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Calculating angle measure is a fundamental skill in mathematics and geometry. Whether you are working on a geometry problem, designing a building, or analyzing data, understanding how to calculate angle measure is essential. An angle is formed by two lines that meet at a point, and it is measured in degrees. Being able to calculate and determine the measurement of an angle is crucial for solving geometric problems and understanding spatial relationships. In this guide, we will explore various methods and techniques to accurately calculate angle measure, allowing you to navigate the world of angles with confidence and precision.

This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

This article has been viewed 179,886 times.

In geometry, an angle is the space formed between two rays (or lines) originating from the same point (or vertex). Angles are usually measured in degrees, with a full circle equivalent to 360 degrees. You can calculate the angle measure in a polygon if you know the shape of the polygon and the measures of the other angles, or know the lengths of the two adjacent sides in the case of a right triangle. Alternatively, you can measure angles with a protractor, or calculate an angle measure with a graphing calculator without using a ruler.

## Steps

### Calculate the measure of an angle in a polygon

**Count the number of sides of the polygon.**To calculate the angles in a polygon, you first need to determine how many sides the polygon has. Note that a polygon always has the same number of sides as the number of corners.

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- For example, a triangle has 3 sides and 3 angles, while a square has 4 sides and 4 angles.

**Find the sum of the measures of all the angles in the polygon.**The formula to calculate the sum of the measures of all the angles in a polygon is: (n – 2) x 180. In this case, “n” is the number of sides of the polygon. The sum of the angle measures of some common polygons is as follows:

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- The angles of a triangle (a 3-sided polygon) have a sum of 180 degrees.
- The angles of a quadrilateral (4-sided polygon) have a sum of 360 degrees.
- The angles of a pentagon (5-sided polygon) have a total measure of 540 degrees.
- The angles of a hexagon (6-sided polygon) have a total measure of 720 degrees.
- The angles of an octagon (8-sided polygon) have a total measure of 1080 degrees.

**Divide the sum of the angle measures of a regular polygon by the number of angles.**A regular polygon is a polygon with equal sides and angles. For example, the measure of each angle in an equilateral triangle is 180 ÷ 3 = 60 degrees, and the measure of each angle in a square is 360 ÷ 4 = 90 degrees.

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- Equilateral triangles and squares are examples of regular polygons, while the Pentagon in Washington, DC is an example of a regular pentagon, and a stop sign is an example of a regular octagon.

**Subtract the sum of the measures of the angles of the irregular polygon from the sum of the measures of the known angles.**If your polygon doesn’t have equal sides and angles then all you need to do is add up all the known angles in the polygon. Then subtract the polygon’s total angle measure from that value to find the unknown angle.

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- For example, if you know the measures of the four angles in a pentagon are 80, 100, 120, and 140 degrees, add the numbers together to get a total of 440. Then add the sum of the angle measures of the pentagon. is 540 degrees minus the value just calculated. 540 – 440 = 100 degrees. So the remaining angle is 100 degrees.

**Tip:** Some polygons provide facts to help you figure out the unknown angle. An isosceles triangle is a triangle with two equal sides and two angles. A parallelogram is a quadrilateral with two equal opposite sides and equal diagonally opposite angles.

### Find the measure of the angle in a right triangle

**Remember that every right triangle has an angle equal to 90 degrees.**By definition, a right triangle always has a 90 degree angle even if people don’t call it a 90 degree triangle. Therefore, you will always know the measure of one angle and can use your trigonometric knowledge to find the other 2 angles.

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**Measure the lengths of the two sides of the triangle.**The longest side of a right triangle is called the hypotenuse. The “adjacent” edge is the edge that is adjacent to the angle you are looking for. The “opposite” edge is the side opposite the angle you’re looking for. Measure the lengths of 2 of the 3 sides so you can find the measure of the remaining angles in the triangle.

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**Tip:** You can use a graphing calculator to solve the equation, or look online for tables that list the values of the sin, cosine, and tangent functions.

**Use the sine function if you know the lengths of the opposite and hypotenuse.**Substitute the values into the equation: sin (x) = opposite side ÷ hypotenuse. Suppose the length of the opposite side is 5 and the length of the hypotenuse is 10. Divide 5 by 10 to get 0.5. Now you know sin(x) = 0.5 is equivalent to the equation x = sin

^{-1}(0.5).

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- If you have a graphing calculator then enter 0.5 and press sin
^{-1}. If you don’t have a graphing calculator, use an online graph to find that value. Both show x = 30 degrees.

**Use the cos function if you know the lengths of the adjacent and hypotenuse.**For this type of problem, use the equation: cos(x) = adjacent side ÷ hypotenuse. If the length of the adjacent side is 1.666 and the length of the hypotenuse is 2.0, then you divide 1.666 by 2 and the result is 0.833. So cos(x) = 0.833 or x = cos

^{-1}(0.833).

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- Enter 0.833 into the graphing calculator and press cos
^{-1}. Alternatively, you can look up this value on the graph of the function cos. The answer is 33.6 degrees.

**Use the tangent function if you know the lengths of the opposite and adjacent sides.**The equation of the tangent function is tangent (x) = opposite side ÷ adjacent side. Let’s say the opposite side length is 75 and the adjacent side length is 100. Divide 75 by 100 to get 0.75. That is, tangent (x) = 0.75 is equivalent to the equation x = tang

^{-1}(0.75).

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- Find this value on the graph of the tangent function, or enter 0.75 into the graphing calculator and press tang
^{-1}. The answer is 36.9 degrees.

## Advice

- Angles are named after their measure of degrees. Like I said, right angles have 90 degrees. Angles whose measure is greater than 0 but less than 90 degrees are acute. Angles whose measure is greater than 90 but less than 180 degrees are obtuse angles. An angle that measures 180 degrees is a flat angle.
- Two angles whose sum is 90 degrees are called supplementary angles (two acute angles in a right triangle are called complementary angles). Two angles whose measure is 180 degrees are called complementary angles.

This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

This article has been viewed 179,886 times.

In geometry, an angle is the space formed between two rays (or lines) originating from the same point (or vertex). Angles are usually measured in degrees, with a full circle equivalent to 360 degrees. You can calculate the angle measure in a polygon if you know the shape of the polygon and the measures of the other angles, or know the lengths of the two adjacent sides in the case of a right triangle. Alternatively, you can measure angles with a protractor, or calculate an angle measure with a graphing calculator without using a ruler.

In conclusion, calculating angle measure is an important skill in geometry and can be approached through various methods. The use of basic trigonometric functions such as sine, cosine, and tangent can assist in finding the measure of angles in right triangles or when given specific side lengths. In addition, the properties of angles in geometric figures such as triangles and quadrilaterals can be utilized to determine unknown angles. Additionally, the use of a protractor or digital angle measuring tools can provide an accurate measurement of angles in various shapes. Overall, understanding and being able to calculate angle measures is crucial in solving geometric problems and is essential in various fields such as construction, engineering, and mathematics.

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