The formula for finding the height of a pyramid is possible to Express the formula then calculate the volume:
V = (S*h)/3, where S is the area of the polytope lying at the base of the pyramid, h is the height of this pyramid.
In this case, h can be calculated as:
h = (3*V)/S.
In that case, if the base of the pyramid has a square, length of its diagonal and the edge length of this pyramid, the height of this pyramid can be expressed from Pythagorean theorem, because the triangle which is formed by an edge of the pyramid, the height and half the diagonal of the square at the base is a right triangle.
The Pythagorean theorem States that the square of the hypotenuse in a right triangle, the is equal to the sum of the squares of the other two sides(a2 = b2 + c2). The face of the pyramid is the hypotenuse, one leg is half diagonal of the square. Then the length of the unknown side (height) is by the formula:
b2 = a2 - c2;
c2 = a2 - b2.
To both situations was the most clear and understandable, you can consider a couple of examples.
Example 1: Area of base of the pyramid 46 cm2, its volume is 120 cm3. Based on these data, the height of the pyramid is this:
h = 3*120/46 = 7.83 cm
Answer: the height of the pyramid will be approximately 7.83 cm
Example 2: From the pyramid, which lies at the base a regular polygon is a square, its diagonal is 14 cm, the length of the edge is 15 cm According to the to find the height of the pyramid, you need to use the following formula (which appeared as a consequence of the Pythagorean theorem):
h2 = 152 - 142
h2 = 225 - 196 = 29
h = √29 cm
Answer: the height of the pyramid is √29 cm or approximately 5.4 cm