Updated March 14, 2018 By Michael O. Smathers
The study of geometry requires you to deal with angles and their relation to other measurements, such as distance. When looking at straight lines, calculating the distance between two points is straightforward: simply measure the distance with a ruler, and use the Pythagorean Theorem when dealing with right triangles. When working with a circle, however, there is no instrument to accurately measure a curve. Therefore, you may have to calculate the distance between two points on a circle using mathematics.
Measure the circle's radius with a ruler, or record the figure given to you in the math problem. The radius of a circle measures the distance from the center to any point along the outside of the circle.
Multiply this measurement by two to calculate the diameter, or distance through the center of the circle.
Multiply this measurement by pi. Pi is an irrational number, but for most everyday purposes and in school, you can round it to two decimal places: 3.14. The diameter of a circle multiplied by pi gives you the circumference, or the distance around the circle.
Draw two lines from the radius of your circle, each connecting to one of the two points you're using to measure arc distance.
Measure the angle made by those lines with a protractor and record the measurement.
Set the angle you measured as a ratio of 360. According to The Geometer's Sketchpad on the Rice University website, there are 360 degrees in any circle, so any angle you measure can be taken as a ratio to determine the proportion of an arc length.
Cross-multiply your numbers using the equation: a/C = T/360. A is your arc length, C is your circumference and T is the angle you measured. Multiply C by T. Set the result equal to 360 times a. Divide both sides of the equation by 360 to solve for a.
1 Expert Answer
Jon G. answered • 10/30/17 Patient knowledgeable STEM educator/former healthcare practitioner
Hi Bryson from Memphis, TN. Hope you have a great week at school. Great Geometry/Algebra problem. You must be working on the Distance Formula for coordinate points. do you know what the Distance formula is? If you don't here it is: So let's identify what things you know and what we need to calculate. So you know what one endpoint or coordinate points are which is: (-2, 2) which we can label (x1, y1) and we have the other coordinates, (5, 5) which we can label (x2, y2) Now we we can substitute the values into our Distance Formula. And this gives us What? Your calculations will give you the length or distance of the diameter of the circle. But your problem is to calculate for the radius, which is HALF of the diameter. So, to do that, we do what? √(x2 - x1)2 + (y2 - y1)2 = length of the radius Let me know if you need an other help.
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Recall the: Distance Formula: The distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is $$ D=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$ Example: The radius of your circle is the distance between the points $(-1,4)$ and $(3,-2)$. Using the Distance Formula: $$ D= \sqrt{ \bigl(3-(-1)\bigr)^2 + (-2-4)^2 } = \sqrt{ 4^2 + (-6)^2 } = \sqrt{ 16+36 } = \sqrt{ 52 } . $$ What is the equation of the circle? It is important to realize the "equation of the circle" is: a point $(x,y)$ is on the circle if and only if the coordinates of the point $x$ and $y$ satisfy the equation. So, how to get the equation? What is the relationship between the $x$ and $y$ coordinates of a point on the circle? Well, let $(x,y)$ be a point on the circle. The big idea is: $$ \text{The distance from the point }(x,y)\text{ to the center }(-1,4)\text{ is }\sqrt{52}.$$
Or $$ 52=(x+1)^2 +(y-4)^2. $$ The shortcut would be to just use the following formula (But it's important to realize why you'd use it and where it comes from): Equation of a Circle The equation of the circle with center located at $(a,b)$ and with radius $r$ is $$ r^2=(x-a)^2 +(y-b)^2 $$ Note that the radius squared is on the left-hand side of the equation.
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