Why do you use 1/3 in volume formulas?
any solid that comes to a vertex or point has the formula
V = (1/3) (base area) x (height)
I you know Calculus you could take the region bounded by the x-axis, the y-axis and a line with x-intercept of h and y-intercept of r and rotate it about the x-axis
The result will be a cone with radius r and height h
the equation of that line is
y = (-r/h)x + r
the generated volume
= π[integral]y^2 dx
= π[integral] (r^2x^2/h^2 - 2r^2x/h + r^2) dx
= π [r^2x^3/(3h^2) - r^2x^2/h + r^2x] from 0 to h
= π(r^2h/3 - r^2h + r^2h - 0)
= π(r^2)(h)/3
= (1/3)(πr^2)(h) or (1/3)basearea x height
When I was still teaching, I had a cylinder and a cone, both with the same radius and height.
In class we would fill the cone with water and pour it into the cylinder, and do that until the cylinder was full.
We were able to fill and pour THREE times, showing that the volume of the cone was
(1/3) of the volume of the cylinder.
notice
volume of cylinder = πr^2h
volume of cone = (1/3)πr^2h
The fraction 1/3 often appears in volume formulas because it represents the relationship between the dimensions of a three-dimensional shape and its volume. Specifically, when calculating the volume of certain shapes, such as cones and pyramids, the formula involves multiplying the base area by the height and then scaling it by 1/3.
To understand why 1/3 is used, let's consider the volume formulas for a cone and a pyramid:
1. Cone: The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where V represents the volume, π is a mathematical constant (approximately equal to 3.14159), r is the radius of the base, and h is the height.
2. Pyramid: The volume of a pyramid can be determined by the formula V = (1/3)Bh, where V is the volume, B represents the base area, and h is the height.
So, both formulas have a factor of 1/3 applied. The reason behind this lies in the geometric proportionality of similar shapes. When you compare similar shapes, such as a small cone and a larger cone that are scaled versions of each other, you'll find that the ratio between their volumes is equal to the ratio between their corresponding dimensions cubed.
Since similar shapes have corresponding dimensions that are in proportion to each other, the ratio of the volumes is given by (V1/V2) = (r1^2h1)/(r2^2h2) for cones and (V1/V2) = (B1h1)/(B2h2) for pyramids. By simplifying these ratios, you'll arrive at the expression 1/3, which indicates that the smaller shape has one-third of the volume of the larger shape.
Overall, the use of 1/3 in volume formulas for cones and pyramids is based on the principle of proportional scaling in similar shapes. It is a way to express the relationship between the dimensions and volumes of these three-dimensional figures.