45-45-90 Triangle
A triangle that has angle measures of 45º, 45º, and 90º. The three sides in these triangle always fit into a specific proportion. If the side opposite either 45º angle is labeled x, then the side opposite the other 45º angle is also x, and the side opposite the 90º angle is x√2.
30-60-90 Triangle
A triangle that has angle measures of 30º, 60º, and 90º. The three sides in these triangles always fit into a specific proportion. If the side opposite the 30º angle is labeled x, then the side opposite the 60º is x√3, and the side opposite the 90º angle is 2x.
Isosceles Triangle
A triangle in which two of the three angles are equal; in addition, the sides opposite the two angles are equal in length.
Equilateral Triangle
A triangle in which all three angles are equal; in addition, all three sides are of equal length.
Area of an Equilateral Triangle
In addition to the standard area formula for triangles, equilateral triangles have a special formula for area: S^2 * √3 / 4, where S is the length of any side of the equilateral triangle.
Relationship of the Sides of a Triangle
The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. For example, given a triangle with sides 3, 4, and 5, 4 is an acceptable length because it is larger than the positive difference of 5 and 3 (5 - 3 = 2), and it is also smaller than the sum of 5 and 3 (5 + 3 = 8). By contrast, given a triangle with sides 3, 4, and 8, 8 is not an acceptable length because it is larger than the sum of the other two sides (3 + 4 = 7). The dimensions 3, 4, and 8, then, do not form a triangle.
Diagonal of a Square
The diagonal of any square can be found by multiplying the length of one side by the square root of 2.
Main Diagonal of a Cube
The main diagonal of a cube is the one that cuts through the center of the cube; the diagonal of a face of a cube is not the main diagonal. The main diagonal of any cube can be found my multiplying the length of one side by the square root of 3.
Main Diagonal of a Rectangular Solid
The main diagonal of a rectangular solid is the one that cuts through the center of the solid; the diagonal of a face of the rectangular solid is not the main diagonal. The main diagonal of a rectangular solid can be found by using the “Deluxe” Pythagorean Theorem: x^2 + y^2 + z^2 = d^2, where x, y, and z are the length, width, and height of the rectangular solid, and d is the main diagonal.
Similar Triangles
Triangles in which the three angles are identical. It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º. See Proportionality of Similar Triangles (below) for additional information about similar triangles.
Proportionality of Similar Triangles
In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
