The easiest way to measure angles is to use a protractor. However, if this is not available, you can determine the size of the angle using simple triangular geometry principles. To do this, you'll need a science calculator. Most smartphones come with this calculator, but if you don't have one, you can also download a free calculator app or use an online calculator. The calculation depends on whether you are measuring acute (less than 90 degrees), obtuse (more than 90 degrees, but less than 180), or reflex angles (more than 180 degrees but less than 360).
Step
Method 1 of 3: Acute Angle
Step 1. Draw a vertical line connecting the two line rays
To determine the degree of an acute angle, connect the 2 rays to form a triangle. Align the short end of the ruler with the bottom ray, then draw a vertical line until it intersects the other ray using the long side of the ruler.
This vertical line produces a right angle. The angle formed by the side (the bottom ray of the angle) of the triangle and the opposite side (the vertical line) is 90 degrees
Step 2. Measure the side length to find the horizontal value (run)
Place the end of the ruler at the point of an acute angle. Measure the length of the side from the acute corner to the point where it intersects the opposite side.
The length of this line is the horizontal value in the slope equation, i.e. slope = vertical/horizontal. If the length obtained is 7, the equation will be "slope = vertical/7."
Step 3. Measure the length of the opposite side to find the vertical (rise)
Place the end of the ruler at the point of a right angle, parallel to the side of the triangle. Measure the length of the vertical line from the vertex of the right angle to the point where the line intersects the top ray of the angle (the hypotenuse of the triangle).
This number is the vertical value in the slope equation. If the result is 5, plug it into the equation so that "slope = 5/7."
Step 4. Divide the vertical by the horizontal to find the slope of the angle
The slope is the steepness of the diagonal line, or hypotenuse, of your triangle. Once you know the number, you can calculate the acute angle.
To continue the previous example, the equation "slope = 5/7" returns "slope = 0.71428571."
Tip:
Do not round the number before calculating the angle in degrees as this will reduce the accuracy of the calculation.
Step 5. Use a calculator to determine the measure of an angle
Type the slope value into the science calculator, then press the inverse tangent key (tan-1). The result is the size of the angle in degrees.
Completing the example above, a slope of 0.71428571, will result in an angle of 35.5 degrees
Method 2 of 3: Obtuse Angle
Step 1. Extend the bottom ray of the corner in a straight line
Mark the vertices with dots, then use the long side of the ruler to draw a straight line that continues the bottom rays of the corner. Make sure the bottom ray of the corner and its extension line form a straight line below the obtuse angle.
Make sure the line is completely straight. If the line slopes slightly up or down, the equation may be inaccurate
Tip:
If you're working with lined paper, align the short end of the ruler with the edge of the paper to ensure that the line extension is completely straight.
Step 2. Draw a vertical line connecting the top beam and the extension line
Align the short end of the ruler with the bottom ray so that the long side of the ruler intersects with the top ray. Draw a line through the long side of the ruler so that it forms a vertical line connecting the top ray and the line extending the bottom ray of the angle.
If it's correct, you've created a right angle under the obtuse angle you want to measure; the ray above the obtuse angle is now the hypotenuse of the right triangle
Step 3. Measure the side length (underline) of the vertex
Place the ruler parallel to the bottom line, with the tip at the point of a right angle. Measure the length of the line from the vertex of the right angle to the vertex of the obtuse angle.
Now you determine the slope of the acute angle of the triangle, which can be used to calculate the measure of the acute angle. The underscore is the horizontal value in the equation "slope = vertical/horizontal."
Step 4. Measure the length of the vertical line
Align the short end of the ruler with the bottom line (side) of the triangle. Measure the length of the line from the point where the vertical line intersects the top ray of the obtuse angle. The result is the length of the vertical line.
The length of the vertical line is the vertical value in the equation "slope = vertical/horizontal." Once you know the values for the vertical and horizontal, you can calculate the slope value and the size of the acute angle
Step 5. Find the slope of the acute angle
Divide the vertical value by the horizontal value to determine the slope of the acute angle. You will use this value to calculate the degree of an acute angle.
For example, the equation "slope = 2/4" will return "slope = 0.5"
Step 6. Calculate the degrees of the acute angle
Enter the slope value in the science calculator, then press the inverse tangent button (tan-1). The value shown is the acute angle in degrees.
Continuing the example above, if the slope of the line is 0.5, it means that the acute angle is 26.565 degrees
Step 7. Subtract 180 degrees from the acute angle
A straight line has an angle of 180 degrees. So, the sum of the calculated acute and obtuse angles should also be 180 degrees. Subtract 180 degrees from the acute angle to get the obtuse angle.
Continuing the example above, if the acute angle is 26.565 degrees, the obtuse angle is 153, 435 degrees (180 – 26, 565 = 153, 435)
Method 3 of 3: Reflex Angle
Step 1. Identify the acute angle that corresponds to the reflex angle
A reflex angle is an angle that is greater than 180 degrees but less than 360. This means that you should see an acute angle in the rays of the reflex angle.
By determining the size of the acute angle, you can calculate the size of the reflex angle. You can use the basic slope formula and the inverse tangent function in a science calculator to find the value of an acute angle
Tip:
If you're confused because the angle is upside down, turn the paper and ignore the reflex angle until the last step.
Step 2. Draw a vertical line connecting the rays of the acute angle
Line up the short end of the ruler with the corner rays that are horizontal, rather than diagonal. Then, draw a vertical line that intersects the horizontal rays of the corner.
The horizontal ray becomes the side of the triangle, and the vertical line becomes the opposite side of the acute angle that you want to calculate
Step 3. Measure the vertical and horizontal acute angles
In the "slope = vertical/horizontal" equation, vertical is the length of the vertical line, or opposite side of the triangle. Horizontal is the length of the horizontal line, or side of the triangle.
Measure the horizontal line from the vertex to the point where it intersects the vertical line. Measure the length of the vertical line from where it meets the horizontal line to the point where it intersects with the diagonal line
Step 4. Divide the vertical by the horizontal to get the slope of the acute angle
Plug the obtained vertical and horizontal line length values into the slope formula. Divide the length of the vertical line by the horizontal line to get the degree of slope of the angle.
For example, if your horizontal line is 8 and your vertical line is 4, the equation would be "slope = 4/8." The slope of your angle is 0.5
Step 5. Use a calculator to find the degrees of an acute angle
Type the obtained slope value into the science calculator, then press the inverse tangent key (tan-1). The value shown is the smaller acute angle of the triangle.
To continue the example, if the slope is 0.5, the acute angle is 26.565 degrees
Step 6. Subtract 360 by the measure of the acute angle
A circle has an angle of 360 degrees. Since a reflex angle is an angle that is greater than 180 degrees, you relate it to the part of the circle. The sum of the reflex angle and the smaller acute angle should be 360 degrees.
To continue the example, if the small acute angle obtained is 26.565 degrees, the reflex angle is 333.435 degrees
Tips
- Make sure the science calculator's trigonometric functions are set to measure in degrees, and not radians.
- Slope is the relationship between vertical and horizontal. The unit of measurement used to calculate the length of the two lines is irrelevant; just make sure you use the same units for both lines. In other words, if you're measuring the length of one line in centimeters, it's a good idea to measure the other in centimeters as well.
