What is the surface area of a square pyramid with a height of 12 mm and a base that measures 10 mm on each side? Round to the nearest tenth if necessary.

The surface area of the square pyramid is [tex]\(360 \, \text{mm}^2\)[/tex]

To calculate the surface area of a square pyramid, we need to find the area of its base and the area of its four triangular faces. Given the following:

The base side length [tex](\(s\))[/tex] is [tex]10\ mm[/tex].

The height of the pyramid [tex](\(h\))[/tex] is [tex]12 \ mm[/tex]

Step-by-Step Calculation:

1. Area of the base:

The base is a square with side length [tex]\(s = 10\) mm.[/tex]

[tex]\[\text{Area of the base} = s^2 = 10^2 = 100 \, \text{mm}^2\][/tex]

2. Slant height of the pyramid:

To find the slant height [tex](\(l\))[/tex], we need to use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle where one leg is half the side length of the base [tex](\(\frac{s}{2} = \frac{10}{2} = 5\) mm)[/tex] and the other leg is the height of the pyramid [tex](\(h = 12\) mm)[/tex].

[tex]\[l = \sqrt{\left(\frac{s}{2}\right)^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{mm}\][/tex]

3 Area of the four triangular faces:

Each triangular face has a base of [tex]10\ mm[/tex] and a slant height of [tex]13 \ mm[/tex].

[tex]\[\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 13 = 65 \, \text{mm}^2\][/tex]

Since there are four triangular faces:

[tex]\[\text{Total area of the four triangular faces} = 4 \times 65 = 260 \, \text{mm}^2\][/tex]

4. Total surface area:

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

[tex]\[\text{Total surface area} = \text{Area of the base} + \text{Total area of the four triangular faces} = 100 + 260 = 360 \, \text{mm}^2\][/tex]

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