In the lesson, we will remember what units of measurement are, find out what units can be used to measure angles, get acquainted with such a unit of measurement as a degree, learn how to measure angles in degrees and draw them with a protractor. We will also learn about other angle units that are used in different situations.

If you have difficulty understanding the topic, we recommend that you watch the lesson and

Some things can be measured, some cannot. For example, friendship or love cannot be measured. And the distance, weight, temperature is quite possible. In order to measure something, everyone needs to agree on the units of measurement.

Meter, inch, arshin - these are such agreements when measuring length. The reference meter is kept in France, in the Chamber of Weights and Measures. Kilogram, pound, pood are conventions for measuring mass. The reference kilogram is also stored in the Chamber of Weights and Measures.

Units of measurement are designed for specific quantities. Do not measure weight in seconds, but time in arshins.

The situation is the same in geometry. There are centimeters for measuring lengths of segments, but they are not suitable for measuring angles. There are units for measuring angles. In this lesson, we will look at one of them, namely degrees.

Divide the full angle into 360 equal parts. For this it is convenient to use a circle. Let's divide it into 360 parts and connect each resulting division with the center. We get 360 equal angles (see Fig. 1).

Rice. 1. A circle divided into 360 equal angles

One such small angle is called an angle of 1° (see Fig. 2).

Rice. 2. 1 degree

It doesn't matter what size the circle we're dividing will be. Divide both circles into 360 parts, we get equal angles at 1°, although the sides of one angle are visually longer than those of the other (see Fig. 3).

Rice. 3. Angles are equal

The sides of the corners can be continued indefinitely, this does not change the size of the corner (see Fig. 4).

Rice. 4. A more explicit example of equality of angles

The magnitude of any angle is the number of times an angle of 1° fits into it.

Here we see an angle of 13° (see Fig. 5).

Rice. 5. Angle 13°

It is clear that full angle consists of 360 such angles. That is, it is equal to 360° (see Fig. 6).

Rice. 6. Full angle

Expanded angle is half a full angle. It is equal (see Fig. 7).

Rice. 7. Expanded angle

Right angle is half turned and equals 90° (see Fig. 8).

Rice. 8. Right angle

There is no need to store the standard of the degree somewhere. If necessary, you can always divide the full angle into 360 parts, or the expanded one into 180, or the straight line into 90.

A ruler is needed in order to measure the existing segment or draw a segment of the desired length. To measure an angle or draw an angle of the desired size, we also use a ruler, but not a straight line, but a round one. It is called a protractor (see Fig. 9).

Rice. 9. Protractor

The units of measurement on it are degrees. The scale starts at zero and ends at 180°. That is, the maximum angle that we can measure or draw is 180° rotated.

Protractors can be of different sizes, but this does not affect the size of the angles they measure. For a larger protractor at the corners, you need to draw the sides longer.

1. Measure a couple of angles.

The straight part of the protractor is aligned with one side of the corner, the center of the protractor with the top of the corner. We look where the second side of the corner turned out to be - 54 ° (see Fig. 10, 11).

Rice. 10. Angle measurement

Let's do the same with the second angle, 137°.

Rice. 11. Angle measurement

If the side of the corner does not reach the scale, then it must first be extended.

2. Draw the angles 29°, 81° and 140°.

First, draw one side of the corner along the ruler (see Fig. 12).

Rice. 12. Building one side of the corner

We mark the top. Compatible with transporter. We mark with a dot the desired value of the angle - 29 ° (see Fig. 13).

Rice. 13. Using a protractor to draw angles

We remove the protractor. We connect the resulting point with the vertex (see Fig. 14).

Rice. 14. Angle 29°

In the same way, we build two other corners (see Fig. 15).

Rice. 15. Building corners

So, we discussed with you that people agreed to use degrees to measure angles. Degree is a full angle.

A tool for measuring and constructing angles is a protractor.

You can not use the names of the corners - full, deployed, straight. We can just say - 360 degrees, 180 or 90 degrees.

In fact, it happens when we measure some quantities with units that seem to be not intended for them, “foreign” units.

Is it possible to measure distance in minutes? Yes, we often use this method. "It's 5 minutes from my house to school." To be more precise, then "5 minutes on foot." Here we use a well-known value - the speed of a pedestrian. And the value "5 minutes" actually means "the distance that a pedestrian walks in 5 minutes." Pedestrian speed - 5 km / h, 5 minutes is hours, multiply one by the other. We get about 400 meters. Not very accurate, but convenient.

Exactly on the same principle, another unit of distance is arranged - a light year. A light year is the distance that light travels in 1 year. This unit is used to measure the distance between stars.

A very common example of using a "foreign" unit of measurement is to measure weight in kilograms. In fact, a kilogram is a unit of mass, and weight is another physical quantity. If you want to know in more detail what is the difference between mass and weight, and why measuring weight in kilograms is not correct, then type “mass and weight” in the search engine and get a lot of explanations about this.

Atmospheric pressure is still measured in millimeters (millimeters of mercury).

Although the angle has its own “native” units of measurement - degrees, which we go through in this lesson, it can still be measured using linear quantities, such as centimeters. If you need to measure the angle, then you can complete it to a triangle, so that one angle is right, and divide the length of one side by the other.

We get the value of the angle, which is called the tangent.

If you increase the triangle, then nothing will change (see Fig. 16).

Rice. 16. Tangent

After all, how many times one side has increased, so much the other.

That is, quantities can often be measured in “foreign” units, but this is a little more complicated, some additional agreements are needed there.

There are other units for measuring angles.

1. Minutes and seconds.

Just as a meter can be divided into decimeters, centimeters, millimeters for more accurate measurements, so degrees can be divided into smaller units.

If an angle of 1 ° is divided into 60 equal parts, then the value of the resulting angle is called a minute, 1 ′.

If a minute is divided into 60 parts, then the resulting value is called a second. A second is already a very small value, but it can also be divided further.

Why did they begin to divide the full angle into 360 parts, because it is not very convenient? In ancient Babylon there was a sexagesimal system (we have a decimal system). They were comfortable dividing by 60.

2. Grads.

To make the measurement of angles closer to our decimal number system, grads have been proposed. To do this, the right angle is divided into 100 parts. The resulting value is called deg. The full angle is then 400 degrees. The system did not take root, and now it is not used.

3. Radian.

If we take two radii of a circle so that a piece of a circle between them is also equal to the radius, then we will take the angle between the radii as a new unit of measurement. It is called 1 rad (radian). This measure is used on a par with the degree. It has its advantages and disadvantages compared to degrees (see Fig. 17).

Rice. 17. Radians

For example, now a full angle (the whole circle) does not consist of an integer number of unit angles. A full angle consists of more than 6 unit angles. Not very convenient, but now the length of the arc (part of the circle) and the angle are well connected. If we take a circle with a radius of 1 cm, then the angle value coincides with the length of the arc. Angle 1 rad - arc 1 cm, angle 2 rad - arc length 2 cm.

Bibliography

1. Zubareva I.I., Mordkovich A.G. Maths. Grade 5 - M.: Mnemosyne, 2013.
2. Vilenkin N.Ya. etc. Mathematics. 5 cells - M.: Mnemosyne, 2013.
3. Erina T.M. Mathematics 5th grade Slave. notebook for study Vilenkina, 2013. - M.: Mnemosyne, 2013.

1. Shkolo.ru ().
2. Cleverstudents.ru ().
3. Festival.1september.ru ().

Homework

1. Zubareva I.I., Mordkovich A.G. Maths. Grade 5 - M.: Mnemosyne, 2013. Pp. 144 No. 522.
2. Draw angles: 23°, 167°, 84°.
3. Ershova A.P., Goloborodko V.V. Independent and test papers in Mathematics for Grade 5 (5th ed.) - 2010. Pp. 163 no. 3.

How to measure an angle?

When solving problems in geometry, we constantly encounter different figures - flat and three-dimensional. Most geometric shapes are made up of sides and angles. The exception is a circle, ellipse, ball. Measuring angles is an important part of the solution, sometimes only the correct measurement helps to solve the problem and find the right answer.

What is the correct way to measure the angle?

A protractor is a special device, somewhat similar to a ruler, which has an arc with a scale applied to it. The numbers on this scale represent degrees. And, as you know, angles are measured in degrees.

How to measure an angle with a protractor

It is enough to attach the center of the protractor to the point - the top of the corner. The center of the protractor is the small point under the arc on the straight line. A small hole is often made in it so that it is convenient to put it on the top of the corner.

How to measure an angle without a protractor

In some special cases, the angle can be counted from the cells in the notebook. For example, a 90° angle is drawn as 2 perpendicular lines coming from the same point. They coincide with the cells of the notebook both horizontally and vertically. If one side of the corner coincides with the cells horizontally, and the other crosses each cell exactly diagonally, then this angle will be equal to 45 °.

There is also a way to measure the angle using a watch. If you suddenly forgot the protractor, and you have a watch with hands on your hand, then try to attach the center of the watch to the top of the corner, align one side with the number 6, and extend the second with a dotted line. One minute is equal to 6 degrees.

The method, of course, is not very convenient, but sometimes it saves.

Back in school, every person came across such a thing as a protractor. But, unfortunately, not everyone, both at school and at an older age, knows how to use it correctly. This manual will teach everyone how to use this item correctly.

Let's start with what a protractor is. A protractor is a tool for measuring the degree value of angles. Most often, such a tool has a semicircular shape. But there are exceptions, namely protractors, which are completely round (360 degrees).

How to use a protractor correctly

• First you need to understand what this tool is. It is a semicircular object (as mentioned above, maybe round) with a small hole in the middle, which is called the reference point. It is the latter (reference point) that needs to be aligned with the top of the triangle.
• Further, to measure the required angle, it is necessary to place the base of the protractor parallel to the side of the angle or the leg of the triangle. To do this, you need to select the side of the triangle (it will be the base one, it is with this side that the base of the protractor must be combined). Do not confuse the baseline of the angle and the base of the protractor (they are completely different things).
• Once you have combined the reference point with the top of the corner, and the base of the protractor with the leg, you can safely measure the angle. Then the second leg of the triangle will point to a scale with numbers on the semicircle of the protractor. The main thing is not to get confused with the numbers, because by doing this, you will make a mistake in measuring the required angle.
• It should be understood that the larger (blunter) the angle, the greater its degree value. The maximum angle can be 180 degrees.
• The smallest (acute) angles can only be less than 90 degrees, all that are greater than this value are already considered large (obtuse).

How to correctly measure the degree value of an angle with a protractor

To do this, you need to do the following:

• The center point (reference point) must be aligned with the top of the angle to be measured. The protractor must be fixed in place with a pen, pencil or any other object. After that, it must be rotated so that one of the sides of the corner coincides with the base of the protractor (the semicircle with the degree scale should look up).
• Now you need to look at what number on the semicircle the second side of the angle points to (it must intersect the semicircle of the protractor). You need to see what number this line passes through.
• If the line cannot be extended (it still does not reach the semicircle of the protractor), you should take a piece of paper or a ruler and combine it with the side that does not reach the semicircle. In this case, the ruler should “extend” the second side of the angle until it intersects with the semicircle indicated by the degrees.

How to draw an angle with a protractor

• First you need to draw a line with a protractor. This line will be the base. It is on it that you will be guided in order to draw the second one. For better convenience, it must be placed horizontally.
• On this line, you need to mark a point that will be the top of your corner. Such a point must be aligned with the reference point on the protractor.
• Next, you need to combine the base line of the angle with the base of the protractor. After that, you should look at the semicircle of the protractor and select the desired degree value. On paper, draw a dot next to this value. To such a point it will be necessary to draw a second line from the top of the corner.
• After that, you can safely put the protractor aside.
• Now you need to pick up a ruler and connect the top of the corner and the point that has already been drawn near the degree value you need.

It is done! You got an angle with the degree value you need. In the end, I would like to say that using a protractor is quite simple if you correctly use all these tips. But if you managed to get a little confused with the degree scales, it is worth remembering that they are absolutely the same, only they go in opposite directions for the convenience of using the protractor.

measure angle means to find its value. The angle value shows how many times the angle selected for the unit of measure fits into the given angle.

The unit of measure for angles is usually degrees. Degree is the angle equal to the part of the straight angle. To indicate degrees in the text, the ° sign is used, which is placed in the upper right corner of the number indicating the number of degrees (for example, 60 °).

Measuring angles with a protractor

A special device is used to measure angles - protractor:

The protractor has two scales - internal and external. The reference point for the inner and outer scales is located on different sides. To get the correct measurement result, the degrees must start from the correct side.

The measurement of angles is carried out as follows: the protractor is placed on the corner so that the top of the corner coincides with the center of the protractor, and one of the sides of the corner passes through the zero division on the scale. Then the other side of the angle will indicate the angle value in degrees:

They say: corner BOC is 60 degrees, angle MON equals 120 degrees and write: ∠ BOC= 60°, ∠ MON= 120°.

For a more accurate measurement of angles, fractions of a degree are used: minutes and seconds. Minute is an angle equal to a fraction of a degree. Second is an angle equal to a fraction of a minute. Minutes are marked with " , and seconds are sign "" . The sign of minutes and seconds is placed in the upper right corner of the number. For example, if the angle has a value of 50 degrees 34 minutes and 19 seconds, then write:

50°34 " 19""

Angle measurement properties

If the ray divides a given angle into two parts (two angles), then the value of this angle is equal to the sum of the values ​​of the two resulting angles.

Consider the angle AOB:

Ray OD divides it into two angles: ∠ AOD and ∠ DOB. Thus, ∠ AOB = ∠AOD + ∠DOB.

The straight angle is 180°.

Any angle has a certain value greater than zero.

Back in school, every person came across such a thing as a protractor. But, unfortunately, not everyone, both at school and at an older age, knows how to use it correctly. This manual will teach everyone how to use this item correctly.

Let's start with what a protractor is. A protractor is a tool for measuring the degree value of angles. Most often, such a tool has a semicircular shape. But there are exceptions, namely protractors, which are completely round (360 degrees).

How to use a protractor correctly

• First you need to understand what this tool is. It is a semicircular object (as mentioned above, maybe round) with a small hole in the middle, which is called the reference point. It is the latter (reference point) that needs to be aligned with the top of the triangle.
• Further, to measure the required angle, it is necessary to place the base of the protractor parallel to the side of the angle or the leg of the triangle. To do this, you need to select the side of the triangle (it will be the base one, it is with this side that the base of the protractor must be combined). Do not confuse the baseline of the angle and the base of the protractor (they are completely different things).
• Once you have combined the reference point with the top of the corner, and the base of the protractor with the leg, you can safely measure the angle. Then the second leg of the triangle will point to a scale with numbers on the semicircle of the protractor. The main thing is not to get confused with the numbers, because by doing this, you will make a mistake in measuring the required angle.
• It should be understood that the larger (blunter) the angle, the greater its degree value. The maximum angle can be 180 degrees.
• The smallest (acute) angles can only be less than 90 degrees, all that are greater than this value are already considered large (obtuse).

How to correctly measure the degree value of an angle with a protractor

To do this, you need to do the following:

• The center point (reference point) must be aligned with the top of the angle to be measured. The protractor must be fixed in place with a pen, pencil or any other object. After that, it must be rotated so that one of the sides of the corner coincides with the base of the protractor (the semicircle with the degree scale should look up).
• Now you need to look at what number on the semicircle the second side of the angle points to (it must intersect the semicircle of the protractor). You need to see what number this line passes through.
• If the line cannot be extended (it still does not reach the semicircle of the protractor), you should take a piece of paper or a ruler and combine it with the side that does not reach the semicircle. In this case, the ruler should “extend” the second side of the angle until it intersects with the semicircle indicated by the degrees.

How to draw an angle with a protractor

• First you need to draw a line with a protractor. This line will be the base. It is on it that you will be guided in order to draw the second one. For better convenience, it must be placed horizontally.
• On this line, you need to mark a point that will be the top of your corner. Such a point must be aligned with the reference point on the protractor.
• Next, you need to combine the base line of the angle with the base of the protractor. After that, you should look at the semicircle of the protractor and select the desired degree value. On paper, draw a dot next to this value. To such a point it will be necessary to draw a second line from the top of the corner.
• After that, you can safely put the protractor aside.
• Now you need to pick up a ruler and connect the top of the corner and the point that has already been drawn near the degree value you need.

It is done! You got an angle with the degree value you need. In the end, I would like to say that using a protractor is quite simple if you correctly use all these tips. But if you managed to get a little confused with the degree scales, it is worth remembering that they are absolutely the same, only they go in opposite directions for the convenience of using the protractor.

How to use a protractor correctly?

I have two answers. One is simple and carries information about exactly how the angle should be measured. Here I decided to post just a video lesson with a virtual protractor, which shows in detail the process of determining the angle.

The second answer concerns the moment, how can you manage to hold the protractor in such a way that you can measure the angle correctly? There are many difficulties and inconveniences here. Either the protractor moves out, or sticks out, does not press against the sheet, and the line of the corner is not clearly visible through the colored plastic. It is even worse when nothing is visible through the protractor, not even the marks of the divisions of the degrees of the angles themselves.

To correctly measure the angle, I advise you to hold any protractor, no matter what you purchase, when measuring with three (two) fingers. The first point is from the bottom right, the second point of holding the protractor is from the bottom left, the third from the top along the radius of the corner. In this position, it is difficult to see the angle, since the finger at the top closes part of the visibility, and without it the protractor crawls away.

Therefore, it is better for inexperienced users to take protractors built into the ruler,

even better from completely transparent plastic.

Such a protractor is conveniently held with two fingers in the center of the ruler - everything is visible and nothing slips anywhere, does not move out, plus excellent visibility through non-colored plastic.

The iron protractor is ideal for advanced users (high school students), has a couple of drawbacks - it slides on paper and can bend.

As you know, transporters measure angles that consist of 2 beams. We combine the 1st beam with the top of the lower ruler, and the 2nd beam will show the angle in degrees, which is located on the upper semicircle.

What is a protractor?

A protractor is an object with which each of us can not only measure angles, but also build them. Outwardly, it resembles a semicircular ruler with a scale and divisions. Below, on a flat surface, is the usual straight ruler for measuring segments. In the upper part there is a semicircle with a double scale for measurements. In each of the directions, the scale is dispersed along the protractor from 0 to 180 degrees.

Terms of use

At school, they explain what a protractor is in mathematics lessons. This is where the need for measurements comes in.

In order for us to find out what one degree is equal to, we need to divide the circle into 360 equal parts. One of these parts will be equal to 1 degree. The size of the circle will not affect the degree in any way! This is easy to check.

Let's draw two circles of different diameters and divide each into 360 equal parts. Then we superimpose the smaller circle on the larger one and see that the lines matched.

We measure the angle

A protractor helps to build and measure the angle. The degree is a common unit used to measure angles. There are several types of angles:

• Spicy. This is called an angle up to 90 degrees.
• A right angle is a 90 degree angle.
• An obtuse angle ranges from 90 to 180 degrees.
• A straight angle is a straight line or 180 degrees.
• A full angle looks like a circle and is 360 degrees.

It's easy to figure out how to measure an angle. In order to find out what the value of the angle is, we need to install the protractor so that its center is located at the top of the angle, and the straight side coincides with one of its sides. The scale will tell us the number of degrees of a given angle. In such a simple way, we can find out what is around the corner in front of us.

To construct an angle with a given degree, you should attach the straight part of the protractor to the line, and its center to the beginning of the line. Subsequently, this point will be the vertex of the corner. Then on the scale we look for given number and make a point. Now the protractor can be removed and connected by a segment to the beginning of the line (top of the corner) with the marked point.

School stationery produced by different companies differ in material, color, size. So: for those whose protractor turned out to be longer than the length of the angle, and it is not possible to determine its value, the side of the angle must be extended using a straight ruler.

Schoolboy set

It is not for nothing that junior students are not familiar with the protractor. When applying it, a certain knowledge base must be laid. For full-fledged work with him in the lesson, the guys study a number of related subjects. Before learning what a protractor is, students must master a straight ruler perfectly, draw straight lines, study addition and subtraction, master a compass, know geometric figures and so on. This whole process takes time, and only after primary school, the student can add a protractor to their toolbox.

Pupils are now offered school stationery in a huge selection. The protractor is no exception. Manufacturers try to satisfy the most demanding needs of customers. Tools are made in various colors. Bright colors are always loved by children. Sometimes even in the same class you cannot find the same protractors, which makes it easier to find them if you lose them. Everyone chooses shapes and sizes to their taste.

Most of these products are made of plastic, and this significantly reduces its cost. But there are wooden and even iron protractors. As practice shows, although metal ones are opaque, they are more practical in the sense that the scale is not erased, and this allows you to use it in action much longer, accurately determining the angles.

The protractor is not as much in demand by schoolchildren as a ruler, but it accompanies students up to final exam. Some of the school's graduates choose specialties that are related to measuring and constructing angles, designing buildings and structures, and working with drawings. By virtue of their professions, they constantly have to deal with protractors and its derivatives. But even former classmates of current engineers, sometimes even with the deepest humanitarian bias, will easily remember the skills of handling this object and determine the number of degrees at any angle.

Outcome

Today, modern children are accustomed to getting any information from the Internet. However, it will not help in any way in measuring angles. Only the ability to use a protractor will make it possible to correctly determine them. For future engineers and designers, this will undoubtedly come in handy in their work, and everyone educated person must have the skills to work with protractors, so everyone should be able to use such a tool!