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Sonic Geometry: Difference between revisions
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==Triangle 180 Degrees== | ==Triangle 180 Degrees== | ||
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. An equilateral triangle has three sides of the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. <ref>[https://en.wikipedia.org/wiki/Triangle Triangle wiki]</ref> | [[File:Regular polygon 3 annotated.svg|thumb|Equilateral triangle with annotation.]] | ||
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. An equilateral triangle has three sides of the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise. In rigorous treatments, a triangle is therefore called a 2-simplex (see also Polytope). Elementary facts about triangles were presented by Euclid, in books 1–4 of his Elements, written around 300 BC. | |||
The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal). The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle, given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees. | |||
<ref>[https://en.wikipedia.org/wiki/Triangle Triangle wiki]</ref> | |||
==Square 360 Degrees== | ==Square 360 Degrees== | ||
[[File:Regular polygon 4 annotated.svg|thumb|Square with annotation.]] | |||
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. | In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. | ||
* Opposite sides of a square are both parallel and equal in length. | * Opposite sides of a square are both parallel and equal in length. | ||
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The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.<ref>[https://en.wikipedia.org/wiki/Heptagon Heptagon wiki]</ref> | The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.<ref>[https://en.wikipedia.org/wiki/Heptagon Heptagon wiki]</ref> | ||
==Octagon 1080 Degrees== | |||
[[File:Regular polygon 8 annotated.svg|thumb|Regular octagon with annotation.]] | |||
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. | |||
If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other). | |||
The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.<ref>[https://en.wikipedia.org/wiki/Octagon Octagon wiki]</ref> | |||
==References== | ==References== | ||