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Home / How to Name the Sides of an Angle

How to Name the Sides of an Angle

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In geometry, an angle is the space between 2 rays (or line segments) with the same endpoint (or vertex). The most common way to measure angles is in degrees, with a full circle measuring 360 degrees. You can calculate the measure of an angle in a polygon if you know the shape of the polygon and the measure of its other angles or, in the case of a right triangle, if you know the measures of two of its sides. Additionally, you can measure angles using a protractor or calculate an angle without a protractor using a graphing calculator.

  1. 1

    Count the number of sides in the polygon. In order to calculate the interior angles of a polygon, you need to first determine how many sides the polygon has. Note that a polygon has the same number of sides as it has angles.[1]

    • For instance, a triangle has 3 sides and 3 interior angles while a square has 4 sides and 4 interior angles.

  2. 2

    Find the total measure of all of the interior angles in the polygon. The formula for finding the total measure of all interior angles in a polygon is: (n – 2) x 180. In this case, n is the number of sides the polygon has. Some common polygon total angle measures are as follows:[2]

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

  3. 3

    Divide the total measure of all of a regular polygon's angles by the number of its angles. A regular polygon is a polygon whose sides are all the same length and whose angles all have the same measure. For instance, the measure of each angle in an equilateral triangle is 180 ÷ 3, or 60 degrees, and the measure of each angle in a square is 360 ÷ 4, or 90 degrees.[3]

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

  4. 4

    Subtract the sum of the known angles from the total measure of the angles for an irregular polygon. If your polygon doesn't have sides of the same length and angles of the same measure, all you need to do is add up all of the known angles in the polygon. Then, subtract that number from the total measure of all of the angles to find the missing angle.[4]

    • For example, if you know that 4 of the angles in a pentagon measure 80, 100, 120, and 140 degrees, add the numbers together to get a sum of 440. Then, subtract this sum from the total angle measure for a pentagon, which is 540 degrees: 540 – 440 = 100 degrees. So, the missing angle is 100 degrees.

    Tip: Some polygons offer "cheats" to help you figure out the measure of the unknown angle. An isosceles triangle is a triangle with 2 sides of equal length and 2 angles of equal measure. A parallelogram is a quadrilateral with opposite sides of equal lengths and angles diagonally opposite each other of equal measure.

  1. 1

    Remember that every right triangle has one angle equal to 90 degrees. By definition, a right triangle will always have one angle that's 90 degrees, even if it's not labeled as such. So, you will always know at least one angle and can use trigonometry to find out the other 2 angles.[5]

  2. 2

    Measure the length of 2 of the triangle's sides. The longest side of a triangle is called the "hypotenuse." The "adjacent" side is adjacent (or next to) to the angle you're trying to determine.[6] The "opposite" side is opposite to the angle you're trying to determine. Measure 2 of the sides so you can determine the measure of the remaining angles in the triangle.[7]

    Tip: You can use a graphing calculator to solve your equations or find a table online that lists the values for various sine, cosine, and tangent functions.

  3. 3

    Use the sine function if you know the length of the opposite side and the hypotenuse. Plug your values into the equation: sine (x) = opposite ÷ hypotenuse. Say that the length of the opposite side is 5 and the length of the hypotenuse is 10. Divide 5 by 10, which is equal to 0.5. Now you know that sine (x) = 0.5 which is the same as x = sine-1 (0.5).[8]

    • If you have a graphing calculator, simply type 0.5 and press sine-1. If you don't have a graphing calculator, use an online chart to find the value. Both will show that x = 30 degrees.

  4. 4

    Use the cosine function if you know the length of the adjacent side and the hypotenuse. For this type of problem, use the equation: cosine (x) = adjacent ÷ hypotenuse. If the length of the adjacent side is 1.666 and the length of the hypotenuse is 2.0, divide 1.666 by 2, which is equal to 0.833. So, cosine (x) = 0.833 or x = cosine-1 (0.833).[9]

    • Plug 0.833 into your graphing calculator and press cosine-1. Alternatively, look up the value in a cosine chart. The answer is 33.6 degrees.

  5. 5

    Use the tangent function if you know the length of the opposite side and the adjacent side. The equation for tangent functions is tangent (x) = opposite ÷ adjacent. Say you know the length of the opposite side is 75 and the length of the adjacent side is 100. Divide 75 by 100, which is 0.75. This means that tangent (x) = 0.75, which is the same as x = tangent-1 (0.75).[10]

    • Find the value in a tangent chart or press 0.75 on your graphing calculator, then tangent-1. This is equal to 36.9 degrees.

Add New Question

  • Question

    How do you find an angle?

    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 models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

    Mario Banuelos, PhD

    Assistant Professor of Mathematics

    Expert Answer

  • Question

    How do I create a 90 degree corner by swinging an arch?

    Donagan

    Pick a convenient point on a line to be the vertex of your 90° angle. Choose two points on the line, one on each side of the vertex and equidistant from the vertex. Use a compass to draw two arcs of the same diameter, each centered on one of those latter points. Draw a line connecting the vertex point with the intersecting point(s) of the arcs. That line describes a 90° angle with the first line.

  • Question

    How do I find the interior angles of a hexagon without base or height or anything?

    Donagan

    The sum of the six interior angles of a regular polygon is (n-2)(180°), where n is the number of sides. Therefore, in a hexagon the sum of the angles is (4)(180°) = 720°. All the angles are equal, so divide 720° by 6 to get 120°, the size of each interior angle.

  • Question

    How do I calculate the angle of a roof as opposed to the vertical wall it leans on?

    Donagan

    For a rough approximation, use a protractor to estimate the angle by holding the protractor in front of you as you view the side of the house. For the exact angle, measure the horizontal run of the roof and its vertical rise. Divide the horizontal measurement by the vertical measurement, which gives you the tangent of the angle you want. Use a trigonometry table to find the angle.

  • Question

    If I I have a pillow wedge that is 24" long and 12" tall, what is the degree of the wedge?

    Donagan

    A right triangle with legs of 24 and 12 has acute angles of 26.6° (opposite the 12 side)(the angle you're looking for), 63.4°(opposite the 24 side), and 90°.

  • Question

    How do I calculate if the angle is (n+11), the second angle is (4n-17), and the third angle is (5n+36)?

    Donagan

    If you are trying to calculate the three angles of a triangle, add together the three angles as expressed in terms of n. Set their sum equal to 180°, then solve for n. Thus, (n+11) + (4n-17) + (5n+36) = 10n + 30 = 180. So n = 15, making the angles equal to 26°, 43°, and 111°.

  • Question

    Translate 2 units down and 6 units to the right?

    Donagan

    If you're looking for the angle, use trigonometry: the angle's tangent is 6/2, or 3.

  • Question

    How do I calculate exterior angles?

    Donagan

    An exterior angle of a triangle is equal to the difference between 180° and the accompanying interior angle. Thus, if an angle of a triangle is 50°, the exterior angle at that vertex is 180° - 50° = 130°.

  • Question

    How can I find angles of a triangle based off of the 3 known side lengths?

    Donagan

    The easiest way is to construct the triangle and then use a protractor to measure the angles. If you can't use that method, you'll have to construct the triangle and do this with any angle: Drop an altitude to the opposite side, thus forming two new triangles. Measure the length of the altitude. For the two angles opposite the altitude, use the sine (opposite side divided by hypotenuse) to find the angles.

  • Question

    What would the angle be on a triangle that is 4" high and the base is 120" long?

    Donagan

    Assuming this is a right triangle and the angle you're looking for is the one opposite the 4" leg, the tangent of that angle is 0.0333. That means the angle is slightly less than 2° (about 1.9°).

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  • Angles are given names according to how many degrees they measure. As noted above, a right angle measures 90 degrees. An angle measuring more than 0 but less than 90 degrees is an acute angle. An angle measuring more than 90 but less than 180 degrees is an obtuse angle. An angle measuring 180 degrees is a straight angle, while an angle measuring more than 180 degrees is a reflex angle.

  • Two angles whose measures add up to 90 degrees are called complementary angles. (The two angles other than the right angle in a right triangle are complementary angles.) Two angles whose measures add up to 180 degrees are called supplementary angles.

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Article SummaryX

To calculate angles in a polygon, first learn what your angles add up to when summed, like 180 degrees in a triangle or 360 degrees in a quadrilateral. Once you know what the angles add up to, add together the angles you know, then subtract the answer from the total measures of the angles for your shape. For example, add 60 and 80 to get 140 for 2 angles in a triangle, then deduct 140 from 180 to work out the third angle in the triangle, which will be 40 degrees. To find out how to calculate angle measure in a right triangle, read on!

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How to Name the Sides of an Angle

Source: https://www.wikihow.com/Calculate-Angles