Question

what is the total surface area of an equilateral triangular pyramid with base side lengths of 6 feet and a height of 10 feet? the total surface area is about square feet.

Explanation:

Step1: Calculate the base area

The base is an equilateral triangle. The formula for the area of an equilateral triangle is \(A_{base}=\frac{\sqrt{3}}{4}a^{2}\), where \(a = 6\) feet.

$$
LATEXBLOCK0
$$

Step2: Calculate the slant height

We first find the height of the base equilateral triangle \(h_{base}=\frac{\sqrt{3}}{2}a=\frac{\sqrt{3}}{2}\times6 = 3\sqrt{3}\) feet. The centroid of the base equilateral triangle divides the base - height in the ratio \(2:1\), so the distance from the centroid to a vertex of the base triangle \(r=\frac{2}{3}\times3\sqrt{3}=2\sqrt{3}\) feet.
Using the Pythagorean theorem to find the slant height \(l\) of the pyramid. Given the height of the pyramid \(H = 10\) feet, \(l=\sqrt{H^{2}+r^{2}}=\sqrt{10^{2}+(2\sqrt{3})^{2}}=\sqrt{100 + 12}=\sqrt{112}=4\sqrt{7}\) feet.

Step3: Calculate the lateral surface area

The lateral surface area of a triangular pyramid (3 congruent triangular faces). The area of each lateral face is \(\frac{1}{2}al\), where \(a = 6\) feet (base side) and \(l = 4\sqrt{7}\) feet (slant height).
The lateral surface area \(A_{lateral}=3\times\frac{1}{2}\times6\times4\sqrt{7}=36\sqrt{7}\) square feet.

Step4: Calculate the total surface area

The total surface area \(A=A_{base}+A_{lateral}\)

$$
LATEXBLOCK1
$$

Answer:

111