Math Antics - Angle Basics
How to Calculate Angles
In geometry, an angle is the space between two 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.
Calculating Angle Measure in a Polygon
Count the number of sides in the polygon.
Find the total measure of all interior angles in the polygon.The formula for finding the total measure of all interior angles in a polygon is (‘’n’’ – 2) x 180, where ‘’n’’ is the number of sides, as well as the number of angles, the polygon has.Some common polygon total angle measures are as follows:
- 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.
Determine if the polygon is a regular polygon.A regular polygon is a polygon whose sides are all the same length and whose angles all have the same measure. 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.
- If the polygon is a regular polygon, simply divide the total measure of all its angles by the number of its angles.Thus, 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. (Although a rectangle is not a regular polygon, by definition, all its angles are also right angles, measuring 90 degrees each.)
- If the polygon is not a regular polygon, you need to know the measures of the other angles in the polygon to calculate the measure of an unknown angle. Proceed to the next step.
Add the measures of the known angles of the polygon together, then subtract it from the total angle measure of the polygon.Most geometry problems of this nature work with triangles or quadrilaterals, because there are fewer numbers to work with, so we’ll do likewise.
- If two of the angles of a triangle have measures of 60 and 80 degrees, add the numbers together to get a sum of 140. Then, subtract this sum from the total angle measure for a triangle, which is 180 degrees: 180 – 140 = 40 degrees. (This kind of triangle, where all the angles have different measures, is called a scalene triangle.)
- You can write the above method out as a formula: ‘’a’’ = 180 – (‘’b’’ + ‘’c’’), where ‘’a’’ is the angle whose measure you’re trying to find, and ‘’b’’ and ‘’c’’ are the angles whose measures you already know. For polygons with more than 3 sides, simply replace “180” with the total angle measure of the polygon and add another term for each additional known angle.
- Some polygons offer “cheats” to help you figure out the measure of the unknown angle. An isosceles triangle is a triangle with two sides of equal length and two angles of equal measure. A parallelogram is a quadrilateral with opposite sides of equal lengths and angles diagonally opposite each other of equal measure.
Calculating Angle Measure in a Right Triangle
Assess what you already know.A right triangle is so named because one of its angles is a right angle. You can find the measure of one of the other angles if you know either of these things:
- The measure of the third angle. In this case, you add its measure to 90, the number of degrees in the right angle and subtract that total from 180.
- The measure of any two of the triangle’s sides. In this case, you can find the measure of the angle using trigonometry.
Determine the right trigonometric function to use.Trigonometric functions are ratios between two of the three sides of the triangle. Although there are six trigonometric functions, the following three are used most often:
- If you know the length of the side opposite the angle and the length of the hypotenuse (the side opposite the right angle), you can use the sine function, which is the length of the opposite side divided by the length of the hypotenuse.
- If you know the length of the side adjacent to the angle and the length of the hypotenuse, you can use the cosine function, which is the length of the adjacent side divided by the length of the hypotenuse.
- If you know the lengths of the opposite and adjacent sides, you can use the tangent function, which is the length of the opposite side divided by the length of the adjacent side.
Find the ratio of the two known sides.For the purposes of this example, we’ll assume we know the side opposite the angle has a length of 5 units and the hypotenuse has a length of 10 units. Because we know the opposite and hypotenuse, the ratio we’re finding is the sine.
- Dividing the opposite’s value of 5 by the hypotenuse’s value of 10 yields 5 / 10 = 0.5.
Find the angle that corresponds to the trig function ratio.Because we’re using the sine to find the angle measure, the angle we’re looking for is called the arcsine or inverse sine. There are two ways to find it:
- In the days before calculators, you would consult a printed table of values for sines, cosines, and tangents for angles from 0 to 90 degrees. Read down the Sine column until you find the value “0.5” and then look at the angle measure corresponding to that sine value.
- With a calculator with trig function capability, you input the sine value (if you didn’t already divide the opposite by the hypotenuse with the calculator to find it) and then press the appropriate key or keys. Depending on the brand of calculator you have, you may press a single key labeled “sin-1” or a key labeled “Inv,” “2ndF,” or “Shift” before pressing the “sin” key.
- Whichever method you use for this example, you should find the angle has a measure of 30 degrees.
QuestionHow do I calculate the angle of a roof as opposed to the vertical wall it leans on?Top AnswererFor 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.Thanks!
QuestionHow do I create a 90 degree corner by swinging an arch?Top AnswererPick 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.Thanks!
QuestionHow do I calculate exterior angles?Top AnswererAn 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°.Thanks!
QuestionHow do I calculate if the angle is (n+11), the second angle is (4n-17), and the third angle is (5n+36)?Top AnswererIf 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°.Thanks!
QuestionTranslate 2 units down and 6 units to the right?Top AnswererIf you're looking for the angle, use trigonometry: the angle's tangent is 6/2, or 3.Thanks!
QuestionHow do I find the interior angles of a hexagon without base or height or anything?Top AnswererThe 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.Thanks!
QuestionIf I I have a pillow wedge that is 24" long and 12" tall, what is the degree of the wedge?Top AnswererA 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°.Thanks!
QuestionWhat would the angle be on a triangle that is 4" high and the base is 120" long?Top AnswererAssuming 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°).Thanks!
QuestionHow can I find angles of a triangle based off of the 3 known side lengths?Top AnswererThe 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.Thanks!
QuestionTo cut a pentagon angle, what is the degree of the cut?Top AnswererThe five internal angles of a regular pentagon are each 108°.Thanks!
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.
- 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.
Things You’ll Need
Trigonometric tables or calculator with trigonometric functions
Sources and Citations
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