Find the total surface area of the given net to the nearest hundredth.
The net of a pyramid is made up of five shapes: one square and four identical isosceles triangles. The area of any square could be calculated by squaring its length. In this case, the length of the square is two centimeters. This means that the area is equal to two squared.
Well two squared is equal to four. And this means that the area of the square is four square centimeters. If we then consider the isosceles triangles, we know that the area of a triangle is equal to the base multiplied by the height divided by two.
In this case, the base is two centimeters. But the height is currently unknown. In order to calculate the height, we need to consider the right-angled triangle shown where the hypotenuse is 3.1 centimeters, the base is one centimeter, and the height is ℎ.
In order to calculate the missing length in any right-angled triangle we need to use Pythagoras’s theorem: 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the hypotenuse or longest side, in this case 3.1 centimeters.
Substituting in our values gives us an equation ℎ squared plus one squared equals 3.1 squared. Well, one squared is one. And 3.1 squared is 9.61. So ℎ squared plus one equals 9.61. Subtracting one from both sides of the equation gives us ℎ squared equals 8.61. And finally, square rooting both sides of the equation gives us ℎ equals 2.934.
This means that the base of the isosceles triangle is two centimeters and its height is 2.934 centimeters. We can therefore calculate the area of the triangle by multiplying two by 2.934 and dividing our answer by two.
This tells us that the area of each of the isosceles triangle is 2.934 square centimeters. As there are four identical isosceles triangles, we need to multiply 2.934 by four. This gives us a total area for the four isosceles triangles of 11.736 square centimeters.
Adding a four square centimeters, the area of the square, to 11.736 gives us 15.736 square centimeters rounding our answer to the nearest hundredth gives us a final answer for the total surface area of the net of 15.74 square centimeters, which was made up of the area of one square and four isosceles triangles.