One vantage, two sight lines

A statue stands on a pedestal, the stand under it. How tall is the statue alone?

Two triangles, one walk

Find the height of pedestal plus statue. Then subtract the pedestal.

Both sight lines start at the same spot. So both triangles have the same walk. The lower line reads the top of the pedestal. The upper line reads the top of the statue.

Each line makes its own right triangle with the ground. Solve them one at a time.

Subtract

Together, the pedestal and the statue make the whole column.

Take the walk as 30 m. The pedestal top reads 30°. Tan 30° = pedestal ÷ 30. So the pedestal is 30 × tan 30° = 30 ÷ √3 m. That is 30√3 ÷ 3 = 10√3 m.

The statue's top reads 60°. Tan 60° = column ÷ 30. So the whole column is 30 × √3 = 30√3 m tall.

Rule: Statue = whole column − pedestal.

So the statue is 30√3 − 10√3 = 20√3 m. Check the triangles: 30² + (10√3)² = 1200 = (20√3)², and 30² + (30√3)² = 3600 = 60².