# 3 Ways to Measure an Angle without a Protractor

The easiest way to measure an angle is with a protractor. However, this can be done using basic principles of triangle geometry. You'll need to use a scientific calculator to solve the equations, and most smartphones have one, but there are also apps you can download or websites you can access for free. The necessary calculations change if the angle is acute (less than 90°), obtuse (greater than 90° and less than 180°), or reflex (greater than 180° and less than 360°).

## Steps

### Method 1 of 3: Acute Angle

#### Step 1. Draw a vertical line connecting the two angle rays and form a triangle

Align the short part of the ruler with the semistraight at the base and then draw a vertical line that cuts through the other, using the long part of the ruler as support.

### This vertical line creates a right angle. The angle formed by the adjacent side (the ray at the base of the angle) and the opposite side (the vertical line) measures 90°

#### Step 2. Measure the length of the adjacent side to find the base measurement

Place the tip of the ruler at the vertex of the angle. Measure the length of the adjacent side from the vertex to the point where it intersects the opposite side.

### This is the base, which will be entered into the slope = height/base equation. For example, when measuring the base and finding 7 cm, the equation so far should be "slope = height/7"

#### Step 3. Measure the length of the opposite side to find the height measurement

Place the edge of the ruler aligned against the adjacent side of the triangle. Measure the length of the vertical line from the point where it meets the adjacent side to where it meets the upper half straight line of the angle (the hypotenuse).

### This value is the height in the equation. For example, when measuring height and finding 5 cm, enter that value into the equation so that it reads "slope = 5/7"

#### Step 4. Divide the height by the base and find the slope of the angle

It is given by the diagonal line, the hypotenuse of the triangle. Once you have that number, you can calculate the acute angle measure.

### To continue the example, the equation slope = 5/7 results in slope = 0.71428571”

#### Tip:

Do not round the number before calculating the angle measure - this decreases the accuracy of the result.

#### Step 5. Use the calculator to determine the angle measure

Enter the slope value into the scientific calculator and then press the inverse tangent button (tan-1). This will give the angle value.

### Method 2 of 3: Obtuse

#### Step 1. Extend the ray at the base of the angle

Mark the vertex with a point and use the ruler to draw a straight line to the left of it. The ray at the base of the angle should be a long line that extends below the other.

### This line must be perfectly straight or it will ruin the precision of the equation

#### Tip:

When working on unlined paper, use the short portion of the ruler aligned with the side of the sheet to ensure the extended line is straight.

#### Step 2. Draw a straight line connecting the top ray and the base

Align the short part of the ruler with the base so that the long part crosses the upper ray and draw a line from the base to it, connecting the two.

### In this way, you create a right angle below the obtuse angle you want to measure, transforming the obtuse angle ray into the hypotenuse of a right triangle

#### Step 3. Measure the length of the base from the vertex

Position the ruler at the base, starting on the vertical line that creates the right angle. Measure the length from this intersection point to the vertex of the original angle.

### You are determining the slope of the acute angle in the triangle made, which you can use to calculate its measurement. The bottom line is the base in the "slope = height/base" equation

#### Step 4. Measure the length of the vertical line

Align the short part of the ruler with the base of the triangle. Measure the height to the point where it cuts the upper ray. This is the length of the height.

### This measurement is the height in the "slope = height/base" equation. Once you have the values ​​for height and base, calculate the slope of the acute angle

#### Step 5. Find the slope of the acute angle

Divide the height by the base to determine the slope. Use this value to calculate the angle measure.

### In the example above, the equation "slope = 2/4" results in "slope = 0.5"

#### Step 6. Calculate the acute angle measurement

Enter the slope value into the scientific calculator and press the inverse tangent button (tan-1). The displayed value is the measurement in degrees.

### Continuing the example, for a slope of 0.5, the acute angle is 26.565°

#### Step 7. Subtract this result from 180

A straight line is an angle of 180°. Since you have drawn a straight line, the sum of the calculated acute angle and the obtuse angle will be 180°. Subtracting the found value from 180 will give the obtuse angle result.

### Method 3 of 3: Reflection

#### Step 1. Identify the smallest acute angle associated with the reflex angle

A reflex angle is greater than 180° and less than 360°. This means that if you look at a reflection angle, you will also see an acute angle opposite it.

### When determining the acute angle measurement, calculate the reflex angle. Use the equation to calculate slope and the inverse tangent function on the scientific calculator to find the acute angle measure

#### Tip:

Ignore the reflex angle until the last step and turn the sheet over if you are confused because the angle is upside down.

#### Step 2. Draw a vertical line connecting the acute angle ray

Align the short part of the ruler with the half straight line that is horizontal. Then draw a vertical line connecting the two.

### The horizontal line becomes the adjacent side of the triangle and the vertical line the opposite side of the acute angle you want to measure

#### Step 3. Measure the height and base of the right triangle

In the "slope = height/base" equation, height is the length of the vertical line, opposite the angle you want to measure. The base is the horizontal line, adjacent to it.

### Measure the horizontal line from the vertex to the point where it meets the other, vertically, and then measure the vertical line from the point where it cuts the horizontal to the point where it meets the diagonal

#### Step 4. Divide the height by the base to find the angle slope

Enter the values ​​found for height and base in the slope equation. By dividing them, the result is the slope of the angle.

### For example, when measuring the horizontal line and finding 8 cm and when measuring the vertical and finding 4 cm, the equation would be "slope = 4/8." The slope of the angle will be 0, 5

#### Step 5. Use the calculator to find the acute angle measurement

Enter the value found for the slope of the angle into the scientific calculator and then press the button for inverse tangent (tan-1). The displayed value is the degree measurement of the acute angle.

### Continuing the example, for a slope of 0.5, the acute angle measurement will be 26.565°

#### Step 6. Subtract this result from 360

A circle is 360°. Since a reflex angle is greater than 180° and less than 360°, you can think of it as a portion of the circle. The sum of the reflex angle and the measured acute angle must be 360°.

## Tips

• Configure the scientific calculator functions to display the result in degrees rather than radians.
• Slope is the relationship between height and base. The unit used to measure the length of the two lines is irrelevant – just ensure that the same unit is used for both. In other words, when measuring one in centimeters, the other must also be measured in centimeters.