The top portion that is removed is actually a smaller right pyramid that is geometrically similar to the whole pyramid.

Since they are similar, the ratio of the corresponding dimensions must be the same. i.e ratio of corresponding sides are same.

Now, area of the horizontal top surface of the frustum = base area of small pyramid = 7cm²


Height of small pyramid = height of big pyramid - height of frustrum

= H cm - 8 cm
= (H - 8) cm


Ratio of height of small to big pyramid
= (H - 8) : H

Since the ratio of their base areas = 7 : 63
= 1 : 9,

Ratio of corresponding dimensions / sides = √1 : √9 = 1 : 3


So (H - 8) : H = 1 : 3

This means that H = 3(H - 8)

H = 3H - 24
24 = 3H - H
24 = 2H

H = 12

Alternatively,

Since ratio of heights = (H - 8) : H,

Ratio of areas = (H - 8)² : H²

But ratio of areas also = 7 : 63 = 1 : 9
(Comparing their base areas)

So (H - 8)² : H² = 1 : 9

H² = 9(H - 8)²

H² = 9(H² - 16H + 64)

H² = 9H² - 144H + 576

8H² - 144H + 576 = 0

H² - 18H + 72 = 0

(H - 6)(H - 12) = 0

So H = 6
(rejected as that would mean H - 8 = -2, which is negative.

The height of small pyramid cannot be negative or 0 i.e H - 8 > 0 → H > 8

Or in another perspective, the height of the right pyramid has to be greater than height of the frustum (H > 8)

or

H = 12 (shown)