Good afternoon, dear guests!
I don’t remember that any teacher in school showed me one general formula for calculating the volume of figures and the area of bodies. And even now, looking at the homework of children, teachers are forced to learn the formulas of all the figures by heart!
My grandfather has been teaching higher mathematics at several universities for 60 years, and he always said that the current presentation of knowledge students greatly confuses a person, there are methods much more effective than today to remember difficult to learn things. And in this small article, I would really like to give you one universal formula, and it is called in mathematics the Simpson formula. (In the first part of the article, the volumes of bodies will be described, in the second - the areas of figures).
So, the volume formula:
V = H / 6 * (B1 + 4 * B2 + B3)where
H - body height;
B1 is the area of the lower base;
B2 - cross-sectional area in the center of the body;
B3 is the area of the upper base.
In order not to be unfounded, everything is proved as follows:
Cylinder and prism (including parallelepiped and cube)
The formula for finding the volume of these figures from the mathematics of the school course: V = S * H
According to Simpson's formula, since the areas of the bases are equal to each other B1 = B2 = B3 we get:
V = H / 6 * (B1 + 4 * B1 + B1) = H / 6 * 6 * B1 = H * B1, Q.E.D!
Cone, pyramid, truncated cone
The formula for finding the volume of a cone and a pyramid from school mathematics is as follows: V = S * H / 3
For the pyramid and cone, according to the Simpson formula, we get:
V = H / 6 * (B1 + 4 * (B1 / 4) + 0) = H / 6 * 2 * B1 = H * B1 / 3, Q.E.D!
For a truncated cone, the school formula is presented under the volumetric body and the layout on the right as evidence:
The truncated pyramid is proved in a similar way.
Ball (Sphere)
For the sphere, the school formula is also presented under the picture and the proof is shown on the right:
You agree that this formula has strong arguments to be calleduniversal? Moreover, it is even suitable for calculating the areas of flat figures, only B1, B2 and B3 - now they take the values not of the area of the bases, but the values of the lengths of the bases!
The same formula, the formula for finding areas:
S = H / 6 * (B1 + 4 * B2 + B3)where
H - body height;
B1 is the length of the lower base;
B2 - the length of the segment in the center of the figure;
B3 is the length of the upper base.
Parallelogram or rectangle
Trapezoid
Triangle
So, remember this one short formula so as not to keep everything in your head!
Brevity is the soul of wit!
Thank you for your patience and attention! Read also my related articles:
Putting your knowledge of geometry into practice! What is missing for children and what is not told at school?
Plotting right angles on the site? Options that not everyone knows!
You don't have to jump over the roof with the measuring tool. How to apply trigonometry in life and construction