Calculate Kite Area: Given Triangle Area 30 cm² and Diagonals 5 cm, 6 cm

To solve this problem, we'll use the properties of triangle and kite areas:

Firstly, we note that the area of triangle $ABD$ is given as $30 \, \text{cm}^2$.

We recognize that triangles $ABD$ and $ADC$ together form the kite with diagonal $AC$, and triangles $ADB$ and $BCD$ form diagonals where two triangles area will be half the kite’s complete area.

Now, find triangle $ABE$: $\frac{1}{2} \times AE \times BD = 30 \implies \frac{1}{2} \times 5 \times BD = 30 \implies BD = 12 \, \text{cm}$

Next, calculate diagonal $AC$ (sum of segments): $AE + EC = AC \implies 5 + 6 = AC \implies AC = 11 \, \text{cm}$

The area of kite $ABCD$ from diagonals $AC$ and $BD$: $\text{Area}_{kite} = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 11 \times 12 = 66 \, \text{cm}^2$

Therefore, the solution to the problem is 66 cm².