Video Transcript
Find the total surface area of the regular pyramid in the given figure, approximating the result to the nearest hundredth.
The total surface area of a pyramid is the sum of the areas of all its faces. This pyramid has a triangular base. And as we’re told that this pyramid is regular, this means that the base is a regular triangle. It is an equilateral triangle. We can see from the figure that the side length of this equilateral triangle is 33.5 centimeters. We’ll think about how to find the area of this triangle in a moment. The pyramid also has three lateral triangular faces, and as this pyramid is regular, the three lateral faces are congruent. From the figure, we can see that each of these triangles has a base of 33.5 centimeters, it is the side length of the equilateral triangle and the base of the pyramid, and a perpendicular height of 38.5 centimeters, which is the slant height of the pyramid.
Using the formula for the area of a triangle, base times perpendicular height over two, the area of each of these faces is 33.5 multiplied by 38.5 over two. As there are three of them, the total lateral area of the pyramid is three times this value. We can work this out on a calculator, giving 1,934.625. And the units for this area are square centimeters. Let’s now think about how we can find the area of the base. We can draw in a perpendicular from one vertex of this triangle to the midpoint of the opposite side. And this will divide the equilateral triangle into two congruent right triangles.
Each of these right triangles has a hypotenuse of 33.5 centimeters and a base of half of this, 16.75 centimeters. As these triangles are right triangles, their three side lengths are related by the Pythagorean theorem, which tells us that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. So, if we call the perpendicular height of this triangle ℎ, we have that ℎ squared plus 16.75 squared is equal to 33.5 squared. Subtracting 16.75 squared from each side, we have that ℎ squared is equal to 33.5 squared minus 16.75 squared, which is 841.6875. ℎ is therefore equal to the square root of this, taking only the positive value as ℎ is a length. In exact form, this is 67 root three over four.
So we now know that this equilateral triangle has a base of 33.5 centimeters and a perpendicular height of 67 root three over four centimeters. Using the area formula base times perpendicular height over two, we have that the area of the triangular base is 33.5 multiplied by 67 root three over four multiplied by one-half. As a decimal, this evaluates to 485.9485 continuing. And once again, the units are square centimeters. The total surface area of the pyramid is then the sum of the base area and the lateral area, which evaluates to 2,420.5735 continuing. All that remains is to round this value to the nearest hundredth, or the second decimal place.
The total surface area of the given regular pyramid approximated to the nearest hundredth is 2,420.57 square centimeters.
