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Apothem of a hexagon Graphs of side, s; apothem, a; and area, A of regular polygons of n sides and circumradius 1, with the base, b of a rectangle with the same area. The green line shows the case n = 6. The apothem (sometimes abbreviated as apo [1]) of a regular polygon is a line
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.
Given a triangle with sides of length a, b, and c, if a 2 + b 2 = c 2, then the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a 2 + b 2 = c 2, there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
Napoleon's theorem: If the triangles centered on L, M, N are equilateral, then so is the green triangle.. In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.
Let ABC be an equilateral triangle whose height is h and whose side is a. Let P be any point inside the triangle, and s, t, u the perpendicular distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA. Now, the areas of these triangles are , , and . They exactly fill the enclosing ...
For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem [4]: p. 72 (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n = 3 case. [5] [6]
In terms of the circumradius R, the area is: [1] = = The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is: =