The need to measure angles has appeared among people since civilization reached a minimum technical level. Everyone knows the phenomenal accuracy of observance of inclination and orientation according to the cardinal points provided by the builders Egyptian pyramids. The modern degree measure of angles is now believed to have been invented by the ancient Akkadians.
What are degrees?
A degree is a generally accepted unit of measurement for angles. In a complete circle 360 degrees. The reason for choosing this particular number is unknown. The Akkadians probably divided the circle into sectors using the angle equilateral triangle, and then the resulting segments were again divided into 60 parts according to their number system. A degree is also divided into 60 minutes, and minutes into 60 seconds. Commonly accepted designations are:
° - angular degrees
’ - minutes,
'' - seconds.
Over the millennia, the degree measure of angles has become firmly established in many areas of human activity. It is still indispensable in all areas of science and technology - from cartography to calculating the orbits of artificial Earth satellites.
What are radians?
Archimedes is credited with the discovery of the constant ratio of the circumference of a circle to its diameter. We call it the number pi. It is irrational, that is, it cannot be expressed as an ordinary or periodic fraction. The most commonly used value for the number π is 3.14, accurate to two decimal places. The length of a circle L with radius R is easily calculated using the formula: L=2πR.
A circle of radius R=1 has a length of 2π. This relation is used in geometry as a formulation of the radian measure of angle.
By definition, a radian is an angle with its vertex at the center of the circle, subtended by an arc with a length equal to the radius of the circle. The international designation for the radian is rad, the domestic designation is rad. It has no dimension.
An arc of a circle with radius R and angular value α radians has length α * R.
Why was it necessary to introduce a new unit of measurement for angle?
The development of science and technology led to the emergence of trigonometry and mathematical analysis, necessary for accurate calculations of mechanical and optical devices. One of its tasks is to measure the length of a curved line. The most common case is determining the length of a circular arc. Using the degree measure of angles for this purpose is extremely inconvenient. The idea of comparing the length of an arc with the radius of a circle arose among many mathematicians, but the term “radian” itself was introduced into scientific use only in the second half of the 19th century. That's it now trigonometric functions in mathematical analysis, the radian measure of angle is used by default.
How to convert degrees to radians
From the formula for the circumference of a circle it follows that 2π radii fit into it. It follows that: 1⁰=2π/360= π/180 rad.
AND simple formula conversion from radians to degrees: 1 rad = 180/π.
Let us have an angle of N degrees. Then the formula for converting from degrees to radians will be: α(radians) = N/(180/π) = N*π/180.
Still have questions?
The answers to them can be found, where the concepts of circumference, radian measure of angles and specific examples shows the conversion of degrees to radians. Knowledge of the above is extremely important for understanding mathematics, without which the existence of modern civilization is impossible.
The need to measure angles has appeared among people since civilization reached a minimum technical level. Everyone knows the phenomenal accuracy of observance of inclination and orientation to the cardinal points, ensured by the builders of the Egyptian pyramids. The modern degree measure of angles is now believed to have been invented by the ancient Akkadians.
What are degrees?
A degree is a generally accepted unit of measurement for angles. In a complete circle 360 degrees. The reason for choosing this particular number is unknown. The Akkadians probably divided the circle into sectors using the angle of an equilateral triangle, and then divided the resulting segments again into 60 parts according to their number system. A degree is also divided into 60 minutes, and minutes into 60 seconds. Commonly accepted designations are:
° - angular degrees
’ - minutes,
'' - seconds.
Over the millennia, the degree measure of angles has become firmly established in many areas of human activity. It is still indispensable in all areas of science and technology - from cartography to calculating the orbits of artificial Earth satellites.
What are radians?
Archimedes is credited with the discovery of the constant ratio of the circumference of a circle to its diameter. We call it the number pi. It is irrational, that is, it cannot be expressed as an ordinary or periodic fraction. The most commonly used value for the number π is 3.14, accurate to two decimal places. The length of a circle L with radius R is easily calculated using the formula: L=2πR.
A circle of radius R=1 has a length of 2π. This relation is used in geometry as a formulation of the radian measure of angle.
By definition, a radian is an angle with its vertex at the center of the circle, subtended by an arc with a length equal to the radius of the circle. The international designation for the radian is rad, the domestic designation is rad. It has no dimension.
An arc of a circle with radius R and angular value α radians has length α * R.
Why was it necessary to introduce a new unit of measurement for angle?
The development of science and technology led to the emergence of trigonometry and mathematical analysis, necessary for accurate calculations of mechanical and optical devices. One of its tasks is to measure the length of a curved line. The most common case is determining the length of a circular arc. Using the degree measure of angles for this purpose is extremely inconvenient. The idea of comparing the length of an arc with the radius of a circle arose among many mathematicians, but the term “radian” itself was introduced into scientific use only in the second half of the 19th century. Nowadays, all trigonometric functions in calculus use the radian angle measure by default.
How to convert degrees to radians
From the formula for the circumference of a circle it follows that 2π radii fit into it. It follows that: 1⁰=2π/360= π/180 rad.
And a simple formula for converting from radians to degrees: 1 rad = 180/π.
Let us have an angle of N degrees. Then the formula for converting from degrees to radians will be: α(radians) = N/(180/π) = N*π/180.
Still have questions?
The answers to them can be found, where the concepts of circumference, radian measure of angles are explained in detail, and the conversion of degrees to radians is shown using specific examples. Knowledge of the above is extremely important for understanding mathematics, without which the existence of modern civilization is impossible.