Different prisms are different from each other. At the same time, they have a lot in common. To find the area of \u200b\u200bthe base of a prism, you need to figure out what kind it looks like.
General theory
A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces - they can vary significantly in size.
When solving problems, it is not only the area of \u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.
Sometimes heights appear in tasks. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.
It should be noted that the area of the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.
triangular prism
It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.
Mathematical notation looks like this: S = ½ av.
To find the area of the base in general view, the formulas are useful: Heron and the one in which half of the side is taken to the height drawn to it.
The first formula should be written like this: S \u003d √ (p (p-a) (p-in) (p-s)). This entry contains a semi-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you want to know the area of the base triangular prism, which is correct, then the triangle is equilateral. It has its own formula: S = ¼ a 2 * √3.
quadrangular prism
Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of \u200b\u200bthe base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = av, where a, b are the sides of the rectangle.
When it comes to a quadrangular prism, then the area of \u200b\u200bthe base right prism calculated by the formula for a square. Because it is he who lies at the base. S \u003d a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: na \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is na opposite to this angle.
If a rhombus lies at the base of the prism, then the same formula will be needed to determine its area as for a parallelogram (since it is a special case of it). But you can also use this one: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with a different number of vertices.
Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of \u200b\u200bthe base of the prism is equal to the area of one such triangle (the formula can be seen above), multiplied by five.
Regular hexagonal prism
According to the principle described for a pentagonal prism, it is possible to divide the base hexagon into 6 equilateral triangles. The formula for the area of the base of such a prism is similar to the previous one. Only in it should be multiplied by six.
The formula will look like this: S = 3/2 and 2 * √3.
Tasks
No. 1. A regular straight line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of \u200b\u200bthe base of the prism and the entire surface.
Solution. The base of a prism is a square, but its side is not known. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (n). x 2 \u003d d 2 - n 2. On the other hand, this segment "x" is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (d 2 - n 2) / 2.
Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.
To find out the area of \u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. The total surface area of the prism is found to be 960 cm 2 .
Answer. The base area of the prism is 144 cm2. The entire surface - 960 cm 2 .
No. 2. Dana At the base lies a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.
Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared times ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of one base of the prism.
Everything side faces identical and are rectangles with sides of 6 and 10 cm. To calculate their areas, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of the side surface is wound 180 cm 2 .
Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 2.
In the fifth century BC ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. WITH physical point To the eye, it looks like time slowing down until it stops completely at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What do I want to focus on Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins available different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. Because we can't compare numbers with different units measurements. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
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correct hexagonal prism - a prism, at the bases of which there are two regular hexagons, and all side faces are strictly perpendicular to these bases.
- A B C D E F A1 B1 C1 D1 E1 F1 - regular hexagonal prism
- a- length of the side of the base of the prism
- h- length of the side edge of the prism
- Smain- base area of the prism
- Sside .- area of the side face of the prism
- Sfull .- square full surface prisms
- Vprisms- prism volume
Base area of the prism
The bases of the prism are regular hexagons with sides a. According to the properties of a regular hexagon, the area of \u200b\u200bthe bases of a prism is
This way
Smain= 3 3 √ 2 ⋅ a2
Thus, it turns out that SA B C D E F= SA1 B1 C1 D1 E1 F1 = 3 3 √ 2 ⋅ a2
Total surface area of the prism
The area of the total surface of the prism is the sum of the areas of the side faces of the prism and the areas of its bases. Each of the side faces of the prism is a rectangle with sides a and h. Therefore, by the properties of the rectangle
S
side .= a ⋅ hA prism has six sides and two bases, so its total surface area is
Sfull .= 6 ⋅ Sside .+ 2 ⋅ Smain= 6 ⋅ a ⋅ h + 2 ⋅ 3 3 √ 2 ⋅ a2
Prism Volume
The volume of a prism is calculated as the product of the area of its base and its height. The height of a regular prism is any of its side edges, for example, the edge A A1 . At the base of a regular hexagonal prism is a regular hexagon whose area is known to us. We get
Vprisms= Smain⋅ A A1 = 3 3 √ 2 ⋅ a2 ⋅ h
Regular hexagon at the bases of a prism
We consider the regular hexagon ABCDEF, which lies at the base of the prism.
Draw segments AD, BE and CF. Let the point O be the intersection of these segments.
According to the properties of a regular hexagon, triangles AOB, BOC, COD, DOE, EOF, FOA are regular triangles. Hence it follows that
A O = O D = E O = O B = C O = O F = a
We draw segment AE intersecting segment CF at point M. Triangle AEO is isosceles, in it A O = O E = a , ∠ E O A = 120 ∘ . According to the properties of an isosceles triangle.
A E = a ⋅ 2 (1 − cos E O A )− − − − − − − − − − − − √ = 3 √ ⋅ a
Similarly, we conclude that A C = C E = 3 √ ⋅ a, F M = M O = 1 2 ⋅ a.
We find E A1
In a triangleA E A1 :
- A A1 = h
- A E = 3 √ ⋅ a- as we just found out
- ∠ E A A1 = 90 ∘
A E A1
E A1 = A A2 1 + A E2 − − − − − − − − − − √ = h2 + 3 ⋅ a2 − − − − − − − − √
If h = a, then E A1 = 2 ⋅ a
F B1
= A C1
= B D1
= C E1
= D F1
=
h2
+
3
⋅
a2
−
−
−
−
−
−
−
−
√
.
In a triangle B E B1 :
- B B1 = h
- B E = 2 ⋅ a- because E O = O B = a
- ∠ E B B1 = 90 ∘ - according to the properties of a regular straight line
Thus, it turns out that the triangle B E B1 rectangular. According to the properties of a right triangle
E B1 = B B2 1 + B E2 − − − − − − − − − − √ = h2 + 4 ⋅ a2 − − − − − − − − √
If h = a, then
E B1 = 5 √ ⋅ a
After similar reasoning, we get that F C1
= A D1
= B E1
= C F1
= D A1
=
h2
+
4
⋅
a2
−
−
−
−
−
−
−
−
√
.
We find O F1
In a triangle F O F1 :
- F F1 = h
- F O = a
- ∠ O F F1 = 90 ∘ - according to the properties of a regular prism
Thus, it turns out that the triangle F O F1 rectangular. According to the properties of a right triangle
O F1 = F F2 1 + O F2 − − − − − − − − − − √ = h2 + a2 − − − − − − √
If h = a, then
From each vertex of the prism, for example, from the vertex A 1 (Fig.), Three diagonals can be drawn (A 1 E, A 1 D, A 1 C).
They are projected onto the plane ABCDEF by the base diagonals (AE, AD, AC). Of the oblique A 1 E, A 1 D, A 1 C, the largest is the one whose projection is the largest. Therefore, the largest of the three diagonals taken is A 1 D (there are more diagonals in the prism equal to A 1 D, but there are no larger ones).
From triangle A 1 AD, where ∠DA 1 A = α
and A 1 D = d
, we find H=AA 1 = d
cos α
,
AD= d
sin α
.
The area of an equilateral triangle AOB is 1/4 AO 2 √3. Hence,
S ocn. \u003d 6 1 / 4 AO 2 √3 \u003d 6 1 / 4 (AD / 2) 2 √3.
Volume V = S H = 3√ 3 / 8 AD 2 AA 1
Answer: 3√3/8 d 3 sin 2 α cos α .
Comment . To represent a regular hexagon (the base of a prism), you can construct an arbitrary BCDO parallelogram. Putting aside the segments OA = OD, OF= OC and OE= OB on the extensions of the lines DO, CO, BO, we obtain the hexagon ABCDEF. Point O represents the center.