what is the surface area of this triangular pyramid? square meters

Explanation:

Step1: Calculate base - area

The base is an equilateral triangle with side length $a = 3$m. The area formula for an equilateral triangle is $A_{base}=\frac{\sqrt{3}}{4}a^{2}$. So $A_{base}=\frac{\sqrt{3}}{4}\times3^{2}=\frac{9\sqrt{3}}{4}\approx\frac{9\times1.732}{4}=3.897$m².

Step2: Calculate area of one lateral - face

The lateral - faces are isosceles triangles with base $a = 3$m and height $h = 4$m. The area formula for a triangle is $A=\frac{1}{2}ah$. So the area of one lateral - face $A_{lateral}=\frac{1}{2}\times3\times4 = 6$m².

Step3: Calculate total surface area

The triangular pyramid has 1 base and 3 lateral - faces. The total surface area $A = A_{base}+3A_{lateral}$. Substitute the values: $A=\frac{9\sqrt{3}}{4}+3\times6\approx3.897 + 18=21.897\approx21.9$m².

Answer:

$21.9$