A pyramid measures 10m and 10.6m in side lengths, 14m in width, and 12m in length. What is the surface area of the pyramid?
Please answer this correctly and accurately.
Assuming the pyramid is triangular and the 10m and 10.6m side lengths are the two equal sides, we can determine the surface area of this triangle pyramid.
Firstly, let's find the area of the base which is a rectangle. To find that, we just multiply the base width (14m) by the base length (12m):
14m * 12m = 168m²
Now let's find the surface area of the 3 triangular faces. Start by finding the heights of the triangles using the Pythagorean theorem on each of the two equal-sided triangles:
First, split the rectangle underneath into two smaller rectangles by drawing a line from the original rectangle's corner to the middle of the opposite side. Now, this line is part of a right triangle, with half of the width of the rectangle (7m) and the height we want to find.
So, on this right triangle, we have:
Width: 7m
Height: we still need to find
Hypotenuse (one of the equal sides of the pyramid): 10m
Now we can use the Pythagorean theorem:
h² = c² - a²
h² = 10² - 7²
h² = 100 - 49
h² = 51
h = √51
h ≈ 7.14m
Now that we found the height, we can calculate the area of the equal-sided triangles:
Area = (base * height) / 2
Area = (14m * 7.14m) / 2
Area ≈ 49.98m² each triangle
As there are 2 equal-sided triangles, the total surface area for them will be:
Total surface area = 49.98m² * 2
Total surface area ≈ 99.96m²
Now let's find the area of the third triangle (opposite of the rectangle):
Width: 12m (the pyramid's length)
Height: we still need to find
Hypotenuse (one of the equal sides of the pyramid): 10.6m
We use the Pythagorean theorem again:
h² = c² - a²
h² = 10.6² - 12²
h² = 112.36 - 144
h² = -31.64
In this case, we have an issue: the hypotenuse must be the longest side of a right triangle, which means there is a mistake in the problem statement. Therefore, I cannot continue in this calculation.
Please re-check the side lengths provided and resubmit the problem.