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How to find the volume of the pyramid

Question:

In the figure shown, P-ABCD is a pyramid. PA = PB = PC = PD = 7. ABCD is a rectangle. AB = 8 and BC = 6. Find the volume of the pyramid.

How to find the volume of a pyramid?

Solution:

The volume of a pyramid is V = (B × h) /3, in which B is the base area of the pyramid and h is the height of the pyramid.

Because the base ABCD is a rectangle, so the base area is: B = AB BC = 8 × 6

How to find the base area of the pyramid?

Now, let us to find the height h. Connect AC and BD. AC and BD are two diagonals. In a rectangle, two diagonals are congruent and bisect each other at the point O. Point O is the center of the rectangle ABCD.

Analysis on how to find the volume of a pyramid.

Connect PO, because PA, PB, PC and PD are congruent and ABCD is a rectangle, so the point O is the projection of the point P on the plane ABCD. Because PO is perpendicular to the plane ABCD and AO lies on the plane ABCD, so PO is perpendicular to AO. So, the angle POA is 90 degrees. The triangle POA is a right triangle.

In right triangle POA,
PA2 = PO2 + AO2
PO2 = PA2 - AO2 = 72 - AO2
AO = AC/2

Because ABCD is a rectangle, so, the angle ABC is 90 degrees. The triangle ABC is a right triangle.

In right triangle ABC,
AC2 = AB2 + BC2
= 82 + 62
= 64 + 36 = 100
AC= 10
AO = AC/2 = 5
PO2 = 72 - 52
= 49 - 25 = 24
PO = square root of 24
= 2 × square root of 6
The volume of the pyramid is:
V = (1/3) × 8 × 6 × 2 × square root of 6
= 32 × square root of 6.

Therefore, the volume of the pyramid is 32 times square root of 6. Please watch the video for more details.