So the answer is that there are equilateral triangles with any area up to (but, I guess, not including), the surface area of the entire sphere. And like plane triangles, angles A, B, and C are also in angular units.. So triangles and are equivalent by dissection.Therefore, spherical triangles with the s Triangles and have a common base and the same area, so they are equivalent by dissection (as shown in Dissection of Two Spherical Triangles with a Common Base). (We say that the sphere is locally flat.) Agreat!many!spherical!triangles!can!be!solved!using!these!two!laws,!but!unlike!planar! Note that for spherical triangles, sides a, b, and c are usually in angular units. For example, start at the north pole. That is, the triangle has 3 sides of given equal length s, each of which is a portion of a great circle. Contributed by: Izidor Hafner (March 2017) Open content licensed under CC BY … If I understand the question… you want to know how many equilateral triangles you could fit into a sphere, where the sides of the triangle are the radius of he sphere. triangles,!some!require!additional!techniques!knownas!the!supplemental! If the triangle is very small (compared to the size of the sphere), the effects of curvature are negligible. By symmetry, both triangles must have the same area. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Therefore in our equilateral triangle, the interior angles are 60 degrees. Sum of interior angles of spherical triangle The sum of the interior angles of a spherical triangle is greater than 180° and less than 540°. For solving the problem in spherical triangle, we need to use spherical geometry/spherical trigonometry. We can shrink the red triangle as small as we want and make the blue triangle as close as we want to the entire sphere. The triangle A ′ ⁢ B ′ ⁢ C ′ is antipodal to A ⁢ B ⁢ C since it can be obtained by reflecting the original one through the center of the sphere. Triangles and also have a common base and the same area, so they are also equivalent by dissection. This Demonstration shows three spherical triangles. Since the area of a sphere is 4 R², if n equilateral spherical triangles tessellate a sphere into n parts, the area of each triangle is: Let us apply this to the spherical equivalent of the icosahedron on the left, and determine the spherical angle ‘a’ at each vertex of the sphere. Suppose an equilateral triangle is drawn on the surface of the earth ... How to find the area of a spherical equilateral triangle? We can construct a triangle with three 90-degree angles. The excess, or area, of small triangles is very small. This Demonstration shows a dissection of a spherical equilateral triangle into three congruent spherical isosceles triangles. Ask Question Asked 3 years, 9 months ago. Equilateral Triangle on the Surface of a Sphere—C.E. For example, consider an equilateral spherical triangle with sides of 60 km on a spherical Earth of radius 6371 km; the side corresponds to an angular distance of 60/6371=.0094, or approximately 10 −2 radians (subtending an angle of 0.57 Note that by continuing the sides of the original triangle into full great circles, another spherical triangle is formed. Mungan, Fall 2004 The purpose of this exercise is to compute the interior angle α and area A of an equilateral triangle on the surface of a sphere of unit radius.