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Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. [1] In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya (section 2.6). [2] Although simple, this formula is only useful if the height can be readily found, which is not always the case.
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 segment from the center to the midpoint of one of its sides.
For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.. Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1]
This (finite) edge and (infinite) line extension are called, respectively, the base and extended base of the altitude. The point at the intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude" or "height", symbol h, is the distance between the foot and ...
Area is the measure of a region's size on a surface.The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.
The circumference is 2 π r, and the area of a triangle is half the base times the height, yielding the area π r 2 for the disk. Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , [ 2 ] but did not identify ...
A skeletal pyramid with its base highlighted. In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured, or on what is considered to be the "bottom" of the figure. [1]
This is because twice the area equals any base times the corresponding height: 2 times the area thus equals both ab and cd where d is the height from the hypotenuse c. The three side lengths of a primitive triangle are coprime, so d = a b / c {\displaystyle d=ab/c} is in fully reduced form; since c cannot equal 1 for any primitive Pythagorean ...