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In geometry, an isosceles triangle (/ aɪ ˈ s ɒ s ə l iː z /) is a triangle that has two sides of equal length or two angles of equal measure. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.
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.
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 difference in area for an equilateral triangle is small, just over 1%, [2] but as Howard Eves pointed out, for an isosceles triangle with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles. [3] In fact, the Malfatti circles are never optimal.
Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles). Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one other side have lengths in the ...
The faces of the icosahedron are eight congruent equilateral triangles with the short side length, and twelve congruent obtuse isosceles triangles with one long edge and two short edges. [8] Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex. [9]
A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will be coplanar and form a cubic volume. This can be seen by adding the 6 edges of tetrahedron inside of a cube.
By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral. The area of the Steiner ellipse equals the area of the triangle times 4 π 3 3 , {\displaystyle {\frac {4\pi }{3{\sqrt {3}}}},} and hence is 4 times the ...
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