{{Short description|Shape with the same area and perimeter}} {{CS1 config|mode=cs1}} [[File:Equable triangle animation.gif|thumb|Every equable triangle has an inradius of 2.<ref name="kilmer">{{cite journal |last=Kilmer |first=Jean E. |journal=The Mathematics Teacher |title=Triangles of Equal Area and Perimeter and Inscribed Circles |year=1988 |jstor=27965678 |volume=81 |issue=1 |pages=65–70|doi=10.5951/MT.81.1.0065 }}</ref><ref name="wilson">{{cite web |last=Wilson |first=Jim |title=Perfect triangles |publisher=University of Georgia |url=http://jwilson.coe.uga.edu/emt725/Perfect/PerTri.html |url-status=dead |archiveurl=https://web.archive.org/web/20120502055601/http://jwilson.coe.uga.edu/emt725/perfect/PerTri.html |archivedate=2012-05-02 }}. See also Wilson's list of [http://jwilson.coe.uga.edu/emt725/Perfect/sol.html solutions]</ref>]]
A two-dimensional '''equable shape''' (or perfect shape) is one whose area is numerically equal to its perimeter.<ref>{{cite book |last=Bradley |first=Christopher J. |date=2005 |title=Challenges in Geometry: For Mathematical Olympians Past and Present |publisher=Oxford University Press |isbn=0-19-856692-1 |page=15}}</ref> For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both with a unitless numerical value of 30.
==Scaling and units== An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then it has an area of {{convert|45|sqft|m2}} and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard). Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. However its common use as GCSE coursework has led to its being an accepted concept. For any shape, there is a similar equable shape: if a shape ''S'' has perimeter ''p'' and area ''A'', then scaling ''S'' by a factor of ''p/A'' leads to an equable shape. Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. In the case of the square, for instance, this equation is
:<math>\displaystyle x^2 = 4x.</math>
Solving this yields that ''x'' = 4, so a 4 × 4 square is equable.
==Tangential polygons== A tangential polygon is a polygon in which the sides are all tangent to a common circle. Since a circle can be defined by any three non-colinear points, all triangles are necessarily tangential. Indeed, every tangential polygon may be triangulated by drawing edges from the circle's center to the polygon's vertices, forming a collection of triangles that all have height equal to the circle's radius; it follows from this decomposition that the total area of a tangential polygon equals half the perimeter times the radius. Thus, a tangential polygon is equable if and only if its inradius is exactly two.<ref name="kilmer"/><ref name="wilson"/>
==Integer dimensions== thumb|450px|The only 5 equable integer triangles,<ref name="dickson">{{harvtxt|Dickson|2005}}, p. 199</ref><ref name="markowitz">{{cite journal |last=Markowitz |first=L. |date=1981 |title=Area = Perimeter |journal=The Mathematics Teacher |volume=74 |issue=3 |pages=222–223|doi=10.5951/MT.74.3.0222 }}</ref> the first 2 being the only 2 equable integer ''right'' triangles.<ref name="bicycle">{{cite book |last1=Konhauser |first1=Joseph D. E. |last2=Velleman |first2=Dan |last3=Wagon |first3=Stan |authorlink3=Stan Wagon |date=1997 |title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries |contribution=95. When does the perimeter equal the area? |volume=18 |series=Dolciani Mathematical Expositions |publisher=Cambridge University Press |isbn=9780883853252 |page=29 |url=https://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29}}</ref> Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10).<ref name="bicycle"></ref>
More generally, the problem of finding all equable triangles with integer sides (that is, equable Heronian triangles) was considered by B. Yates in 1858.<ref>{{cite journal |last=Yates |first=B. |date=1858 |title=Quest 2019 |journal=The Lady's and Gentleman's Diary |page=83}}</ref><ref>{{cite book |last=Dickson |first=Leonard Eugene |authorlink=Leonard Eugene Dickson |date=2005 |title=History of the Theory of Numbers, Volume Il: Diophantine Analysis |publisher=Courier Dover Publications |isbn=9780486442334 |page=195}}</ref> As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).<ref name="dickson"></ref><ref name="markowitz"></ref>
The only equable rectangles with integer sides are the 4 × 4 square and the 3 × 6 rectangle.<ref name="bicycle"/> An integer rectangle is a special type of polyomino, and more generally there exist polyominoes with equal area and perimeter for any even integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.<ref>{{cite book |last=Picciotto |first=Henri |date=1999 |title=Geometry Labs |publisher=MathEducationPage.org |page=208 |url=https://books.google.com/books?id=7gTMKr7TT6gC&pg=PA208}}</ref>
== Equable solids == In three dimensions, a shape is equable when its surface area is numerically equal to its volume. An example is a cube with side length six. As with equable shapes in two dimensions, an equable solid may be found by scaling any solid by an appropriate factor.
==References== {{reflist}}
Category:Geometric shapes