1. **State the problem:** We have a square pyramid with a base area of 64 cm² and a slant height of 9 cm. We need to find the height $a$ of the pyramid. 2. **Recall formulas:** - The base is a square, so if the side length is $s$, then the base area $A = s^2$. - The slant height $l$ is the length from the apex to the midpoint of a base edge. - The height $a$ is the perpendicular distance from the apex to the center of the base. 3. **Find the side length $s$ of the base:** $$s^2 = 64 \implies s = \sqrt{64} = 8 \text{ cm}$$ 4. **Relate height, slant height, and half the base side:** The height $a$, half the base side $\frac{s}{2}$, and the slant height $l$ form a right triangle: $$a^2 + \left(\frac{s}{2}\right)^2 = l^2$$ 5. **Substitute known values:** $$a^2 + \left(\frac{8}{2}\right)^2 = 9^2$$ $$a^2 + 4^2 = 81$$ $$a^2 + 16 = 81$$ 6. **Solve for $a^2$:** $$a^2 = 81 - 16 = 65$$ 7. **Find $a$:** $$a = \sqrt{65} \approx 8.06 \text{ cm}$$ **Final answer:** The height of the pyramid is approximately $8.06$ cm.