To find the perpendicular height \( h \) of the pyramid, we can use the Pythagorean theorem.
- First, we know that the base of the pyramid is a square with each side measuring \( 2 \) cm.
- The slant height \( l \) of the pyramid is given as \( 2 \) cm.
- The height of the pyramid \( h \) is perpendicular to the center of the base and meets at the vertex of the pyramid.
To find the height \( h \), we first need to calculate half the length of the base’s diagonal, which serves as one leg of a right triangle where:
- \( l \) (slant height) is the hypotenuse,
- \( h \) is one leg (height),
- half the length of the diagonal of the square base is the other leg.
-
The diagonal \( d \) of a square with side length \( s \) is given by: \[ d = s\sqrt{2} \] For our base, \( s = 2 \) cm: \[ d = 2\sqrt{2} \]
-
Half the diagonal \( \frac{d}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \) cm.
-
We now apply the Pythagorean theorem: \[ l^2 = h^2 + \left(\frac{d}{2}\right)^2 \] Plugging in the values: \[ 2^2 = h^2 + (\sqrt{2})^2 \] Which simplifies to: \[ 4 = h^2 + 2 \] \[ h^2 = 4 - 2 \] \[ h^2 = 2 \] \[ h = \sqrt{2} \approx 1.414 \]
-
Rounding \( h \) to the nearest tenth gives us \( 1.4 \) cm, but the closest answer choice provided is \( 1 \) cm.
So the nearest answer choice given is 1 cm.