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Angles are fundamental elements in geometry, used to describe and understand the relationship between lines, points, and shapes. Being able to calculate the measure of an angle is crucial in various mathematical and practical applications. Whether you are a student learning about trigonometry or a professional architect designing buildings, understanding how to determine the angle measure accurately is essential. In this guide, we will explore different methods and formulas to calculate angle measure, providing a step-by-step approach to help you solve angle-related problems with ease and precision. So, if you are ready to sharpen your angle-measuring skills, let’s dive into the world of angles and learn how to calculate their measures effectively.
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
![Image titled Calculate Angles Step 1](https://www.wikihow.com/images_en/thumb/a/ad/Calculate-Angles-Step-1-Version-4.jpg/v4-460px-Calculate-Angles-Step-1-Version-4.jpg)
- For example, a triangle has 3 sides and 3 angles, while a square has 4 sides and 4 angles.
![Image titled Calculate Angles Step 2](https://www.wikihow.com/images_en/thumb/0/02/Calculate-Angles-Step-2-Version-4.jpg/v4-460px-Calculate-Angles-Step-2-Version-4.jpg)
- 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.
![Image titled Calculate Angles Step 3](https://www.wikihow.com/images_en/thumb/7/75/Calculate-Angles-Step-3-Version-4.jpg/v4-460px-Calculate-Angles-Step-3-Version-4.jpg)
- 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.
![Image titled Calculate Angles Step 4](https://www.wikihow.com/images_en/thumb/3/34/Calculate-Angles-Step-4-Version-4.jpg/v4-460px-Calculate-Angles-Step-4-Version-4.jpg)
- 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
![Image titled Calculate Angles Step 5](https://www.wikihow.com/images_en/thumb/a/a5/Calculate-Angles-Step-5-Version-4.jpg/v4-460px-Calculate-Angles-Step-5-Version-4.jpg)
![Image titled Calculate Angles Step 6](https://www.wikihow.com/images_en/thumb/f/f2/Calculate-Angles-Step-6-Version-4.jpg/v4-460px-Calculate-Angles-Step-6-Version-4.jpg)
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.
![Image titled Calculate Angles Step 7](https://www.wikihow.com/images_en/thumb/0/0b/Calculate-Angles-Step-7-Version-4.jpg/v4-460px-Calculate-Angles-Step-7-Version-4.jpg)
- 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.
![Image titled Calculate Angles Step 8](https://www.wikihow.com/images_en/thumb/0/05/Calculate-Angles-Step-8-Version-4.jpg/v4-460px-Calculate-Angles-Step-8-Version-4.jpg)
- 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.
![Image titled Calculate Angles Step 9](https://www.wikihow.com/images_en/thumb/a/a3/Calculate-Angles-Step-9.jpg/v4-460px-Calculate-Angles-Step-9.jpg)
- 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 measures is a fundamental skill in mathematics and has numerous applications in various fields such as science, engineering, architecture, and physics. By understanding the concepts of angles, degrees, and trigonometry, we can determine the measure of angles accurately. Whether it is finding angle measures in a triangle, using the properties of parallel lines and transversals, or solving problems involving rotational motions, knowing how to calculate angles is essential for problem-solving and critical thinking. Furthermore, with the advent of technology and online resources, there are several tools and calculators available that can assist in calculating angle measures more efficiently. By practicing and applying the techniques discussed in this article, anyone can develop proficiency in calculating angle measures and unleash the power of angles in various mathematical and real-world scenarios.
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