30-60-90 Triangle - Rules, Formula, Theorem, Sides, Examples - Free Printable

Educational worksheet: 30-60-90 Triangle - Rules, Formula, Theorem, Sides, Examples. Download and print for classroom or home learning activities.

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- The triangle ABC is equilateral, with all sides equal to `a` and all angles equal to 60°.
- The line AD is an altitude from vertex A to side BC, which also bisects angle A (into two 30° angles) and side BC (into two segments of length `a/2` each).
- This creates two congruent right triangles, ABD and ACD, each with angles of 30°, 60°, and 90°.
- In a 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse. Here, BD = `a/2` is opposite the 30° angle at A, and AB = `a` is the hypotenuse, which confirms the ratio.
- The side opposite the 60° angle (AD) is `√3/2` times the hypotenuse. Therefore, the height AD = `(√3/2) * a`.
- The area of triangle ABC can be calculated as `(1/2) * base * height = (1/2) * a * (√3/2 * a) = (√3/4) * a²`.

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