The height of the birdhouse is 17.7cm
How to get the total height?
The total height of the birdhouse will be 9.2cm plus the missing cathetus of the right triangle.
For the right triangle we know that the base measures 9.2cm and the hypotenuse measures 12.5cm
Using the Pythagorean theorem we know that the sum of the squares of the catheti is equal to the square of the hypotenuse.
Then we can write:
h² +9.2² = 12.5²
h = √(12.5² - 9.2²)
h = 8.5
Then the total height of the birdhouse is 9.2cm + 8.5cm = 17.7cm