# Answer to Question #279 in Geometry for Jessy

Question #279

In what units are angles measured? What instrument is used for measurement? Explain how the measurement is made.

Expert's answer

There are several units used to measure angles. Among these units degree and radian are by far the most common.

Degree, denoted by a small superscript circle (°), is 1/360 of a full circle, so one full circle is 360°. An advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates, in astronomy and ballistics.

Radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. Radian has an abbreviation rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used, angles are considered as dimensionless. Radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. Radian is the (derived) unit of angular measurement in the SI system.

Angles are measured with a tool called a protractor. Step-by-step instructions for measuring:

Step 1. Find the center hole on the straight edge of the protractor.

Step 2. Place the hole over the vertex, or point, of the angle you wish to measure.

Step 3. Line up the zero on the straight edge of the protractor with one of the sides of the angle.

Step 4. Find the point where the second side of the angle intersects the curved edge of the protractor.

Step 5. Read the number that is written on the protractor at the point of intersection. This is the measure of the angle in degrees.

Degree, denoted by a small superscript circle (°), is 1/360 of a full circle, so one full circle is 360°. An advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates, in astronomy and ballistics.

Radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. Radian has an abbreviation rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used, angles are considered as dimensionless. Radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. Radian is the (derived) unit of angular measurement in the SI system.

Angles are measured with a tool called a protractor. Step-by-step instructions for measuring:

Step 1. Find the center hole on the straight edge of the protractor.

Step 2. Place the hole over the vertex, or point, of the angle you wish to measure.

Step 3. Line up the zero on the straight edge of the protractor with one of the sides of the angle.

Step 4. Find the point where the second side of the angle intersects the curved edge of the protractor.

Step 5. Read the number that is written on the protractor at the point of intersection. This is the measure of the angle in degrees.

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