Suppose a chord of a circle is 32 millimeters long and is 12 millimeters from the center of the circle. What is the circumference of the circle?
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A radius to a chord cuts the chord in half at a right angle.
So if the radius is r, I see that
r^2 = 12^2 + 16^2
r^2 = 400
r = √400 = 20
circumference = 2πr = 2π(20) = 40π
To find the circumference of a circle, we can use the formula:
Circumference = 2 * π * r
where π (Pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle.
In this case, we are given the length of a chord and the distance of the chord from the center of the circle. To find the radius, we need to use the properties of chords and radii.
We can draw a right triangle with the chord as the base. The distance from the center of the circle to the chord is the height of the triangle. The radius is the hypotenuse of the triangle.
Using the Pythagorean theorem, we can calculate the radius:
r = √(h^2 + (0.5c)^2)
Where h is the distance from the center to the chord and c is the length of the chord.
In this case, h = 12 mm and c = 32 mm:
r = √(12^2 + (0.5*32)^2)
= √(144 + 256)
= √400
= 20 mm
Now that we have the radius, we can calculate the circumference:
Circumference = 2 * π * r
= 2 * 3.14159 * 20
= 125.6636 mm
Therefore, the circumference of the circle is approximately 125.6636 mm.