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A curve of constant width defined by an 8th-degree polynomial. Circles have constant width, equal to their diameter.On the other hand, squares do not: supporting lines parallel to two opposite sides of the square are closer together than supporting lines parallel to a diagonal.
For a circle, the width is the same as the diameter; a circle of width w has perimeter π w. A Reuleaux triangle of width w consists of three arcs of circles of radius w. Each of these arcs has central angle π /3, so the perimeter of the Reuleaux triangle of width w is equal to half the perimeter of a circle of radius w and therefore is equal ...
Its width, approximately 1.54, is the root of a degree-6 polynomial with integer coefficients. [17] [19] [20] Just as it is possible for a circle to be surrounded by six congruent circles that touch it, it is also possible to arrange seven congruent Reuleaux triangles so that they all make contact with a central Reuleaux triangle of the same size.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
The Blaschke–Lebesgue theorem states that this is the unique minimum possible area of a curve of constant width, and the Blaschke–Lebesgue inequality states that every convex set of width has area at least this large, with equality only when the set is bounded by a Reuleaux triangle. [1]
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