A simple extension of the Inscribed Angle Theorem shows that the measure of the angle of intersecting chords in a circle is equal to half the sum of the measure of the two arcs that the angle and its … [Read more...] about Angles of Intersecting Chords

# Circles

In geometry, a circle is defined as the collection of all points that are the same distance from one point, which is the center of the circle.

The distance from the center of the circle to any point on the circle is called the radius, and commonly written 'r'.

If we draw a line from one point on the circle, through its center and on to another point on the circle, directly across from the first point, that line's length will be 2 times r, and is called the circle's diameter, 'd'. d= 2*r

A circle is a geometric shape completely defined by its radius- knowing the radius we can calculate the circle's area, and its circumference.

A circle's circumference, C, is C_{circle}=2*π*r (where r is the radius) , and since the diameter, d, is 2 times r, we can also write C_{circle}=d*π

A circle's area is given by the formula A_{circle}=π*r^{2}

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*Now that we've explained the basic concept of circles in geometry, let's scroll down to work on specific geometry problems relating to this topic.*

## Circle Inscribed in a Quadrilateral

A tangential quadrilateral is a quadrilateral whose four sides are all tangent to a circle inscribed within it. In such a quadrilateral, the sum of lengths of the two opposite sides of the … [Read more...] about Circle Inscribed in a Quadrilateral

## A Circle Inscribed in a Square

We've seen that when a square is inscribed in a circle, we can express all the properties of either the square or circle (area, perimeter, circumference, radius, side length) if we know just the … [Read more...] about A Circle Inscribed in a Square

## Square Inscribed in a Circle

When a square is inscribed in a circle, we can derive formulas for all its properties- length of sides, perimeter, area and length of diagonals, using just the circle's radius. Conversely, we can … [Read more...] about Square Inscribed in a Circle

## Thales’ Theorem

In today's lesson, we will prove Thales' Theorem - the inscribed angle that subtends the diameter of a circle is always a right angle, using the sum of angles in a triangle. Thales of Miletus was a … [Read more...] about Thales’ Theorem

## Kite Inscribed in a Circle

When we inscribe a kite is in a circle, all four of the kite's vertices lie on the circle's circumference. In today's lesson, we will show that in the case of a kite inscribed in a circle, the axis … [Read more...] about Kite Inscribed in a Circle

## Concentric Circles Intersected by a Secant

Two circles that have the same center point are called concentric circles. A secant is a line that interest a circle (or any other curved line) at two or more point. We will now show that a secant … [Read more...] about Concentric Circles Intersected by a Secant

## Finding the Length of a Common Chord

If we know the radii of two intersecting circles, and how far apart their centers are, we can calculate the length of the common chord. Problem Circles O and Q intersect at points A and B. The … [Read more...] about Finding the Length of a Common Chord

## Common Chord of Two Circles

In today's lesson, we will show that a line connecting the centers of two intersecting circles is a perpendicular bisector of the common chord of the two circles, connecting the intersection … [Read more...] about Common Chord of Two Circles

## Distance Between the Centers of Overlapping Congruent Circles

Congruent circles have the same radius length. In today's lesson, we will find the distance between the centers of two overlapping congruent circles, given the length of the overlapping arc, using … [Read more...] about Distance Between the Centers of Overlapping Congruent Circles