This - oldest geometric problem.
Step-by-step instruction
1st method. - Using the “golden” or “Egyptian” triangle. The sides of this triangle have the aspect ratio 3:4:5, and the angle is strictly 90 degrees. This quality was widely used by the ancient Egyptians and other ancient cultures.
Ill.1. Construction of the Golden or Egyptian Triangle
- We manufacture three measurements (or rope compasses - a rope on two nails or pegs) with lengths 3; 4; 5 meters. The ancients often used the method of tying knots with equal distances between them as units of measurement. Unit of length - " nodule».
- We drive a peg at point O and attach the measure “R3 - 3 knots” to it.
- We stretch the rope along the known boundary - towards the proposed point A.
- At the moment of tension on the border line - point A, we drive in a peg.
- Then - again from point O, stretch the measure R4 - along the second border. We don’t drive the peg in yet.
- After this, we stretch the measure R5 - from A to B.
- We drive a peg at the intersection of measurements R2 and R3. – This is the desired point B – third vertex of the golden triangle, with sides 3;4;5 and with a right angle at point O.
2nd method. Using a compass.
The compass may be rope or pedometer. Cm:
Our compass pedometer has a step of 1 meter.
Ill.2. Compass pedometer
Construction - also according to Ill. 1.
- From the reference point - point O - the neighbor's corner, draw a segment of arbitrary length - but larger than the radius of the compass = 1m - in each direction from the center (segment AB).
- We place the leg of the compass at point O.
- We draw a circle with radius (compass pitch) = 1 m. It is enough to draw short arcs - 10-20 centimeters each, at the intersection with the marked segment (through points A and B). With this action we found equidistant points from the center- A and B. The distance from the center does not matter here. You can simply mark these points with a tape measure.
- Next, you need to draw arcs with centers at points A and B, but with a slightly (arbitrarily) larger radius than R=1m. You can reconfigure our compass to a larger radius if it has an adjustable pitch. But for such a small current task, I wouldn’t want to “pull” it. Or when there is no adjustment. Can be done in half a minute rope compass.
- We place the first nail (or the leg of a compass with a radius greater than 1 m) alternately at points A and B. And draw two arcs with the second nail - in a taut state of the rope - so that they intersect with each other. It is possible at two points: C and D, but one is enough - C. And again, short serifs at the intersection at point C will suffice.
- Draw a straight line (segment) through points C and D.
- All! The resulting segment, or straight line, is exact direction on North:). Sorry, - at a right angle.
- The figure shows two cases of boundary discrepancy across a neighbor's property. Ill. 3a shows a case where a neighbor’s fence moves away from the desired direction to its detriment. On 3b - he climbed onto your site. In situation 3a, it is possible to construct two “guide” points: both C and D. In situation 3b, only C.
- Place a peg at corner O, and a temporary peg at point C, and stretch a cord from C to the rear boundary of the site. - So that the cord barely touches peg O. By measuring from point O - in direction D, the length of the side according to the general plan, you will get a reliable rear right corner of the site.
Ill.3. Construction right angle– from the neighbor’s corner, using a pedometer and a rope compass
If you have a compass-pedometer, then you can do without rope altogether. In the previous example, we used the rope one to draw arcs of a larger radius than those of the pedometer. More because these arcs must intersect somewhere. In order for the arcs to be drawn with a pedometer with the same radius - 1m with a guarantee of their intersection, it is necessary that points A and B are inside the circle with R = 1m.
- Then measure these equidistant points roulette- in different directions from the center, but always along line AB (neighbor’s fence line). The closer points A and B are to the center, the farther the guide points C and D are from it, and the more accurate the measurements. In the figure, this distance is taken to be about a quarter of the pedometer radius = 260mm.
Ill.4. Constructing a right angle using a pedometer and tape measure
- This scheme of actions is no less relevant when constructing any rectangle, in particular the contour of a rectangular foundation. You will receive it perfect. Its diagonals, of course, need to be checked, but isn't the effort reduced? – Compared to when the diagonals, corners and sides of the foundation contour are moved back and forth until the corners meet..
Actually, we solved a geometric problem on earth. To make your actions more confident on the site, practice on paper - using a regular compass. Which is basically no different.
Each of us went to school. There a man gets great amount the knowledge that may later be needed in life. Not everyone, of course, can fully appreciate the significance of the knowledge acquired at school, but this is not the point now.
Mathematics. This is a scary word for many., which frightened quite a number of schoolchildren at one time. Numbers, formulas and calculations were amenable to only the most inquisitive. And every year this complex subject became more and more difficult.
In high school, geometry appears and everything becomes even more complicated and incomprehensible. Perhaps many at least once in their lives, but in their hearts, cursed a science they did not understand and wondered why it was needed at all, and whether it would be needed in life.
Perhaps in Everyday life It was not possible to apply the knowledge acquired at school. It is unlikely that it was necessary to calculate logarithms and quadratic equations or prove that two parallel ones will never converge. But where knowledge of geometry and mathematics may certainly be needed is in construction and when carrying out repairs.
This article will focus on calculating the right angle, which is required during the construction of buildings. Precision in building construction must be complied with, because only exact calculations can eliminate distortions and instability in the organization of the entire building. Calculating a right angle during construction is not such a difficult process, which will require knowledge and application of some simple rules mathematics and geometry. This will be discussed in more detail below.
Is it really a right angle?
Perhaps some readers who read the title of this article will object that a right angle cannot always be obtained, and even and precise right angles are not always used in construction.
And, in principle, they are right. It is very difficult to obtain, especially if there is unevenness in the foundation on which the building is being constructed. But even taking this into account, under no circumstances can one conclude that the calculation of a right angle can be done simply “by eye”. In any case, if it is not possible to calculate the ideal right angle, then it is necessary to achieve the closest value to the ideal angle of 90 degrees. And this can be achieved using simple tools and not the most complex mathematical knowledge and knowledge of geometry.
What is needed to determine a right angle?
So, what tools will you need to use to check the right angle? It’s worth noting right away that no equipment or serious tools are required for this. You will need to use very simple things that can be found in almost every household. And even if you don’t have them on hand, you can easily purchase them in the store. There will be no difficulties with this.
To calculate a right angle you need to take:
- Pencil;
- Construction tape.
That's all. It's that simple.
How can you calculate a right angle?
So, this article will describe the 3-4-5 principle when determining a 90 degree angle. There is nothing complicated about this. You just need to think a little and delve into all the calculations that can help in checking the angle.
So, the following steps need to be outlined:
Conclusion
Here's how easy it is to calculate a right angle without using any construction tools and instruments. You can use the simplest, but at the same time very effective remedy , which, coupled with the use of existing knowledge and simple calculations, can help make measurements.
When using the suggested values, the key is the final measurement between the two marks that were made earlier. The distance, which will be exactly 5 meters, will seem to be straight. If the value is more or less than 5 meters, this will mean that it is not straight.
TO
High-quality repairs and finishing imply good room geometry. Without verified geometry, at least in the most in the right places, good repair it won't be possible to do. Here I will tell you how to make a 90 degree angle between walls with your own hands and where it is really needed. You can also read the article at the link → and how to check the correctness of the geometry, and what will happen if the geometry is broken.
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Where you need a 90 degree angle between walls
90 degree angles are generally not displayed everywhere in the apartment. In economical repairs, and in most euro-renovations, 90-degree angles are needed only in two places:
- in the corner where the kitchen furniture will hang/stand,
- and in the bathroom, where the bathtub itself will be located, in two adjacent corners(or in one, if the shower stall is in the corner). Or in all 4 corners of the bathroom, since the sink will be there, washing machine and so on.
In other cases, everything is at the request of the customer or the person carrying out the repairs on their own.
How to check and set angles
The angle is easy to check construction square, can be purchased at the store, if you are going to draw out the corners, you will need it.
Just lean the square against the inner corner. We will not consider external corners for now, for the sake of understanding the process itself. After understanding how to level internal corners at 90 degrees with your own hands, external ones will not be a problem for you.
Let's see what happens. If everything is fine, there are no gaps between the square, then relax. If the gap exceeds 5 mm, then you should be wary and find out how to align such an angle in a straight line so that both the bathtub and cabinets hang well. The fact is that a gap of 5 mm under a small, even half-meter (the length of at least one edge) square, for the entire length of the wall turns out to be quite large and at the end of the wall can reach 5 cm.
Making a square yourself
You can build a square yourself, and of any size. It is most convenient to make such a square from plasterboard profiles 27*28 mm (rigid or semi-rigid).
We use the Egyptian triangle rule, in which: if the legs of the angle are equal to 3 and 4 parts, and the hypotenuse is 5 parts, then the angle will be rectangular (a right angle between the legs).
Cut and bend required length profile in the middle (the sides of our square do not have to be equal to the 3 and 4 parts we have defined, the rule is only needed to make a right angle). We bend it and take it as 1 part, for example, 30 cm. The larger you make the part, the “straighter” the angle will be.
General rules for any foundation
Select a starting point. The first side of our foundation needs to be tied to some object on our site.
Example. Let's make sure that our foundation (house) is parallel to one of the sides of the fence. Therefore, we stretch the first string equidistant from this side of the fence to the distance we need.
Construction of a right angle (90⁰). As an example, we will consider a rectangular foundation in which all angles are as close as possible to 90⁰.
There are several ways to do this. We will look at 2 main ones. © www.site
Method 1. Golden triangle rule
To construct a right angle we will use the Pythagorean theorem.
In order not to go deep into geometry, let's try to describe it more simply. So that between two segments a And b To make an angle of 90⁰, you need to add the lengths of these segments and derive the root of this sum. The resulting number will be the length of our diagonal connecting our segments. It is very easy to do the calculation using a calculator.
Usually, when marking the foundation, the dimensions of the sides are taken so that when taken from the root, a whole number is obtained. Example: 3x4x5; 6x8x10.
If you have a tape measure, then in general there will be no problems if you take segments that are different from those in common use. For example: 3x3x4.24; 2x2x2.83; 4x6x7.21
If we made measurements in meters, then the values turn out to be very clear: 4m24cm; 2m83cm; 7m21cm.
Calculator
It is also worth noting that measurements can be made in any length measurement system; the main thing is to use the aspect ratio we know: 3x4x5 meters, 3x4x5 centimeters, etc. That is, even if you don’t have a tool for measuring length, you can take, for example, a staff (the length of the staff does not matter) and measure it with it (3 staff x 4 staff x 5 staff).
Now let's see how to put this into practice.
Instructions for marking a rectangular foundation
Method 1. Rules of the golden triangle (i.e. Pythagoras)
Let's look at the example of building a rectangular foundation with dimensions 6x8m using the golden triangle (so-called Pythagoras).
1. Mark the first side of the foundation. This is the easiest part in constructing our rectangle. The main thing to remember. If we want our foundation (house) to be parallel to one of the sides of the fence or other object on the site or beyond, then we make the first line of our foundation equidistant from the object we have chosen. We described this procedure above. To place the first string, you can use pegs firmly fixed in the ground, but in ideal For this purpose, use cast-offs. We will use it. We will make the distance between the cast-offs for this side 14 m: between the cast-offs and future corners, 3 m and 8 m under the foundation.
2. Pull the second string as perpendicular to the first as possible. In practice, it is difficult to pull it perfectly perpendicular, so in the figure we also showed it slightly deflected.
3. We fasten both strings at the intersection point. You can fasten it with a staple or tape. The main thing is to be reliable.
4. We begin to form a right angle using the Pythagorean theorem. We will build right triangle with legs 3 by 4 meters and a hypotenuse of 5 meters. To begin with, we measure 4 meters from the intersection of the strings on the first string, and 3 meters on the second. Place marks on the lace using tape (clothespin, etc.).
5. Connect both marks with a tape measure. We fix one end of the tape measure at the 4 meter mark and lead it towards the 3 meter mark on the other string.
6. If we have a right triangle, then both marks should converge at a distance of 5 meters. In our case, the marks did not match. Therefore, in our case, we move the string to the right until the 3 m mark coincides with the 5 m division of the tape measure.
7. As a result, we got a right triangle with an angle of 90⁰ between the two strings.
8. We don’t need any more marks and they can be removed.
9. Let's start building a rectangle. We measure on both strings the lengths of the sides of our foundation to be 6 and 8 meters, respectively. We put marks on the strings.
10. Pull the third string as perpendicular to the first string as possible. We fasten both strings at the 8 m mark.
11. Pull the fourth string as perpendicular to the second string as possible. We fasten both strings at 6 meter marks.
12. We make marks on the third string 6 meters and on the fourth 8 meters.
13. To get a quadrilateral with right angles in our case, it is necessary that both marks on the third and fourth strings coincide. To do this, move both strings until the marks connect.
14. As a result, if everything was measured correctly, then we should get a regular rectangle. Let's check if it turned out by measuring the diagonals.
15. We measure the lengths of the diagonals. If they are the same, as in our case, we have a regular rectangle. The diagonals have the same length in an isosceles trapezoid. But we know one angle of 90⁰, and in an isosceles trapezoid there are no such angles.
16. Ready marking of a rectangular foundation using the Pythagorean theorem. © www.site
Method 2. Web
A very simple way to make markings in the form of a rectangle with corners of 90⁰. The most important thing we need is twine that does not stretch, and the accuracy of your measurements using a tape measure.
1. Cut the pieces of twine that we will need to form the markings. IN in this example we are building a foundation with sides 6 by 8 meters. Also, to correctly construct a rectangle, we will need equal diagonals, which for a 6 by 8 meter rectangle will be equal to 10 meters (i.e. Pythagoras is described above). You also need to take a reserve length of string for fastening.
2. We connect our “web” as in the picture. We fasten the sides with diagonals in 4 places in the corners. The diagonals themselves do not need to be fastened at the intersection point.
3. Pull the first string (points 1,2). We will secure it with pegs. The main thing is that the pegs hold firmly in the ground and do not move away when our structure is pulled. This important point need to be taken into account.
4. We tighten corner 3. The main condition is that string 1-3 and diagonal 2-3 do not sag and are as tight as possible. After fixing with a peg at point 3, we have an angle at point 1 of 90⁰.
5. Pull corner 4 and install the peg. We make sure that the twine at points 2-4, 3-4 and diagonal 1-4 do not sag and are as tight as possible.
6. If all conditions are met, then the result should be a rectangle with angles as close as possible to 90⁰.
Marking for the foundation of the house
We make a two-tier cast-off. The lower tier is the level of the pillars.
The upper tier of cast-off is the level of the grillage.
Create a rectangle for the outer contour using the so-called Pythagoras. Then we retreat by an amount equal to the width of the tape and make an internal contour.
The easiest way to mark. We build a rectangle according to the dimensions of the foundation using the Pythagorean theorem to find the right angle. © www.site
From the author
In this article, we looked at how to make markings for the foundation with your own hands by constructing a rectangle with angles of 90⁰. In general, there is nothing complicated about the markup. The cost of the issue is the cost of twine, boards for casting (an economical option - pegs) and the ability to use a tape measure.