The square base of a pyramid has an area of 144 square feet. If the slant height of the pyramid is 10 feet, what is the height of the pyramid

Question

Very hard question for me

 

Answer 1

Hi Gilbert, this is a good question.  Since you mention the 3D Pythagorean theorem, I'll show the 3D approach and also a way to just use the 2D Pythagorean theorem.

 

3D version

The 3D Pythagorean theorem is similar to the 2D version, except we add a term for the vertical distance.

 

X² + Y² + Z² = Distance²

 

Here we have 3D so X is the width between the two points, Y is the length between two points and Z is the height.

Since the Area of the square is 144 sq ft, we know

 

A = s²

 

so each side of the square is 12 ft.  Since the top of the pyramid is directly over the center of the square,

 

X = Y = 12/2 = 6 ft

 

We also know that the slant height is the distance in the 3D Pythagorean theorem, so we want to solve for Z.

 

X² + Y² + Z² = Distance²

becomes

6²+ 6²+ Z² = 10²

 

Solving for Z gives

 

Z = √28 = 2√7

 

2D version

We can break the problem up into two 2D problems and use the 2D Pythogorean theorem twice.

The first time, we need to find the distance from one of the corners to the center of the square. We have

 

6²+ 6² = d²

So d = √72

 

Next we build a right triangle between the top of the pyramid, the center of the square, and one of the corners of the square.  I'll use the letter Z to represent the height of the pyramid, and we know the hypotenuse of this triangle is the slant height of the pyramid.

 

√72² + Z² = 10²

 

Solving gives the same answer as the 3D version, Z = 2√7