Introduction
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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. Dividing both circles into 360 parts, we get equal angles of 1°, although the sides of one angle are visually longer than 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
- Zubareva I.I., Mordkovich A.G. Mathematics. Grade 5 - M.: Mnemosyne, 2013.
- Vilenkin N.Ya. etc. Mathematics. 5 cells - M.: Mnemosyne, 2013.
- Erina T.M. Mathematics 5th grade Slave. notebook for study Vilenkina, 2013. - M.: Mnemosyne, 2013.
- Shkolo.ru ().
- Cleverstudents.ru ().
- Festival.1september.ru ().
Homework
- Zubareva I.I., Mordkovich A.G. Mathematics. Grade 5 - M.: Mnemosyne, 2013. Pp. 144 No. 522.
- Draw angles: 23°, 167°, 84°.
- Ershova A.P., Goloborodko V.V. Independent and control work in mathematics for grade 5 (5th ed.) - 2010. Pp. 163 no. 3.
Protractor, also known as Smart Protractor, is an application for measuring the angle or inclination of an object that is part of the Smart Tools.
Smart Tools is a toolkit of six applications aimed at performing various tasks. With some you can measure the speed or distance, with the help of others (compass) - to determine the location of the server. The same solution should be used for measuring angles.
Usage
The protractor will become an indispensable assistant in cases where the user needs to measure the slope of a certain object, but the necessary tool is not at hand. This application is useful even in such banal and everyday cases, when it became necessary to hang a picture on the wall of the living room. Do you want maximum accuracy? Smart Protrator will provide it for you!
Opportunities
The software includes three different goniometers with their own "purpose". A standard goniometer is needed in order to determine the angle of an object that can fit on the surface of the gadget's screen. The second protractor contains an additional pendulum showing the deviation of the screen from the center. Well, the third tool should be used when you want to measure the angle of inclination of massive objects that will not fit on your smartphone. In this case, a kind of conductor that helps to identify the angle of inclination is the camera, which, of course, must be working.
Please note that the free version of the Protractor for Android provides only the first two goniometers. Purchasing Pro gives you access to a third goniometer and also adds a special ruler to these elements.
Key features
- includes three different goniometers for different purposes;
- the free version has somewhat limited functionality;
- activation Pro adds a fourth element for measuring the angle of inclination;
- the application has a simple interface that is easy to master;
- Protractor requires Android 4.0 or later;
- You can download and use the app for free.
Protractor
The protractor is the most common tool for measuring angles. The term "protractor", which in French means "transfer".
This tool may have a different appearance, but any protractor has a scale that is located on a semicircle. The center of this scale is marked with a dash or hole.
Now let's look at this semicircular object and find a hole in the center of its flat corner. The hole in the protractor is the reference point, and from this point begins the reference line, which runs along the entire flat part of the protractor.
The protractor refers to a fairly simple hand tool with which you can draw and measure any angle. Having learned how to use it, you will not have problems in solving problems related to measuring angles.
Angle measurement
Measurement of angles occurs by comparing the angle being measured with the angle, which serves as a unit of measurement. To do this, you need to gradually fill the inner part of the measured angle with single angles, tightly stacking one to the other. The number of angles laid gives the measure of the angle being measured. The most convenient and common tool for measuring angles is a protractor.
The angle unit can be any angle or other common unit of measure.
Such a unit can be one degree, which is 1/180 of a straightened angle. Therefore, one hundred and eighty angles equal to one degree can be placed in a developed angle.
For a more illustrative example, we take a model of half a pie or pizza, which are cut into one hundred and eighty equal pieces and stacked tightly one to the other. At the same time, we see that the sides of the corners are combined with the side of the expanded corner, and when we reach the last one, we will see that it coincides with the other side of the expanded corner.
So, in the example of the picture below, we see that in one sixth of the expanded angle, an angle of one degree can be laid thirty times, and in half of the expanded angle - 90 times.
If it is necessary to measure an angle that is less than one degree, then in such cases other units of measurement are used, such as minutes or seconds.
The degree measure of an angle is a positive number that shows us the number of times that a degree or its parts fit into a given angle.
But now you have already learned that to measure angles, they most often use such a common unit of measurement as a degree. Its angle is one hundred and eightieth of a straight angle.
For the convenience of measuring angles, a drawing tool such as a protractor is used.
How to measure an angle with a protractor
And now let's start measuring the angle using a tool such as a protractor. To measure an angle with a protractor, you need to follow simple steps. For this you need:
First, mark at the top of the angle you will be measuring the starting point of reference.
Secondly, you need to combine the bottom line of the angle with the line from which the reference will be made.
Next, it is necessary to mark on paper the located numbers that would correspond to the measured angle.
Your next step is to draw a line that intersects the protractor markup. The number located at the place of this intersection indicates the angle in degrees.
Since some protractors have two oppositely directed scales, this allows you to measure angles that are directed in different directions.
