Diagonal of a parallelepiped. Formula
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Aporia of Zeno"]. Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his 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 catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl 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 a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have 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 about a flying arrow:
A flying arrow is motionless, since at every 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 a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs 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 whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to 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 research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing 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. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the 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 is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps 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'll 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 that’s why they are shamans, 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 page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to 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 easily.
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 us 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 into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems 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 the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've 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 the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. 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 operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
In geometry, the following types of parallelepipeds are distinguished: rectangular parallelepiped (the faces of the parallelepiped are rectangles); a right parallelepiped (its side faces act as rectangles); inclined parallelepiped (its side faces act as perpendiculars); a cube is a parallelepiped with absolutely identical dimensions, and the faces of the cube are squares. Parallelepipeds can be either inclined or straight.
The main elements of a parallelepiped are that two faces of the presented geometric figure that do not have a common edge are opposite, and those that do are adjacent. The vertices of the parallelepiped, which do not belong to the same face, act opposite to each other. A parallelepiped has a dimension - these are three edges that have a common vertex.
The line segment that connects opposite vertices is called a diagonal. The four diagonals of a parallelepiped, intersecting at one point, are simultaneously divided in half.
In order to determine the diagonal of a parallelepiped, you need to determine the sides and edges, which are known from the conditions of the problem. With three known ribs A , IN , WITH draw a diagonal in the parallelepiped. According to the property of a parallelepiped, which says that all its angles are right, the diagonal is determined. Construct a diagonal from one of the faces of the parallelepiped. The diagonals must be drawn in such a way that the diagonal of the face, the desired diagonal of the parallelepiped and the known edge create a triangle. After a triangle is formed, find the length of this diagonal. The diagonal in the other resulting triangle acts as the hypotenuse, so it can be found using the Pythagorean theorem, which must be taken under the square root. This way we know the value of the second diagonal. In order to find the first diagonal of the parallelepiped in the formed right triangle, it is also necessary to find the unknown hypotenuse (using the Pythagorean theorem). Using the same example, sequentially find the remaining three diagonals existing in the parallelepiped, performing additional constructions of diagonals that form right triangles and solve using the Pythagorean theorem.
A rectangular parallelepiped (PP) is nothing more than a prism, the base of which is a rectangle. For a PP, all diagonals are equal, which means that any of its diagonals is calculated using the formula:
a, c - sides of the base of the PP;
c is its height.
Another definition can be given by considering the Cartesian rectangular coordinate system:
The PP diagonal is the radius vector of any point in space specified by x, y and z coordinates in the Cartesian coordinate system. This radius vector to the point is drawn from the origin. And the coordinates of the point will be the projections of the radius vector (diagonals of the PP) onto the coordinate axes.
1055;projections coincide with the vertices of this parallelepiped.
Parallelepiped and its types
If we literally translate its name from ancient Greek, it turns out that it is a figure consisting of parallel planes. There are the following equivalent definitions of a parallelepiped:
- a prism with a base in the form of a parallelogram;
- a polyhedron, each face of which is a parallelogram.
Its types are distinguished depending on what figure lies at its base and how the lateral ribs are directed. In general, we talk about inclined parallelepiped, whose base and all faces are parallelograms. If the side faces of the previous view become rectangles, then it will need to be called direct. And rectangular and the base also has 90º angles.
Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, is the main difference between mathematicians and artists. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the ribs is completely invisible.
About the introduced notations
In the formulas below, the notations indicated in the table are valid.
Formulas for an inclined parallelepiped
First and second for areas:
The third is to calculate the volume of a parallelepiped:
Since the base is a parallelogram, to calculate its area you will need to use the appropriate expressions.
Formulas for a rectangular parallelepiped
Similar to the first point - two formulas for areas:
And one more for volume:
First task
Condition. Given a rectangular parallelepiped, the volume of which needs to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.
Solution. To answer the problem question, you will need to know all the sides in three right triangles. They will give the necessary values of the edges by which you need to calculate the volume.
First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from where the main diagonal of the parallelogram was drawn. The angle between them will be what is needed.
The first triangle that will give one of the values of the sides of the base will be the following. It contains the required side and two drawn diagonals. It's rectangular. Now you need to use the ratio of the opposite leg (side of the base) and the hypotenuse (diagonal). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be designated by the letter “a”.
The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, side edge to diagonal. It is equal to the cosine of 45º. That is, “c” is calculated as the product of the diagonal and the cosine of 45º.
c = 18 * 1/√2 = 9 √2 (cm).
In the same triangle you need to find another leg. This is necessary in order to then calculate the third unknown - “in”. Let it be designated by the letter “x”. It can be easily calculated using the Pythagorean theorem:
x = √(18 2 - (9√2) 2) = 9√2 (cm).
Now we need to consider another right triangle. It contains the already known sides “c”, “x” and the one that needs to be counted, “b”:
in = √((9√2) 2 - 9 2 = 9 (cm).
All three quantities are known. You can use the formula for volume and calculate it:
V = 9 * 9 * 9√2 = 729√2 (cm 3).
Answer: the volume of the parallelepiped is 729√2 cm 3.
Second task
Condition. You need to find the volume of a parallelepiped. In it, the sides of the parallelogram, which lies at the base, are known to be 3 and 6 cm, as well as its acute angle - 45º. The side rib has a slope to the base of 30º and is equal to 4 cm.
Solution. To answer the question of the problem, you need to take the formula that was written for the volume of an inclined parallelepiped. But both quantities are unknown in it.
The area of the base, that is, of a parallelogram, will be determined by a formula in which you need to multiply the known sides and the sine of the acute angle between them.
S o = 3 * 6 sin 45º = 18 * (√2)/2 = 9 √2 (cm 2).
The second unknown quantity is height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle in which the height is the leg and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite the unknown height. This means that we can use the ratio of the leg to the hypotenuse.
n = 4 * sin 30º = 4 * 1/2 = 2.
Now all the values are known and the volume can be calculated:
V = 9 √2 * 2 = 18 √2 (cm 3).
Answer: the volume is 18 √2 cm 3.
Third task
Condition. Find the volume of a parallelepiped if it is known that it is straight. The sides of its base form a parallelogram and are equal to 2 and 3 cm. The acute angle between them is 60º. The smaller diagonal of the parallelepiped is equal to the larger diagonal of the base.
Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.
Since the smaller diagonal of the parallelepiped coincides in size with the larger base, they can be designated by the same letter d. The largest angle of a parallelogram is 120º, since it forms 180º with the acute one. Let the second diagonal of the base be designated by the letter “x”. Now for the two diagonals of the base we can write the cosine theorems:
d 2 = a 2 + b 2 - 2av cos 120º,
x 2 = a 2 + b 2 - 2ab cos 60º.
It makes no sense to find values without squares, since later they will be raised to the second power again. After substituting the data, we get:
d 2 = 2 2 + 3 2 - 2 * 2 * 3 cos 120º = 4 + 9 + 12 * ½ = 19,
x 2 = a 2 + b 2 - 2ab cos 60º = 4 + 9 - 12 * ½ = 7.
Now the height, which is also the side edge of the parallelepiped, will turn out to be a leg in the triangle. The hypotenuse will be the known diagonal of the body, and the second leg will be “x”. We can write the Pythagorean Theorem:
n 2 = d 2 - x 2 = 19 - 7 = 12.
Hence: n = √12 = 2√3 (cm).
Now the second unknown quantity is the area of the base. It can be calculated using the formula mentioned in the second problem.
S o = 2 * 3 sin 60º = 6 * √3/2 = 3√3 (cm 2).
Combining everything into the volume formula, we get:
V = 3√3 * 2√3 = 18 (cm 3).
Answer: V = 18 cm 3.
Fourth task
Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; the side faces are rhombuses; one of the vertices located above the base is equidistant from all the vertices lying at the base.
Solution. First you need to understand the condition. There are no questions with the first point about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.
To determine the volume, you will need a formula written for an inclined parallelepiped. There are again no known quantities in it. However, the area of the base is easy to calculate because it is a square.
S o = 5 2 = 25 (cm 2).
The situation with height is a little more complicated. It will be like this in three figures: a parallelepiped, a quadrangular pyramid and an isosceles triangle. This last circumstance should be taken advantage of.
Since it is the height, it is a leg in a right triangle. The hypotenuse in it will be a known edge, and the second leg is equal to half the diagonal of the square (the height is also the median). And the diagonal of the base is easy to find:
d = √(2 * 5 2) = 5√2 (cm).
n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).
V = 25 * 2.5 √2 = 62.5 √2 (cm 3).
Answer: 62.5 √2 (cm 3).
The prism is called parallelepiped, if its bases are parallelograms. Cm. Fig.1.
Properties of a parallelepiped:
The opposite faces of a parallelepiped are parallel (that is, they lie in parallel planes) and equal.
The diagonals of a parallelepiped intersect at one point and are bisected by this point.
Adjacent faces of a parallelepiped– two faces that have a common edge.
Opposite faces of a parallelepiped– faces that do not have common edges.
Opposite vertices of a parallelepiped– two vertices that do not belong to the same face.
Diagonal of a parallelepiped– a segment that connects opposite vertices.
If the lateral edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.
A right parallelepiped whose bases are rectangles is called rectangular. A prism, all of whose faces are squares, is called cube.
Parallelepiped- a prism whose bases are parallelograms.
Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the plane of the base.
Rectangular parallelepiped is a right parallelepiped whose bases are rectangles.
Cube– a rectangular parallelepiped with equal edges.
parallelepiped called a prism whose base is a parallelogram; Thus, a parallelepiped has six faces and all of them are parallelograms.
Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and are divided in half at it. Any face can be taken as a base; the volume is equal to the product of the area of the base and the height: V = Sh.
A parallelepiped whose four lateral faces are rectangles is called a straight parallelepiped.
A right parallelepiped whose six faces are rectangles is called rectangular. Cm. Fig.2.
The volume (V) of a right parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .
For a rectangular parallelepiped, in addition, the formula holds V=abc, where a,b,c are edges.
The diagonal (d) of a rectangular parallelepiped is related to its edges by the relation d 2 = a 2 + b 2 + c 2 .
Rectangular parallelepiped- a parallelepiped whose side edges are perpendicular to the bases, and the bases are rectangles.
Properties of a rectangular parallelepiped:
In a rectangular parallelepiped, all six faces are rectangles.
All dihedral angles of a rectangular parallelepiped are right.
The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (the lengths of three edges that have a common vertex).
The diagonals of a rectangular parallelepiped are equal.
A rectangular parallelepiped, all of whose faces are squares, is called a cube. All edges of the cube are equal; the volume (V) of a cube is expressed by the formula V=a 3, where a is the edge of the cube.
It will be useful for high school students to learn how to solve Unified State Examination problems to find the volume and other unknown parameters of a rectangular parallelepiped. The experience of previous years confirms the fact that such tasks are quite difficult for many graduates.
At the same time, high school students with any level of training should understand how to find the volume or area of a rectangular parallelepiped. Only in this case will they be able to count on receiving competitive scores based on the results of passing the unified state exam in mathematics.
Key points to remember
- The parallelograms that make up a parallelepiped are its faces, their sides are its edges. The vertices of these figures are considered the vertices of the polyhedron itself.
- All diagonals of a rectangular parallelepiped are equal. Since this is a straight polyhedron, the side faces are rectangles.
- Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
- The lateral edges of a rectangular parallelepiped are perpendicular to the base. Therefore, they are its heights.
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a, towards the base of the PP;
with its height.
- what do we need to know, what data do we have?
- what properties does a rectangular parallelepiped have?
- does the Pythagorean Theorem apply here? How?
- Is there enough data to apply the Pythagorean theorem, or are some other calculations needed?
Diagonal square, of a square parallelepiped (see properties of a square parallelepiped) is equal to the sum of the squares of its three different sides (width, height, thickness), and, accordingly, the diagonals of a square parallelepiped are equal to the root of this sum.
I remember the school curriculum in geometry, we can say this: the diagonal of a parallelepiped is equal to the square root obtained from the sum of its three sides (they are designated by small letters a, b, c).
The length of the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of its sides.
As far as I know from the school curriculum, grade 9, if I’m not mistaken, and if memory serves, the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of all three sides.
the square of the diagonal is equal to the sum of the squares of the width, height and length, based on this formula we get the answer, the diagonal is equal to the square root of the sum of its three different dimensions, with letters they denote ncz abc
A rectangular parallelepiped (PP) is nothing more than a prism, the base of which is a rectangle. For a PP, all diagonals are equal, which means that any of its diagonals is calculated using the formula:
Another definition can be given by considering the Cartesian rectangular coordinate system:
The PP diagonal is the radius vector of any point in space specified by x, y and z coordinates in the Cartesian coordinate system. This radius vector to the point is drawn from the origin. And the coordinates of the point will be the projections of the radius vector (diagonals of the PP) onto the coordinate axes. The projections coincide with the vertices of this parallelepiped.
A rectangular parallelepiped is a type of polyhedron consisting of 6 faces, at the base of which is a rectangle. A diagonal is a line segment that connects opposite vertices of a parallelogram.
The formula for finding the length of a diagonal is that the square of the diagonal is equal to the sum of the squares of the three dimensions of the parallelogram.
I found a good diagram-table on the Internet with a complete listing of everything that is in the parallelepiped. There is a formula to find the diagonal, which is denoted by d.
There is an image of the edge, vertex and other important things for the parallelepiped.
If the length, height and width (a,b,c) of a rectangular parallelepiped are known, then the formula for calculating the diagonal will look like this:
Typically, teachers do not offer their students a bare formula, but make efforts so that they can derive it on their own by asking leading questions:
Usually, after answering the questions posed, students can easily derive this formula on their own.
The diagonals of a rectangular parallelepiped are equal. As well as the diagonals of its opposite faces. The length of the diagonal can be calculated by knowing the length of the edges of the parallelogram emanating from one vertex. This length is equal to the square root of the sum of the squares of the lengths of its edges.
A cuboid is one of the so-called polyhedra, which consists of 6 faces, each of which is a rectangle. A diagonal is a segment that connects opposite vertices of a parallelogram. If the length, width and height of a rectangular parallelepiped are taken to be a, b, c, respectively, then the formula for its diagonal (D) will look like this: D^2=a^2+b^2+c^2.
Diagonal of a rectangular parallelepiped is a segment connecting its opposite vertices. So we have cuboid with diagonal d and sides a, b, c. One of the properties of a parallelepiped is that the square diagonal length d is equal to the sum of the squares of its three dimensions a, b, c. Hence the conclusion is that diagonal length can be easily calculated using the following formula:
