Find the surface area of the pyramid with an equilateral triangle for a base.

To find the surface area of a pyramid with an equilateral triangle as its base, we need to calculate both the area of the base and the areas of the triangular lateral faces.<br /><br />Let's assume the side length of the equilateral triangle base is \( s \).<br /><br />### Step 1: Calculate the Area of the Base<br /><br />The formula for the area of an equilateral triangle is:<br /><br />\[<br />A_{\text{base}} = \frac{\sqrt{3}}{4} s^2<br />\]<br /><br />### Step 2: Calculate the Area of the Lateral Faces<br /><br />A pyramid with an equilateral triangle base has three identical triangular lateral faces. To find the area of one lateral face, we need the slant height (\( l \)) of the pyramid.<br /><br />Assuming you have the height (\( h \)) of the pyramid (the perpendicular distance from the apex to the center of the base), you can find the slant height using the Pythagorean theorem in the right triangle formed by the height of the pyramid, half the side length of the base, and the slant height:<br /><br />\[<br />l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}<br />\]<br /><br />The area of one lateral face (a triangle) is given by:<br /><br />\[<br />A_{\text{lateral}} = \frac{1}{2} \times s \times l<br />\]<br /><br />Since there are three lateral faces, the total lateral surface area is:<br /><br />\[<br />A_{\text{total lateral}} = 3 \times A_{\text{lateral}} = \frac{3}{2} \times s \times l<br />\]<br /><br />### Step 3: Calculate the Total Surface Area<br /><br />The total surface area of the pyramid is the sum of the base area and the total lateral surface area:<br /><br />\[<br />A_{\text{total}} = A_{\text{base}} + A_{\text{total lateral}}<br />\]<br /><br />Substituting the expressions we found:<br /><br />\[<br />A_{\text{total}} = \frac{\sqrt{3}}{4} s^2 + \frac{3}{2} \times s \times \sqrt{h^2 + \left(\frac{s}{2}\right)^2}<br />\]<br /><br />This formula gives you the total surface area of the pyramid with an equilateral triangle base, given the side length \( s \) and the height \( h \) of the pyramid. If you have specific values for \( s \) and \( h \), you can substitute them into this formula to get the numerical result.