![]() ![]() 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). More generally, the problem of finding all equable triangles with integer sides (that is, equable Heronian triangles) was considered by B. 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). Integer dimensions Ĭombining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. All triangles are tangential, so in particular the equable triangles are exactly the triangles with inradius two. Thus, a tangential polygon is equable if and only if its inradius is two. 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. Solving this yields that x = 4, so a 4 × 4 square is equable.Ī tangential polygon is a polygon in which the sides are all tangent to a common circle. In the case of the square, for instance, this equation is Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. 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. However its common use as GCSE coursework has led to its being an accepted concept. Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then it has an area of 45 square feet (4.2 m 2) and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard). For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both have a unitless numerical value of 30.Īn area cannot be equal to a length except relative to a particular unit of measurement. A two-dimensional equable shape (or perfect shape) is one whose area is numerically equal to its perimeter. ![]()
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