How to set 90 degrees. Egyptian triangle

At finishing works and construction sometimes requires clear geometry: perpendicular walls and other structures that require a right angle of 90 degrees. An ordinary square cannot check or mark corners with sides of several meters. The described method is excellent for marking or checking any angles - the length of the sides is not limited. The main tool for measurements is a tape measure.

We will look at accurately marking right angles, as well as a method for checking already marked corners on walls and other objects.

Pythagorean theorem

The theorem is based on the statement that In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This is written as a formula:

a²+b²=c²

Sides a and b are legs, between which the angle is exactly 90 degrees. Therefore, side c is the hypotenuse.

By substituting two known quantities into this formula, we can calculate the third, unknown one. Therefore, we can mark right angles and also check them.

The Pythagorean theorem is also known as the “Egyptian triangle”. This is a triangle with sides 3, 4 and 5, and it does not matter in what units the lengths are. Between sides 3 and 4 is exactly ninety degrees. Let's check this statement with the above formula: a²+b²=c² = (3×3)+(4×4) = 9+16 = (5×5) = 25 - everything converges!

Now let's put the theorem into practice.

Checking right angle Let's start with the simplest thing - checking a right angle using the Pythagorean theorem. The most common example in finishing and construction is checking perpendicularity

So, we take any tested internal angle. On the walls (at the same height) or on the floor, mark segments of arbitrary lengths on both walls. The length of these segments is arbitrary; if possible, you need to mark as many as possible, but so that it is convenient to measure the diagonal between the marks on the walls. For example, we marked 2.5 meters (or 250 cm) on one wall and 3 meters (or 300 cm) on the other. Now we square the length of the segment of each wall (multiply by itself) and add up the resulting products. It looks like this: (2.5×2.5)+(3×3)=15.25 - this is the diagonal squared. Now we need to take the square root of this number √15.25≈3.90 - 3.9 meters should be the diagonal between our marks. If the measurement with a tape measure shows a different diagonal length, the angle being checked is rotated and has a deviation from 90°.

Right angle diagonal calculator

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Length a

Length b

Diagonal c

Extracting the square root has never attracted me - an ordinary person cannot do without a calculator, to to the same, not at all mobile devices calculators can extract it. Therefore, you can use a simplified method. You just need to remember: at a right angle with sides exactly 100 centimeters, the diagonal is 141.4 cm. Thus, for a right angle with sides of 2 m, the diagonal is 282.8 cm. That is, for each meter of the plane there are 141.4 cm. This method has one drawback: from the measured angle it is necessary to set off the same distances on both walls and these segments must be multiples of a meter. I won’t claim it, but in my humble experience, it’s much more convenient. Although you should not forget about the original method completely - in some cases it is very relevant.

The question immediately arises: which deviation from the calculated length of the diagonal is considered normal (error), and which is not? If the angle being tested with marked sides of 1 m is 89°, then the diagonal will decrease to 140 cm. From understanding this dependence, we can draw an objective conclusion that an error of a few millimeters in the diagonal of 141.4 cm will not give a deviation of one whole degree.

How to check the outer corner? Checking the external corner is essentially no different, you just need to extend the lines of each wall on the floor (or ground, using a cord) and measure the resulting internal angle in the usual way.

How to mark a right angle with a tape measure

The marking can be based both on the general Pythagorean theorem and on the principle of the “Egyptian triangle”. However, this is only in theory, lines are simply drawn on paper, but “catching” all selected sizes with stretched cords or lines on the floor is a more difficult task.

Therefore, I propose a simplified method based on the diagonal of 141.4 cm for a triangle with sides of 100 cm. The entire marking sequence is shown in the pictures below. It is important not to forget: the diagonal of 141.4 cm must be multiplied by the number of meters in segment A-B. Sections A-B and A-B must be equal and correspond to an integer number in meters.

Pictures enlarge by clicking!

How to mark an acute angle

Much less often there is a need to create acute angles, in particular 45°. To form such figures, the formulas are more complex, but this is not the most problematic. It is much more difficult to connect all the lines drawn or stretched with cords - this is not an easy task. Therefore, I suggest using a simplified method. First, a right angle of 90° is marked, and then the diagonal 141.4 is divided into the required number of equal parts. For example, to get 45°, you need to divide the diagonal in half and draw a line from point A through the division point. This way we get two 45 degree angles. If you divide the diagonal into 3 parts, you get three angles of 30 degrees. I think the algorithm is clear to you. Actually, I told everything that I could tell, I hope I presented everything in understandable language and you will no longer have questions about how to mark and check right angles. It is worth adding that any finisher or builder should be able to do this, because relying on a construction square small size

- unprofessional. Those who are engaged in independent construction know that before the construction of a structure begins, they must mark out the foundation with their own hands. Here we consider the case of starting work on the construction of a pile screw foundation

on the site, for a number of horticultural reasons, not cleared of useful plants. This made it difficult to work on marking the future foundation, but these difficulties were easily overcome with the help of a simple device for setting right angles.

How to mark the foundation with your own hands Usually marking the foundation in done by eye using a tape measure. First, posts marking the corners of the walls are placed at distances of the length and width of the future building. Then the diagonals of the resulting rectangle are measured and the process of rearranging two adjacent pillars begins until the diagonal measurements are aligned. According to the basics of geometry, a rectangle is a figure whose two diagonals are equal to each other. But it was precisely because of the fit that measuring the diagonals during the fitting process was difficult. Landings made it difficult to tighten the tape measure and obscured the rangefinder laser. But this difficulty can be overcome.

1. Before starting work, you must have minimal knowledge of geometry and know the solution to the Pythagorean theorem :). Let me remind you of the theorem. The square of the hypotenuse is equal to the sum of the squares of the legs in a right triangle.

2. Stretch a cord between two pegs indicating the first wall of the foundation. If the side of the foundation, for example, is 6 meters, then the distance between the pegs should be at least 8 meters.

3. Let's make a device for setting a right angle on the ground. To do this, you must purchase packaging. non-stretchy cord or use a steel cable. In total you will need about 13 meters of cord.

4. We tie the ends of the cord folded together so that the length of the resulting loop is 6 meters. Accuracy in tying and sizing is important.

5. Take a permanent felt-tip pen and, using a tape measure, make marks from the center of the knot at a distance of 3 meters in one direction and at a distance of 4 meters in the other direction. So we got rope right triangle. This invention will allow you to calculate the direction of a 90° angle by simply stretching the triangle.

Marking the first wall Lifehack kit Sides of a triangle

6. To work on the ground, we will need thin wooden pegs or pieces of thin reinforcement.

7. We install one peg to indicate the corner of the foundation on the marking line made earlier in step 2.

8. Take a rope life hack. We place the knot on the peg indicating the angle and stretch the sides of the rope triangle by driving the first peg at a distance of 4 meters into the wall markings of step 2. The bend of the cord should be at the marker mark of 4 meters.

9. Place the peg at the 3 meter mark. One side of the rectangle is parallel to the marking of the first wall, and the second side indicates the direction of the marking at a 90° angle for the second wall. Pythagorean theorem in action - see photo.

10. We stretch the marking cord for the second wall, parallel to the side of the triangle.

11. Conduct similar actions to mark the third wall.

12. We mark the lengths of the second and third walls on the markings and carry out control at one of the angles of the correct direction of the fourth wall. If the length of the wall in the markings was 6 meters and its direction crossed the marking points of walls two and three, then we can say that measuring the diagonals will give an equal result. If alignment does not work, check again that the markings are installed correctly.

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 exactly 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. Constructing a right angle - from the neighbor’s angle, using a compass-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.

Slopes at 90 degrees

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for increase ]

Then move on. When you place beacons on this wall, plaster it. And set up beacons again. When you have plastered all the walls, use a wide spatula to remove excess mortar from the corners. Corners should be smooth and clean. Your angles will be exactly 90 degrees - this is guaranteed.

Be sure to place a perforated corner on the outer corners. By level, of course. Apply a liquid layer of plaster to the right and left of the corner. Stretch it out with a big rule. The role of the lighthouse will be played by the corner, and the role of the second lighthouse will be the end of the rule itself. This will make your walls perfectly smooth.

Some deviations from the norm are allowed, but not more than 1mm. Try to have fewer holes and scratches. Then it will be much easier to putty and the putty consumption will be minimal. Beacons should not be removed from the bathroom and toilet. Yes, and there is no need to go through the liquid layer. There will still be tiles there.

If you have 90 degree angles in your bathroom, the tiles will look simply amazing. Because perfect angles- it is beautiful. Wallpaper or paint on the walls of a room with perfectly straight corners will also look perfect, without errors.

Technologies

Sgraffito technique - a step towards perfection of your interior
Recently, color has been used very often as a finish. decorative plaster, which is perfect for finishing building facades and various architectural elements

Silk plaster - a highlight in room design
Many designers have recently often used silk plasters for wall decoration. What are they? Let's figure out what their beauty and zest are?

Plaster: types, purpose, work technique
Plaster is a material intended for maintaining construction work. The technology for applying plaster is similar to the technology for putties with a slight difference - plaster is not sanded with abrasive materials

Plastering surfaces by machine - advantages
Plastering walls is one of the important stages room decoration. For small volumes of work, plaster is applied manually, but for objects larger than 300 m2, machine application of plaster is required

Preparing the surface for plastering
Plastering is considered the main work of leveling the surface, and is also a preparation for the next stage of repair. The plastering technology itself also requires a preparatory stage

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:

1. in the corner where the kitchen furniture will hang/stand,
2. and in the bathroom, where the bath 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

It’s easy to check the angle with a construction square; you can buy it at the store; if you’re going to cut out the corners, you’ll 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 align internal corners at 90 degrees with your own hands, external corners will not pose 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 27*28 mm plasterboard profiles (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).

We cut and bend the profile in the middle to the required length (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.