Introduction: How to Find the Radius for a Circle of Which the Length of a Chord and the Distance Between Its Midpoint and the Top of the Circle Are Known

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This may seem like a very specific thing to try to find, but it does have some practical applications.

I am currently working on a tubular bridge where I want the width of the floor (Represented by the chord) to 1.3m and the distance from the floor to the top of the tube to be 2.15m (Max height of person). There are many possibilities for a radius where the width and height are equal or greater to those constants. However, there is only 1 radius measurement where both criteria are satisfied. I also want to use the least material possible, so I'm looking for the smallest possible radius.

In this Instructable, I will show you how to calculate the radius of a circle where the length of a chord is known as well as the distance between its midpoint and the top of the circle.

Supplies

  • Paper
  • Pencil
  • Ruler
  • Compass
  • Protractor

Step 1: Understand the Context

This diagram is all we know at the start. We want to find the radius of the circle that passes through points A, B, and D. We also want to find the angle between the points of chord and the center of the circle in order to easily replicate the scenario. In the second diagram, this angle is AEB.


There are 2 approaches to finding the radius and the value of AEB, which we will call theta. The first is the easier one, and is to find the circumscribed circle of the triangle ADB (Refer to 3rd diagram). This one is more geometric. Then there's the trigonometric algebraic approach, which relies on exact values and is a bit more involved, but is more interesting.

Step 2: Circumscribed Circle Method

In this method, we will use GeoGebra (or equivalent software) to draw a model of the situation. Then we will use it to find the circumscribed circle of the triangle and ultimately measure the radius. This could also be done on paper, but for good accuracy, software needs to be used.

Step 3: Build the Initial Situation

Build the initial situation in a geometry software such as GeoGebra using your own measurements for width and height. This should just look like a T, with the chord and height with given lengths. Us a segment and perpendicular bisector to do so.

Step 4: Create the Triangle

Create a triangle using the extremity points of the T.

Step 5: Perpendicular Bisectors

Create a perpendicular bisector in each of the triangle's edges.

Step 6: Intersection and Circumscribed Circle

Use the intersect feature to create a point at the intersection of the 3 points (really 2 but they all meet at one point).

Create a center circle at using this point of intersection. It should pass through the 3 points of the triangle.

Step 7: Measuring the Radius and Angle

Measure the radius of the circle by creating a segment between the center and one of the chord's extremity points.

With width = 1.3m and height = 2.15m, it gives 1.173 as the radius of the circle.

For the angle of the sector that the chord is in, just create another segment and measure the angle. GeoGebra gives us 67.289°.

These results are actually very accurate. The software does all the algebra for you and delivers precise results. We will see that the results are the same in the second method, but slightly more accurate.

Step 8: Algebraic Trigonometry Method

In this method, we will use algebra with exact values to get an incredibly accurate answer. It involves trigonometric relationships (sine, cosine, Pythagorean theory) and some equation solving.

Throughout the explanation, I will be referencing letters from the diagram above (Not the one in intro).

Step 9: Calculating CB

Using Pythagorean theory, calculate CB

Step 10: Calculate Angle MCB

Use the tangent to calculate MCB.

Step 11: Express Angle COB in Function of the Radius

Use cosine law and the measurements found in previous steps to calculate CÔB.

Step 12: Find the Sine of COB

Using the Pythagorean relationship between sine and cosine, calculate CÔB's sine, still in function of the radius, r.

Step 13: Use Sine Law to Solve for the Radius

Using the sine of CÔB found in the last step, use sine law to relate it to other known measurements in the triangle. Then, solve for the radius, because sine CÔB is in function of the radius. You can either solve it on paper or use a numeric solver.

Step 14: Calculate Theta

Use the smaller triangle (MOB) and the sine to calculate theta now that you know the radius.

Step 15: Results of 2nd Method

With width = 1.3m and height = 2.15m:

The radius' exact value is 1009/860 meters ≈ 1.173.

The exact value of theta is 2 * arcsin(559/1009) ≈ 1.1744 rad ≈ 67.286°.

This method is more accurate than the first because it guarantees us exact values. Rounded to 3 decimals, the radius is the same in both methods. In contrast, the value of theta differs slightly, proving that the algebraic approach is slightly better.

Step 16: Congratulations!

You can now calculate the radius of a circle where you know the length of a chord and its distance from the top of the circle. The skills involved also have many other applications. Geometry software can be a great tool for verifying calculations too!


Works Cited

“Circumcircle of a Triangle.” GeoGebra, www.geogebra.org/m/q9mxARZM. Accessed 16 June 2023.

GeoGebra. “Geometry - GeoGebra.” Geogebra.org, 2019, www.geogebra.org/geometry.