulas, depending on which information we know about the figure.

    If we know the lengths of all four sides, we can use the formula: Area = s(s-a)(s-b)(s-c) where s is half of the perimeter and a, b, and c are the lengths of the three sides not adjacent to s.

    If we only know the length of two sides and the angle between them (or vice versa), we can use Heron’s Formula: Area = sqrt[s(s-a)(s-b)(s-c)] where a, b, and c are as above and s is again half the perimeter. This formula works for any type of triangle, not just right triangles.

    If we know all four sides but not any angles, we can use another variation on Heron’s Formula: Area = 1/2 ab sin C where C is any angle and a and b are opposite sides. The angle must be given in radians for this formula to work correctly.

    Examples of Quadrilaterals

    A quadrilateral is a four-sided closed figure. The opposite sides of a quadrilateral are parallel and the sum of its angles is 360 degrees. There are many types of quadrilaterals, including parallelograms, rectangles, rhombuses, and squares.

    A parallelogram is a quadrilateral with two pairs of parallel sides. The most common type of parallelogram is the rectangle, which has four right angles.

    A rhombus is a quadrilateral with all four sides the same length. A square is a special type of rhombus in which all four angles are also right angles.

    Conclusion

    There you have it! These are just a few examples of quadrilaterals and how they are defined. As you can see, there is a lot of flexibility when it comes to the shapes and sizes of quadrilaterals. So whether you’re looking for a simple shape to use in your math homework or you’re trying to identify a complex figure in your geometry class, hopefully this article has helped you out.

    Quadrilateral

    Named laminae

    diamond | golden rectangle | golden rhombus | half rectangle | half square | isosceles trapezoid | kite | lozenge | monokite | parallelogram | quarter rectangle | quarter square | rectangle | regular diamond | rhombus | right trapezoid | square | trapezoid | inverted isosceles trapezoid (total: 19)

    Definitions

    Definitions Diamond

    Definitions Golden rectangle

    Definitions Golden rhombus

    Defining inequalities

    abs(x)/a + abs(y)/b<=1

    -a/2<=x<=a/2 and -(a ϕ)/2<=y<=(a ϕ)/2

    2 (1 + sqrt(5)) a + sqrt(2 (5 + sqrt(5))) (2 x + sqrt(5) y + y)>=0 and 2 (1 + sqrt(5)) a + sqrt(2 (5 + sqrt(5))) (-2 x + sqrt(5) y + y)>=0 and 2 sqrt(5 (5 + sqrt(5))) a>=5 sqrt(2) (2 x + sqrt(5) y + y) and 2 (1 + sqrt(5)) a>=sqrt(2 (5 + sqrt(5))) (-2 x + sqrt(5) y + y)

    -a/2<=x<=a/2 and 0<=y<=b/2

    -a/2<=x<=a/2 and 0<=y<=a/2

    0<=y<=h and a y + b h>=b y + 2 h x and a y + b h>=b y - 2 h x

    h (sqrt(a^2 - h^2) + x) + y sqrt(a^2 - h^2)>=0 and sqrt((b - h) (b + h)) (h + y)>=h x and h sqrt(b^2 - h^2)>=y sqrt(b^2 - h^2) + h x and h (sqrt(a^2 - h^2) + x)>=y sqrt(a^2 - h^2)

    y>=0 and a + y>=x and sqrt(2) a>=2 y and x>=y

    sqrt(3) y>=3 x and 2 sqrt(3) a>=x + sqrt(3) y and 2 sqrt(3) a + x>=sqrt(3) y and 3 x + sqrt(3) y>=0

    y>=0 and b sin(A) + y cos(A)>=x sin(A) and a sin(A)>=y and x sin(A)>=y cos(A)

    Alternate names

    isosceles trapezium

    equiangular rectangle

    diamond | equilateral parallelogram | rhomb

    right trapezium

    trapezium

    inverted isosceles trapezium

    Lamina properties

    diamond | (-a, 0) | (0, -b) | (a, 0) | (0, b) golden rectangle | (a/2, -(ϕ a)/2) | (a/2, (ϕ a)/2) | (-a/2, (ϕ a)/2) | (-a/2, -(ϕ a)/2) golden rhombus | (-a/(2 sqrt(5/8 - sqrt(5)/8)), 0) | (0, -a/(2 sqrt(5/8 + sqrt(5)/8))) | (a/(2 sqrt(5/8 - sqrt(5)/8)), 0) | (0, a/(2 sqrt(5/8 + sqrt(5)/8))) half rectangle | (-a/2, 0) | (a/2, 0) | (a/2, b/2) | (-a/2, b/2) half square | (-a/2, 0) | (a/2, 0) | (a/2, a/2) | (-a/2, a/2) isosceles trapezoid | (-b/2, 0) | (b/2, 0) | (a/2, h) | (-a/2, h) kite | (-sqrt(a^2 - h^2), 0) | (0, -h) | (sqrt(b^2 - h^2), 0) | (0, h) lozenge | (0, 0) | (a, 0) | ((1 + 1/sqrt(2)) a, a/sqrt(2)) | (a/sqrt(2), a/sqrt(2)) monokite | (0, 0) | ((sqrt(3) a)/2, (3 a)/2) | (0, 2 a) | (-(sqrt(3) a)/2, (3 a)/2) parallelogram | (0, 0) | (b, 0) | (cos(A) a + b, sin(A) a) | (cos(A) a, sin(A) a) quarter rectangle | (0, 0) | (a/2, 0) | (a/2, b/2) | (0, b/2) quarter square | (0, 0) | (a/2, 0) | (a/2, a/2) | (0, a/2) rectangle | (-a/2, -b/2) | (a/2, -b/2) | (a/2, b/2) | (-a/2, b/2) regular diamond | (-a, 0) | (0, -a) | (a, 0) | (0, a) rhombus | (-cos(θ) a, 0) | (0, -sin(θ) a) | (cos(θ) a, 0) | (0, sin(θ) a) right trapezoid | (0, 0) | (a, 0) | (a, h_2) | (0, h_1) square | (-a/2, -a/2) | (a/2, -a/2) | (a/2, a/2) | (-a/2, a/2) trapezoid | (0, 0) | (b, 0) | ((a^2 - b^2 - c^2 + d^2)/(2 (a - b)), sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (-a + b))) | (-(a^2 - 2 a b + b^2 + c^2 - d^2)/(2 (a - b)), sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (-a + b))) inverted isosceles trapezoid | (-b/2, 0) | (b/2, 0) | (a/2, sqrt(-1/4 (-a + b)^2 + c^2)) | (-a/2, sqrt(-1/4 (-a + b)^2 + c^2))

    diamond | 4 golden rectangle | 4 golden rhombus | 4 half rectangle | 4 half square | 4 isosceles trapezoid | 4 kite | 4 lozenge | 4 monokite | 4 parallelogram | 4 quarter rectangle | 4 quarter square | 4 rectangle | 4 regular diamond | 4 rhombus | 4 right trapezoid | 4 square | 4 trapezoid | 4 inverted isosceles trapezoid | 4

    diamond | a>0 and b>0 golden rectangle | a>0 golden rhombus | a>0 half rectangle | a>0 and b>0 half square | a>0 isosceles trapezoid | 0<a<b and h>0 kite | 0<h<a and 0<h<b lozenge | a>0 monokite | a>0 parallelogram | a>0 and b>0 and 0<A<π quarter rectangle | a>0 and b>0 quarter square | a>0 rectangle | a>0 and b>0 regular diamond | a>0 rhombus | a>0 and 0<θ<π/2 right trapezoid | a>0 and h_1>0 and h_2>0 square | a>0 trapezoid | a>0 and b>0 and c>0 and d>0 and (a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)>0 inverted isosceles trapezoid | 0<b<a and c>(a - b)/2

    diamond | 2 a | 2 b golden rectangle | a sqrt(ϕ^2 + 1) | a sqrt(ϕ^2 + 1) golden rhombus | a/sqrt(5/8 - sqrt(5)/8) | a/sqrt(5/8 + sqrt(5)/8) isosceles trapezoid | 1/2 sqrt((a + b)^2 + 4 h^2) | 1/2 sqrt((a + b)^2 + 4 h^2) kite | sqrt(a^2 - h^2) + sqrt(b^2 - h^2) | 2 h lozenge | sqrt(2 + sqrt(2)) a | sqrt(2 - sqrt(2)) a monokite | 2 a | sqrt(3) a parallelogram | sqrt(a^2 - 2 a b cos(A) + b^2) | sqrt(a^2 + 2 a b cos(A) + b^2) rectangle | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) regular diamond | 2 a | 2 a rhombus | 2 a cos(θ) | 2 a sin(θ) right trapezoid | sqrt(a^2 + h_1^2) | sqrt(a^2 + h_2^2) square | sqrt(2) a | sqrt(2) a trapezoid | sqrt((a^2 (-b) + a b^2 - a c^2 + b d^2)/(b - a)) | sqrt((a^2 (-b) + a b^2 - a d^2 + b c^2)/(b - a)) inverted isosceles trapezoid | sqrt(a b + c^2) | sqrt(a b + c^2)

    square | r = a/2

    square | h = 1/2 (sqrt(2) - 1) a

    golden rectangle | a ϕ half rectangle | b/2 half square | a/2 isosceles trapezoid | h monokite | 2 a parallelogram | a sin(A) quarter rectangle | b/2 quarter square | a/2 rectangle | b square | a trapezoid | sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (b - a)) inverted isosceles trapezoid | sqrt(c^2 - 1/4 (b - a)^2)

    diamond | A = 2 a b golden rectangle | A = a^2 ϕ golden rhombus | A = (2 a^2)/sqrt(5) half rectangle | A = (a b)/2 half square | A = a^2/2 isosceles trapezoid | A = 1/2 h (a + b) kite | A = h (sqrt(a^2 - h^2) + sqrt(b^2 - h^2)) lozenge | A = a^2/sqrt(2) monokite | A = sqrt(3) a^2 parallelogram | A = a b sin(A) quarter rectangle | A = (a b)/4 quarter square | A = a^2/4 rectangle | A = a b regular diamond | A = 2 a^2 rhombus | A = a^2 sin(2 θ) right trapezoid | A = 1/2 a (h_1 + h_2) square | A = a^2 trapezoid | A = ((a + b) sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)))/(4 (b - a)) inverted isosceles trapezoid | A = 1/4 (a + b) sqrt(4 c^2 - (b - a)^2)

    diamond | x^_ = (0, 0) golden rectangle | x^_ = (0, 0) golden rhombus | x^_ = (0, 0) half rectangle | x^_ = (0, b/4) half square | x^_ = (0, a/4) isosceles trapezoid | x^_ = (0, ((2 a + b) h)/(3 (a + b))) kite | x^_ = (1/3 (-sqrt(a^2 - h^2) + sqrt(b^2 - h^2)), 0) lozenge | x^_ = (1/4 (2 + sqrt(2)) a, a/(2 sqrt(2))) monokite | x^_ = (0, (7 a)/6) parallelogram | x^_ = (1/2 (cos(A) a + b), 1/2 sin(A) a) quarter rectangle | x^_ = (a/4, b/4) quarter square | x^_ = (a/4, a/4) rectangle | x^_ = (0, 0) regular diamond | x^_ = (0, 0) rhombus | x^_ = (0, 0) right trapezoid | x^_ = ((a (h_1 + 2 h_2))/(3 (h_1 + h_2)), 1/3 (h_2 + h_1^2/(h_1 + h_2))) square | x^_ = (0, 0) trapezoid | x^_ = (b/2 + ((2 a + b) (c^2 - d^2))/(6 (-a^2 + b^2)), ((2 a + b) sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)))/(6 (-a^2 + b^2))) inverted isosceles trapezoid | x^_ = (0, ((2 a + b) sqrt(-(-a + b)^2 + 4 c^2))/(6 (a + b)))

    Mechanical properties

    diamond | J_x invisible comma x = (a b^3)/3 golden rectangle | J_x invisible comma x = (a^4 ϕ^3)/12 golden rhombus | J_x invisible comma x = (a^4 ϕ)/(3 (ϕ^2 + 1)^2) half rectangle | J_x invisible comma x = (a b^3)/24 half square | J_x invisible comma x = a^4/24 isosceles trapezoid | J_x invisible comma x = 1/12 h^3 (3 a + b) kite | J_x invisible comma x = 1/6 h^3 (sqrt(a^2 - h^2) + sqrt(b^2 - h^2)) lozenge | J_x invisible comma x = a^4/(6 sqrt(2)) monokite | J_x invisible comma x = (37 a^4)/(8 sqrt(3)) parallelogram | J_x invisible comma x = 1/3 a^3 b sin^3(A) quarter rectangle | J_x invisible comma x = (a b^3)/48 quarter square | J_x invisible comma x = a^4/48 rectangle | J_x invisible comma x = (a b^3)/12 regular diamond | J_x invisible comma x = a^4/3 rhombus | J_x invisible comma x = 1/3 a^4 sin^3(θ) cos(θ) right trapezoid | J_x invisible comma x = 1/12 a (h_1 + h_2) (h_1^2 + h_2^2) square | J_x invisible comma x = a^4/12 trapezoid | J_x invisible comma x = -((3 a + b) ((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))^(3/2))/(96 (a - b)^3) inverted isosceles trapezoid | J_x invisible comma x = 1/96 (3 a + b) (4 c^2 - (b - a)^2)^(3/2)

    diamond | J_y invisible comma y = (a^3 b)/3 golden rectangle | J_y invisible comma y = (a^4 ϕ)/12 golden rhombus | J_y invisible comma y = (a^4 ϕ^3)/(3 (ϕ^2 + 1)^2) half rectangle | J_y invisible comma y = (a^3 b)/24 half square | J_y invisible comma y = a^4/24 isosceles trapezoid | J_y invisible comma y = 1/48 h (a + b) (a^2 + b^2) kite | J_y invisible comma y = 1/6 h ((a^2 - h^2)^(3/2) + (b^2 - h^2)^(3/2)) lozenge | J_y invisible comma y = 1/4 (1 + sqrt(2)) a^4 monokite | J_y invisible comma y = (sqrt(3) a^4)/8 parallelogram | J_y invisible comma y = 1/6 a b sin(A) (a^2 cos(2 A) + a^2 + 3 a b cos(A) + 2 b^2) quarter rectangle | J_y invisible comma y = (a^3 b)/48 quarter square | J_y invisible comma y = a^4/48 rectangle | J_y invisible comma y = (a^3 b)/12 regular diamond | J_y invisible comma y = a^4/3 rhombus | J_y invisible comma y = 1/3 a^4 sin(θ) cos^3(θ) right trapezoid | J_y invisible comma y = 1/12 a^3 (h_1 + 3 h_2) square | J_y invisible comma y = a^4/12 trapezoid | J_y invisible comma y = -(sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)) (a^5 - a^4 b + 6 a^3 b^2 - 6 a^2 b^3 - 8 a^2 b c^2 + 8 a^2 b d^2 - 7 a b^4 + 4 a b^2 c^2 - 4 a b^2 d^2 + 3 a c^4 - 6 a c^2 d^2 + 3 a d^4 + 7 b^5 + 4 b^3 c^2 - 4 b^3 d^2 + b c^4 - 2 b c^2 d^2 + b d^4))/(96 (a - b)^3) inverted isosceles trapezoid | J_y invisible comma y = 1/96 (a + b) (a^2 + b^2) sqrt(4 c^2 - (b - a)^2)

    diamond | J_zz = 1/3 a b (a^2 + b^2) golden rectangle | J_zz = 1/12 a^4 ϕ (ϕ^2 + 1) golden rhombus | J_zz = (a^4 ϕ)/(3 (ϕ^2 + 1)) half rectangle | J_zz = 1/24 a b (a^2 + b^2) half square | J_zz = a^4/12 isosceles trapezoid | J_zz = 1/48 h ((a + b) (a^2 + b^2) + 4 h^2 (3 a + b)) kite | J_zz = 1/6 h (a^2 sqrt(a^2 - h^2) + b^2 sqrt(b^2 - h^2)) lozenge | J_zz = 1/12 (3 + 4 sqrt(2)) a^4 monokite | J_zz = (5 a^4)/sqrt(3) parallelogram | J_zz = 1/6 a b sin(A) (2 (a^2 + b^2) + 3 a b cos(A)) quarter rectangle | J_zz = 1/48 a b (a^2 + b^2) quarter square | J_zz = a^4/24 rectangle | J_zz = 1/12 a b (a^2 + b^2) regular diamond | J_zz = (2 a^4)/3 rhombus | J_zz = 1/6 a^4 sin(2 θ) right trapezoid | J_zz = 1/12 a (a^2 (h_1 + 3 h_2) + (h_1 + h_2) (h_1^2 + h_2^2)) square | J_zz = a^4/6 trapezoid | J_zz = (sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)) (a^4 - 4 a^3 b - 3 a^2 (c^2 + d^2) + a b (6 c^2 - 2 d^2) + b^2 (3 b^2 + 3 c^2 - d^2)))/(48 (a - b)^2) inverted isosceles trapezoid | J_zz = 1/96 (a + b) (a^2 + b^2) sqrt(4 c^2 - (a - b)^2) + 1/96 (3 a + b) (4 c^2 - (a - b)^2)^(3/2)

    diamond | J_x invisible comma y = 0 golden rectangle | J_x invisible comma y = 0 golden rhombus | J_x invisible comma y = 0 half rectangle | J_x invisible comma y = 0 half square | J_x invisible comma y = 0 isosceles trapezoid | J_x invisible comma y = 0 kite | J_x invisible comma y = 0 lozenge | J_x invisible comma y = -1/24 (3 + 2 sqrt(2)) a^4 monokite | J_x invisible comma y = 0 parallelogram | J_x invisible comma y = -1/12 a^2 b sin^2(A) (4 a cos(A) + 3 b) quarter rectangle | J_x invisible comma y = -1/64 a^2 b^2 quarter square | J_x invisible comma y = -a^4/64 rectangle | J_x invisible comma y = 0 regular diamond | J_x invisible comma y = 0 rhombus | J_x invisible comma y = 0 right trapezoid | J_x invisible comma y = -1/24 a^2 (h_1^2 + 2 h_2 h_1 + 3 h_2^2) square | J_x invisible comma y = 0 trapezoid | J_x invisible comma y = ((a - b - c - d) (a - b + c - d) (a - b - c + d) (a - b + c + d) (2 b (2 a^2 + c^2) - (a + b) (2 b^2 + 3 c^2) + d^2 (3 a + b)))/(96 (a - b)^3) inverted isosceles trapezoid | J_x invisible comma y = 0

    diamond | r_x = b/sqrt(6) r_y = a/sqrt(6) golden rectangle | r_x = 1/2 sqrt(1/6 (2 + sqrt(5))) a r_y = a/(2 sqrt(3)) golden rhombus | r_x = a/sqrt(6 (ϕ^2 + 1)) r_y = a/sqrt(6 (1/ϕ^2 + 1)) half rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) half square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3)) isosceles trapezoid | r_x = h sqrt((3 a + b)/(6 a + 6 b)) r_y = sqrt(a^2/24 + b^2/24) lozenge | r_x = a/sqrt(6) r_y = 1/2 sqrt(2 + sqrt(2)) a monokite | r_x = 1/2 sqrt(37/6) a r_y = a/(2 sqrt(2)) quarter rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) quarter square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3)) rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) regular diamond | r_x = a/sqrt(6) r_y = a/sqrt(6) rhombus | r_x = (a sin(θ))/sqrt(6) r_y = (a cos(θ))/sqrt(6) right trapezoid | r_x = sqrt(h_1^2 + h_2^2)/sqrt(6) r_y = (a sqrt((h_1 + 3 h_2)/(h_1 + h_2)))/sqrt(6) square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3)) inverted isosceles trapezoid | r_x = sqrt(-((3 a + b) ((b - a)^2 - 4 c^2))/(a + b))/(2 sqrt(6)) r_y = sqrt(a^2 + b^2)/(2 sqrt(6))

    golden rectangle | K = 1/3 a^4 ϕ (1 - (192 sum_(n=1)^∞ tanh(1/2 π (2 n - 1) ϕ)/(2 n - 1)^5)/(π^5 ϕ)) rectangle | K = 1/3 a^3 b (1 - (192 a sum_(n=1)^∞ tanh((π b (2 n - 1))/(2 a))/(2 n - 1)^5)/(π^5 b)) square | K = 1/3 a^4 (1 - (192 sum_(n=1)^∞ tanh(1/2 π (2 n - 1))/(2 n - 1)^5)/π^5)

    Distance properties

    diamond | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) golden rectangle | a ϕ | a | a ϕ | a golden rhombus | a | a | a | a half rectangle | a | b/2 | a | b/2 half square | a | a/2 | a | a/2 isosceles trapezoid | b | 1/2 sqrt((a - b)^2 + 4 h^2) | a | 1/2 sqrt((a - b)^2 + 4 h^2) kite | a | b | b | a lozenge | a | a | a | a monokite | sqrt(3) a | a | a | sqrt(3) a parallelogram | b | a | b | a quarter rectangle | a/2 | b/2 | a/2 | b/2 quarter square | a/2 | a/2 | a/2 | a/2 rectangle | a | b | a | b regular diamond | sqrt(2) a | sqrt(2) a | sqrt(2) a | sqrt(2) a rhombus | a | a | a | a right trapezoid | a | h_2 | sqrt(a^2 + (h_2 - h_1)^2) | h_1 square | a | a | a | a trapezoid | b | d | a | c inverted isosceles trapezoid | b | c | a | c

    diamond | p = 4 sqrt(a^2 + b^2) golden rectangle | p = 2 a (ϕ + 1) golden rhombus | p = 4 a half rectangle | p = 2 a + b half square | p = 3 a isosceles trapezoid | p = sqrt((a - b)^2 + 4 h^2) + a + b kite | p = 2 (a + b) lozenge | p = 4 a monokite | p = (2 + 2 sqrt(3)) a parallelogram | p = 2 (a + b) quarter rectangle | p = a + b quarter square | p = 2 a rectangle | p = 2 (a + b) regular diamond | p = 4 sqrt(2) a rhombus | p = 4 a right trapezoid | p = sqrt(a^2 + (h_2 - h_1)^2) + a + h_1 + h_2 square | p = 4 a trapezoid | p = a + b + c + d inverted isosceles trapezoid | p = a + b + 2 c

    diamond | r = (a b)/sqrt(a^2 + b^2) golden rhombus | r = a/sqrt(5) kite | r = (h (sqrt(a^2 - h^2) + sqrt(b^2 - h^2)))/sqrt((sqrt(a^2 - h^2) + sqrt(b^2 - h^2))^2 + 4 h^2) lozenge | r = a/(2 sqrt(2)) monokite | r = 1/2 (3 - sqrt(3)) a quarter square | r = a/4 regular diamond | r = a/sqrt(2) rhombus | r = a sin(θ) cos(θ) square | r = a/2

    golden rectangle | R = 1/2 a sqrt(ϕ^2 + 1) half rectangle | R = 1/4 sqrt(4 a^2 + b^2) half square | R = (sqrt(5) a)/4 monokite | R = a quarter rectangle | R = 1/4 sqrt(a^2 + b^2) quarter square | R = a/(2 sqrt(2)) rectangle | R = 1/2 sqrt(a^2 + b^2) square | R = a/sqrt(2)

    diamond | 2 max(a, b) golden rectangle | a sqrt(ϕ^2 + 1) golden rhombus | a/sqrt(5/8 - sqrt(5)/8) half rectangle | (sqrt(5) a)/2 half square | (sqrt(5) a)/2 isosceles trapezoid | max(b, 1/2 sqrt((a + b)^2 + 4 h^2)) kite | max(a, b, 2 h, sqrt(a^2 - h^2) + sqrt(b^2 - h^2)) lozenge | sqrt(2 + sqrt(2)) a monokite | 2 a parallelogram | max(sqrt(a^2 - 2 a b cos(A) + b^2), sqrt(a^2 + 2 a b cos(A) + b^2)) quarter rectangle | 1/2 sqrt(a^2 + b^2) quarter square | a/sqrt(2) rectangle | sqrt(a^2 + b^2) regular diamond | 2 a rhombus | 2 a max(cos(θ), sin(θ)) right trapezoid | max(sqrt(a^2 + h_1^2), sqrt(a^2 + h_2^2)) square | sqrt(2) a trapezoid | max(b, sqrt((a^2 b - a b^2 + a d^2 - b c^2)/(a - b)), sqrt((a^2 b - a b^2 + a c^2 - b d^2)/(a - b))) inverted isosceles trapezoid | max(b, sqrt(a b + c^2))

    diamond | χ = 1 golden rectangle | χ = 1 golden rhombus | χ = 1 half rectangle | χ = 1 half square | χ = 1 isosceles trapezoid | χ = 1 kite | χ = 1 lozenge | χ = 1 monokite | χ = 1 parallelogram | χ = 1 quarter rectangle | χ = 1 quarter square | χ = 1 rectangle | χ = 1 regular diamond | χ = 1 rhombus | χ = 1 right trapezoid | χ = 1 square | χ = 1 trapezoid | χ = 1 inverted isosceles trapezoid | χ = 1

    monokite | s^_ = 1/6 a (4 + 3 log(3)) square | s^_ = 1/15 a (2 + sqrt(2) + 5 sinh^(-1)(1))

    square | A^_ = (11 a^2)/144

    Common classes

    convex laminae | polygonal laminae | quadrilateral laminae

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