Drawing the altitude of an equilateral triangle decomposes the equilateral triangle into 2 congruent triangles. They are right triangles with acute angles of 30 and 60 degrees. These congruent angles make all triangles with angles 30, 60, and 90 degrees similar by the Angle-Angle Triangle Similarity Theorem.

If we consider a right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side 1 unit long, then the hypotenuse must be 2 units long since the triangle can be thought of as half of an equilateral triangle. Call the length of the altitude . By the Pythagorean Theorem, we can say , so .

Now, consider another right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side  units long. By the Angle-Angle Triangle Similarity Theorem, it must be similar to the right triangle with angles 30, 60, and 90 degrees and with sides 1, , and 2 units long. The scale factor is , so a triangle with angles 30, 60, and 90 degrees has side lengths  and  units long.

Triangle with a vertex above bottom horizontal side. A vertical line drawn from top vertex to horizontal side forming a right angle. The bottom right angle is labeled 60 degrees. The right side is labeled 2 y. The vertical line is labeled y times square root of 3. The right side top angle is labeled 30 degrees. The horizontal side, right of the vertical line to the angle is labeled y.

Triangle A B C. Point D lies on A B. A line is drawn from point C to point D with a right angle symbol as it hits point D. Side B C is labeled 5. Angle C B A is labeled 60 degrees. Angle B C D is labeled 30 degrees.

Triangle E G H. Point F lies on G H. A line is drawn from point E to point F with a right angle symbol as it hits point F. E F is labeled 4. Angle E G H is labeled 60 degrees. Angle G E F is labeled 30 degrees.

In triangle , so . That means  is units and  is units.

In triangle , so . That means  is units and  is , or , units.