Two posts and one tower
The foot of a tower is hidden. Two readings from one road can still give its height.
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- A tower, a far post and a near post on one line.
- From the far post, the top reads 30°.
- From the near post, the top reads 60°.
- Both triangles share one height.
Two triangles, one height
A post is a marked spot on the road where you stand. A reading is the angle you measure there. Each post makes its own right triangle with the tower. Both triangles have the same tower as their opposite side.
Let the near post be y metres from the tower. Its reading is 60°. Tan 60° = height ÷ y. So height = y × tan 60° = y√3.
The posts stand 60 m apart. Both posts are on one side, in line with the tower. So the far post is y + 60 metres out. Its reading is 30°. Tan 30° = height ÷ (y + 60). So height = (y + 60) × tan 30° = (y + 60) ÷ √3.
Where the two agree
A tower has only one height. So write y√3 = (y + 60) ÷ √3. Multiply both sides by √3. Then 3y = y + 60. Subtract y from both sides: 2y = 60. So y = 30.
The near post is 30 m out. The height is y√3 = 30√3 m.
Rule: Two readings, one tower: the two heights must match.
Check both triangles. Near: 30² + (30√3)² = 3600 = 60². Far: 90² + (30√3)² = 10800 = (60√3)².
Watch out!
The bigger angle is read at the nearer post. A closer post must look up more steeply at the same top.