Volume of Pyramids (The 1:3 Rule)
Volume of pyramids and the 1:3 rule is a Grade 7 geometry concept in Big Ideas Math Advanced 2, Chapter 14: Surface Area and Volume. A pyramid holds exactly one-third the volume of a prism with the same base and height, so the formula is V equals one-third times B times h where B is the base area. For a square pyramid with 6 cm sides and height 9 cm, volume equals one-third times 36 times 9 equals 108 cubic centimeters.
Key Concepts
Property A pyramid is pointy, meaning it holds a lot less space than a prism with the exact same base and height. In fact, a pyramid holds exactly one third (1/3) the volume of its matching prism! $$V = \frac{1}{3}Bh$$.
Examples Square Pyramid: A pyramid has a square base with a side length of 6 cm, and a straight up height of 9 cm. Big B (Area of the square): 6 x 6 = 36 sq cm. Volume: V = 1/3 x 36 x 9 = 108 cubic cm. Triangular Pyramid: A triangular pyramid has a base area of 15 square meters and a height of 8 meters. Volume: V = 1/3 x 15 x 8 = 40 cubic meters.
Explanation If you have a hollow pyramid and a hollow prism with the exact same base and height, it will take exactly 3 pyramids full of water to fill the prism perfectly to the top. This is one of the coolest relationships in geometry! Because of this, the formula for a pyramid is exactly the same as a prism ($V = Bh$), you just have to remember to divide your final answer by 3.
Common Questions
What is the formula for the volume of a pyramid?
V equals one-third times B times h, where B is the area of the base and h is the perpendicular height from base to apex.
Why does the pyramid volume formula have one-third?
A pyramid holds exactly one-third of the volume of a prism with the same base and height. Three pyramids would fill one prism completely.
How do you find the volume of a square pyramid?
Calculate the base area B as side squared, then apply V equals one-third times B times h. For a pyramid with base side 6 cm and height 9 cm, B equals 36 and V equals one-third times 36 times 9 equals 108 cubic cm.
What textbook covers pyramid volume in Grade 7?
Big Ideas Math Advanced 2, Chapter 14: Surface Area and Volume covers the volume of pyramids including the one-third rule.