The problem asks for the surface area of a pyramid with a square base, given its net. The net consists of a square with side length $15$ cm and four congruent triangles, each with base $15$ cm and height $20$ cm. We need to find the total surface area of this pyramid.

1. Problem Description

The problem asks for the surface area of a pyramid with a square base, given its net. The net consists of a square with side length $15$ cm and four congruent triangles, each with base $15$ cm and height $20$ cm. We need to find the total surface area of this pyramid.

2. Solution Steps

First, find the area of the square base.

The area of a square is given by

$Area_{square} = side^2$.

Since the side length of the square is $15$ cm, its area is:

$Area_{square} = (15 cm)^2 = 225 cm^2$.

Next, find the area of one of the triangular faces.

The area of a triangle is given by

$Area_{triangle} = \frac{1}{2} \times base \times height$.

The base of each triangle is $15$ cm and the height is $20$ cm. Therefore, the area of one triangle is:

$Area_{triangle} = \frac{1}{2} \times 15 cm \times 20 cm = 150 cm^2$.

Since there are four identical triangular faces, the total area of the triangular faces is

$Area_{triangles} = 4 \times Area_{triangle} = 4 \times 150 cm^2 = 600 cm^2$.

Finally, the total surface area of the pyramid is the sum of the area of the square base and the total area of the four triangular faces:

$Area_{total} = Area_{square} + Area_{triangles} = 225 cm^2 + 600 cm^2 = 825 cm^2$.

3. Final Answer

$825$ cm$^2$