In geometry, an angle is defined as the portion of plane or space between two rays originating from the same point or vertex. The unit of measurement most used to indicate the width of an angle are degrees and the angle with maximum width, the round angle, is equal to 360 °. Knowing the shape of the polygon and the measurement of the other angles, it is possible to calculate the width of a specific angle. In some particular cases, for example in the case of a right triangle, it is possible to calculate the width of an angle knowing the measure of the two sides that identify it. In reality, you can physically measure the width of an angle using a protractor. If you have a graphing calculator available, you can use it to calculate the width of an angle based on the data at your disposal.
Steps
Method 1 of 2: Calculate the Interior Angles of a Polygon
Step 1. Count the number of sides that make up the polygon under examination
In order to calculate the width of its internal angles, you will first have to determine the number of sides that compose it. Note that the number of interior angles of a polygon corresponds to the number of its sides.
For example, a triangle has 3 sides, so it will have 3 internal angles. A square has 4 sides, so it will have 4 internal corners
Step 2. Calculate the total width of all internal angles of the polygon
The formula for calculating the total sum of all internal angles of a polygon is as follows: (n - 2) x 180. In this case the variable n represents the number of sides that make up the polygon. Below is the list of the sums of the internal angles of the most popular polygons:
- The sum of the internal angles of a triangle (a polygon made up of 3 sides) is equal to 180 °;
- The sum of the internal angles of a quadrilateral (a polygon made up of 4 sides) is equal to 360 °;
- The sum of the internal angles of a pentagon (a polygon made up of 5 sides) is equal to 540 °;
- The sum of the internal angles of a hexagon (a polygon made up of 6 sides) is equal to 720 °;
- The sum of the internal angles of an octagon (a polygon made up of 8 sides) is equal to 1,080 °.
Step 3. Divide the sum of all interior angles of a regular polygon by the number of its angles
A polygon is defined as regular when its sides all have the same length and its internal angles the same width. For example, the width of each internal angle of an equilateral triangle will be equal to 180 ÷ 3, that is 60 °; while the width of each internal corner of a square will be equal to 360 ÷ 4, that is 90 °.
Equilateral triangles and squares are just a few examples of regular polygons. The Pentagon building erected in Washington D. C. is an example of a regular pentagon, while the stop road sign is an example of a regular octagon
Step 4. In the case of an irregular polygon, you can calculate the width of one angle by subtracting the width of the other known angles from the total sum of the interior angles
In the case of a polygon whose sides do not all have the same length, and whose angles therefore will not all have the same width, to calculate the width of a specific angle you will need to know the sum of all known internal angles, after which you will have to subtract the value obtained from the total width of the internal angles of the polygon under examination (information you already know).
For example, if 4 corners of a pentagon measure 80 °, 100 °, 120 ° and 140 ° respectively, their sum will be 440 °. Knowing that the sum of all interior angles of a pentagon is 540 °, you can calculate the amplitude of the remaining angle by performing a simple subtraction: 540 - 440 = 100 °. At this point you can say that the unknown angle of the example pentagon has an amplitude of 100 °
Advise:
some particular polygons have peculiarities that can help you quickly and easily calculate the width of an unknown angle. For example, an isosceles triangle is characterized by 2 sides of the same length and consequently by two angles with the same width. A parallelogram is a quadrilateral whose opposite sides have the same length, so the opposite angles will also have the same width.
Method 2 of 2: Calculate the Angles of a Right Triangle
Step 1. Remember that all right triangles are characterized by having an internal angle of 90 °
By definition, a right triangle has an internal angle with a width of 90 ° even when not explicitly specified. In this case, knowing the width of one angle, you can use the trigonometric functions to calculate the width of the other two angles.
Step 2. Measure the length of the two sides of the triangle
The longer side of a right triangle is called a "hypotenuse". "Adjacent" is defined as the cathetus or the side which is adjacent to the angle that you have to calculate, while "opposite" is defined as the cathetus or the side opposite to the angle you want to calculate. By obtaining the measurement of two sides of the triangle you will be able to calculate the width of the angles of the triangle that you do not yet know.
Advise:
you can use a graphing calculator to quickly solve equations. Alternatively, you can search for an online table that summarizes the values of the various trigonometric functions (sine, cosine and tangent).
Step 3. If you know the length of the opposite side and the hypotenuse, you can use the trig function "sine"
The complete formula you will need to use is the following: sin (x) = opposite_side ÷ hypotenuse. Assume that the length of the opposite side of the triangle under consideration is 5 units and that the length of the hypotenuse is equal to 10 units. Start by dividing 5 by 10 to get 0, 5. Now you know that sin (x) = 0, 5, so solving the equation for "x" you get x = sin-1 (0, 5).
If you have a graphing calculator, enter the value 0, 5 and press the trigonometric function key "sin-1". If you don't have a graphing calculator, you can use one of the many websites that list trigonometric function tables to get the value of the inverse sine function. In both cases you will get that" x "is equal to 30 °.
Step 4. If you know the length of the adjacent side and the hypotenuse, you can use the "cosine" trig function
In this case you will have to use the following formula: cos (x) = adjacent_side ÷ hypotenuse. Assume that the length of the side adjacent to the angle you need to calculate is 1. 666 units and that the length of the hypotenuse is 2. Start by dividing 1. 666 by 2, resulting in 0.833. Now you know what cos (x) = 0.833, so solving the equation for "x", you get x = cos-1 (0, 833).
Now you can solve the equation by typing the value 0.833 into a graphing calculator and pressing the "cos" function key-1". If you don't have a graphing calculator you can use one of the many websites that list trigonometric function tables to get the value of the inverse cosine function. In this case the final result will be 33.6 °.
Step 5. If you know the length of the side adjacent and the side opposite the angle you need to calculate, you can use the "tangent" trig function
In this case you will have to use the following formula: tan (x) = opposite_side ÷ adjacent_side. Assume that the length of the opposite side is equal to 75 units and that the length of the adjacent side is equal to 100 units. Start by dividing 75 by 100, resulting in 0.75. Entering the value obtained in the initial formula and solving the equation based on "x" you will get: tan (x) = 0.75, that is x = tan-1 (0, 75).
Calculate the value of the inverse function of the tangent using one of the many websites related to trigonometric functions or use a graphing calculator by typing the value 0, 75 and pressing the "tan-1". The value you get will be 36.9 °.
Advice
- There are different types of angles whose names vary according to the width. As mentioned earlier in the article, an angle is said to be right when it has a width of 90 °. An angle is acute when its amplitude is greater than 0 ° but less than 90 °. An angle is said to be obtuse when its amplitude is greater than 90 ° but less than 180 °. An angle is said to be flat when its width is equal to 180 °. An angle is defined as concave when its width is greater than 180 °.
- Two angles are said to be complementary when their sum is equal to 90 ° (for example the two non right angles of a right triangle are always complementary). Two angles are said to be additional when their sum is equal to 180 °.
