How to Estimate Angle Measures

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

In real life it takes special equipment to accurately measure angles, but you can make a pretty good guess if you know a few tricks. We'll discuss some of those in this lesson.

Estimating Angle Measures

You've created the perfect ramp for your skateboard jump. It gives you just that amount of lift you need, your tricks are working perfectly, and your landings look like they belong in the Olympics. Then the community planning committee announces that your ramp is too steep; it has to be less than 30°. After you get over the desire to throw a fit, you stop and think, ''Wait a minute! Are they right? Is my ramp really steeper than 30°?''

Estimating an angle means trying to make an educated guess about an angle measurement. In this lesson, we'll explore some tricks that make this easy.

What Type of Angle Is It?

The four quadrants
Getting in the Ballpark

When you're estimating an angle, first decide which quadrant it would land in, if you put one side on the 'starting point', the 0° line between Q1 and Q4, and then let it stretch counterclockwise around the quadrants. Some angle types are listed below:

Angle Type Measure Range Lands in Quadrant
acute 0° to 90° Q1
right 90° between Q1 & Q2
obtuse 90° to 180° Q2
straight 180° between Q2 & Q3
reflex 180° to 360° anywhere in Q3 or Q4

First, let's eliminate the easy ones. If your angle lines up perfectly with the 0° line and one of the other lines between the quadrants, then you don't need to estimate.

  1. Right (90°) angle
  2. Straight line (180° angle)
  3. 3/4 of a circle (270° angle)

Assuming it's not one of those, then it's time to go to work. If it's not acute (between 0° and 90°), then mentally cut out as many 90° blocks as you can, (setting them aside to add in later) until you are left with an angle that is less than 90°.

Getting closer to a Q2 angle
Q2 angle

Getting closer to a Q3 angle
Q3 angle

Getting closer to a Q4 angle
Q4 angle

Acute angles
Acute Angles

Now, mentally rotate your remaining angle so it lines up with the picture above (the area of the angle should span across Q1, with one side of it resting on the 0° 'starting' line). 45° is the center between 0° and 90°. Is your remaining angle larger or smaller than that? Is it a 60°? A 30°? Perhaps a 75° or a 15°? If it doesn't really match any of those, then try and estimate within 5°:

  • larger than 75°: use 80° or 85° (depending on which seems closer)
  • between 60° and 75°, use 65° or 70°
  • between 45° and 60°, use 50° or 55°
  • between 30° and 45°, use 35° or 40°
  • between 15° and 30°, use 20° or 25°
  • between 0° and 15°, use 5° or 10°

Adding in the 90° Blocks

Once you've picked the closest acute angle, make sure you add 90° for each 90° block you removed (if any). For example, say you removed two 90° blocks from your angle and had 25° left. Your final angle would be:

25° + 90° + 90° = 205°.

Using the Tangent

Since our skateboard ramp example isn't an angle on paper, we'll use the tangent (the ratio between height and length) to get us close.

Tangent/Slope Ratio Approx. Angle
1/5 11°
1/3 18°
1/2 27°
2/3 34°
1 45°
2 63°
3 72°
4 76°
5 79°
10 84°
20 87°

Now obviously you probably don't want to memorize all of these, but it helps to at least know that a tangent of 1 is 45°, a 1/2 is 27°, and a 3 is 72°. Okay, so here's the plan for estimating an acute angle outside (if it's not acute, mentally chop off 90° blocks again, until you get down to acute):

  1. Take a stick, your arm, or whatever, and measure the height and the length.
  2. Divide the height by the length. That will give you the tangent.
  3. Remember what you can of the tangent/angle table. Using your tangent ratio, you should be able to get a pretty good idea of what the angle is.

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