The height of the right quadrangular pyramid is 10 cm and makes an angle

The height of the right quadrangular pyramid is 10 cm and makes an angle of 45 degrees with the lateral edge to find the total surface area of the pyramid.

The solution of the problem:

Let us define the diagonal d of the base.
d / 2 = H = 10 cm.

d = 2 * 10 = 20 cm.

Determine the side of the base a.
2a ^ 2 = d2;

2a ^ 2 = 400;

a ^ 2 = 200;

a = 10√ 2.

Let’s define an edge b.
b ^ 2 = H ^ 2 + (d / 2) ^ 2 = 10 ^ 2 + 10 ^ 2 = 100 + 100 = 200;

b = √200 = 10√2.

Let’s define the apothem with.
c ^ 2 = b ^ 2 – (a / 2) ^ 2;

c ^ 2 = 200 – (10√ 2: 2) 2 = 200 – 50 = 150;

c = 5√6 cm.

Let us define the surface area S.
S = a ^ 2 + 4 * (10√ 2 * 5√6 / 2);

S = 200 + 10√ 2 * 10√ 6;

S = 200 + 200√3 = 200 + 346;

S = 546 cm2

Answer: surface area 546 cm2.