Protractor - Knowledge Hypermarket. Measuring angles

Protractor, also known as Smart Protractor, is an application for measuring the angle or inclination of an object that is included in the Smart Tools set.

Smart Tools is a set of tools of six applications aimed at performing various tasks. With the help of some you can measure speed or distance, with the help of others (compass) you can determine the location of the server. This solution should also be used to measure angles.

Usage

An inclinometer will become an indispensable assistant in cases where the user needs to measure the inclination of a certain object, but does not have the necessary tool at hand. This application will be useful even in such banal and everyday cases when there is a need to hang a picture on the living room wall. Want maximum accuracy? Smart Protrator will provide it to you!

Possibilities

The software includes three different protractors with their own "purpose". A standard protractor 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 that shows the deviation of the screen from the center. Well, the third tool should be used when you need to measure the angle of inclination of massive objects that will not fit on your smartphone. In this case, a kind of guide that helps to identify the angle of inclination is the camera, which, of course, must be working.

Please note that the free version of Protractor for Android provides only the first two protractors for use. Purchasing the Pro gives you access to a third protractor and also adds a special ruler to these items.

Key Features

• includes three different protractors for different purposes;
• the free version has somewhat limited functionality;
• activation Pro adds a fourth element to measure inclination angle;
• the application has a simple interface that is not difficult to master;
• To work with the protractor, Android 4.0 or later is required;
• You can download and use the application completely free of charge.

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Initial value

Converted value

degree radian grad gon minute second zodiacal sector thousandth revolution circle revolution quadrant right angle sextant

More about angles

General information

A plane angle is a geometric figure formed by two intersecting lines. A plane angle consists of two rays with a common origin, and this point is called the vertex of the ray. The rays are called sides of the angle. Angles have many interesting properties, for example, the sum of all angles in a parallelogram is 360°, and in a triangle - 180°.

Types of angles

Direct angles are 90°, spicy- less than 90°, and stupid- on the contrary, more than 90°. Angles equal to 180° are called deployed, angles of 360° are called full, and angles greater than full but less than full are called non-convex. When the sum of two angles is 90°, that is, one angle complements the other to 90°, they are called additional adjacent, and if up to 360° - then conjugated

When the sum of two angles is 90°, that is, one angle complements the other to 90°, they are called additional. If they complement each other up to 180°, they are called adjacent, and if up to 360° - then conjugated. In polygons, the angles inside the polygon are called internal, and those conjugate to them are called external.

Two angles formed by the intersection of two lines that are not adjacent are called vertical. They are equal.

Measuring angles

Angles are measured using a protractor or calculated using a formula by measuring the sides of the angle from the vertex to the arc, and the length of the arc that limits these sides. Angles are usually measured in radians and degrees, although other units exist.

You can measure both angles formed between two straight lines and between curved lines. To measure between curves, tangents are used at the point of intersection of the curves, that is, at the vertex of the angle.

Protractor

A protractor is a tool for measuring angles. Most protractors are shaped like a semicircle or a circle and can measure angles up to 180° and 360°, respectively. Some protractors have an additional rotating ruler built into them for ease of measurement. Scales on protractors are often written in degrees, although sometimes they are also in radians. Protractors are most often used in geometry lessons at school, but they are also used in architecture and engineering, in particular in tool making.

Use of angles in architecture and art

Artists, designers, craftsmen and architects have long used angles to create illusions, accents and other effects. Alternating acute and obtuse angles, or geometric patterns of acute angles, are often used in architecture, mosaics, and stained glass, such as Gothic cathedrals and Islamic mosaics.

One of the famous forms of Islamic fine art is decoration using geometric girih designs. This design is used in mosaics, metal and wood carvings, on paper and fabric. The drawing is created by alternating geometric shapes. Traditionally, five figures are used with strictly defined angles from combinations of 72°, 108°, 144° and 216°. All these angles are divisible by 36°. Each shape is divided into several smaller symmetrical shapes by lines to create a more subtle design. Initially, these figures or mosaic pieces themselves were called girikh, hence the name of the entire style. In Morocco, there is a similar geometric style of mosaic, zullage or zilij. The shape of the terracotta tiles from which this mosaic is made is not observed as strictly as in girikha, and the tiles are often more bizarre in shape than the strict geometric figures in girikha. Despite this, zullyaj artists also use angles to create contrasting and intricate patterns.

In Islamic art and architecture, the rub al-hizb is often used - a symbol in the form of one square superimposed on another at an angle of 45°, as in the illustrations. It can be depicted as a solid figure, or in the form of lines - in this case this symbol is called the Al-Quds star. The Rub al-Hizb is sometimes decorated with small circles at the intersection of the squares. This symbol is used in the coats of arms and on the flags of Muslim countries, for example on the coat of arms of Uzbekistan and on the flag of Azerbaijan. The bases of the tallest twin towers in the world at the time of writing (spring 2013), the Petronas Towers, are built in the form of rub al-hizb. These towers are located in Kuala Lumpur in Malaysia and the country's Prime Minister was involved in their design.

Sharp corners are often used in architecture as decorative elements. They give the building a strict elegance. Obtuse angles, on the contrary, give buildings a cozy appearance. For example, we admire Gothic cathedrals and castles, but they look a little sad and even scary. But we will most likely choose a house for ourselves with a roof with obtuse angles between the slopes. Corners in architecture are also used to strengthen different parts of the building. Architects design the shape, size and angle of inclination depending on the load on the walls that need strengthening. This principle of strengthening by tilting has been used since ancient times. For example, ancient builders learned to build arches without cement or other binding materials, laying stones at a certain angle.

Usually buildings are built vertically, but sometimes there are exceptions. Some buildings are intentionally built to slope, and some lean because of mistakes. One example of leaning buildings is the Taj Mahal in India. The four minarets that surround the main building were built with an inclination from the center, so that in the event of an earthquake they would not fall inward, on the mausoleum, but in the other direction, and would not damage the main building. Sometimes buildings are built at an angle to the ground for decorative purposes. For example, the Leaning Tower of Abu Dhabi or Capital Gate is tilted 18° to the west. And one of the buildings in Stuart Landsborough's Puzzle World in Wanka, New Zealand, tilts 53° to the ground. This building is called the “Leaning Tower”.

Sometimes the leaning of a building is the result of a design error, such as the leaning of the Leaning Tower of Pisa. The builders did not take into account the structure and quality of the soil on which it was built. The tower was supposed to stand straight, but the poor foundation could not support its weight and the building sank, leaning to one side. The tower has been restored many times; the most recent restoration in the 20th century stopped its gradual subsidence and increasing slope. We managed to level it from 5.5° to 4°. The tower of the SuurHusen Church in Germany is also leaning because its wooden foundation rotted on one side after the marshy soil on which it was built drained. At the moment, this tower is tilted more than the Leaning Tower of Pisa - by about 5°.

Do you find it difficult to translate units of measurement from one language to another? Colleagues are ready to help you. Post a question in TCTerms and within a few minutes you will receive an answer.

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

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

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

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

Units of measurement are invented for specific quantities. Weight cannot be measured in seconds, but time cannot be measured in arshins.

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

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

Rice. 1. A circle divided into 360 equal angles

Let's call one such small angle an angle of 1° (see Fig. 2).

Rice. 2. 1 degree

It doesn't matter what size the circle we are dividing is. Let's divide 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 equal angles

The size of any angle is how many 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

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

Rice. 7. Full angle

Right angle is half of the unfolded one and is equal to 90° (see Fig. 8).

Rice. 8. Right angle

There is no need to store the degree standard anywhere. If necessary, you can always divide a full angle into 360 parts, or a rotated angle into 180, or a straight angle into 90.

A ruler is needed to measure an existing segment or draw a segment of the required length. To measure an angle or draw an angle of the required size, we also use a ruler, but not a straight one, 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 we can measure or draw is 180°, unfolded.

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

1. Let's measure a couple of angles.

The straight part of the protractor is aligned with one side of the angle, the center of the protractor with the vertex of the angle. Let's see where the second side of the angle is - 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 angle does not reach the scale, then it must first be extended.

2. Draw angles of 29°, 81° and 140°.

First, we draw one side of the angle using a ruler (see Fig. 12).

Rice. 12. Constructing one side of an angle

We mark the top. Combine with a protractor. We mark the desired angle value with a dot - 29° (see Fig. 13).

Rice. 13. Using a protractor to construct angles

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

Rice. 14. Angle 29°

We build the other two corners in the same way (see Fig. 15).

Rice. 15. Constructing angles

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

A tool for measuring and constructing angles is a protractor.

You don't have to use the names of the angles - full, extended, straight. We can simply say - 360 degrees, 180 or 90 degrees.

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

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

Exactly the same principle applies to another unit of distance measurement - the light year. A light year is the distance that light travels in 1 year. This unit is used to measure the distances 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 measurement of mass, and weight is another physical quantity. If you want to learn more about the difference between mass and weight, and why measuring weight in kilograms is not correct, then type “mass and weight” into the search engine and get a lot of explanations about this.

We still measure atmospheric pressure in millimeters (mm of mercury).

Although the angle has its own “native” units of measurement - degrees, which we will cover in this lesson, it can still be measured using linear quantities, for example centimeters. If you need to measure an 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 angle value, which is called tangent.

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

Rice. 16. Tangent

After all, as much as one side has increased, so has the other.

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

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 are divided into smaller units of measurement.

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

If a minute is divided into 60 parts, 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 even begin to divide a full angle into 360 parts, because it is not very convenient? In ancient Babylon there was a sexagesimal system (we have a decimal system). It was convenient for them to divide by 60.

2. Grads.

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

3. Radian.

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

Rice. 17. Radians

For example, now a complete angle (the entire circle) does not consist of an integer number of unit angles. A full angle consists of more than 6 unit angles. It’s not very convenient, but now the length of the arc (part of a circle) and the angle are well connected. If we take a circle with a radius of 1 cm, then the size of the angle 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. Mathematics. 5th grade. - M.: Mnemosyne, 2013.
2. Vilenkin N.Ya. and others. Mathematics. 5 grades - M.: Mnemosyne, 2013.
3. Erina T.M. Mathematics 5th grade. Slave. notebook for school Vilenkina, 2013. - M.: Mnemosyna, 2013.

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

Homework

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

Measure angle- means finding its magnitude. The magnitude of the angle shows how many times the angle chosen for the unit of measurement fits into a given angle.

Typically, the unit of measurement for angles is a degree. Degree- this is an angle equal to part of a 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

To measure angles, a special device is used - protractor:

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

Angles are measured as follows: the protractor is placed on the angle so that the top of the angle coincides with the center of the protractor, and one of the sides of the angle passes through the zero division on the scale. Then the other side of the angle will indicate the size of the angle in degrees:

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

To measure angles more accurately, fractions of a degree are used: minutes and seconds. Minute is an angle equal to part of a degree. Second is an angle equal to a fraction of a minute. Minutes are indicated by " , a seconds - sign "" . The minutes and seconds sign is placed in the upper right corner of the number. For example, if the angle is 50 degrees 34 minutes and 19 seconds, then write:

50°34 " 19""

Angle Measurement Properties

If a 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.