Angle Between Two Circles

The angle between two circles is defined as the angle formed by two lines drawn from the point of intersection of the circles to the centers of the circles. This concept is important in geometry, particularly when dealing with the properties of circles and their interactions with each other.

Understanding the Angle Between Two Circles

When two circles intersect, they can do so in two points or one point (in the case of tangency). The angle between the two circles at a point of intersection is the angle between the tangents to the circles at that point. Alternatively, it can also be considered as the angle between the radii at the point of intersection.

Formula

If two circles intersect at two points, we can define the angle between them using the following formula:

$$ \cos(\theta) = \frac{r_1^2 + r_2^2 - d^2}{2r_1r_2} $$

where:

• $\theta$ is the angle between the two circles,
• $r_1$ and $r_2$ are the radii of the two circles, and
• $d$ is the distance between the centers of the two circles.

Important Points

• If the circles are touching externally, the angle between them is $0^\circ$.
• If the circles are touching internally, the angle between them is $180^\circ$.
• If the circles do not intersect (i.e., they are separate or one is contained within the other without touching), the concept of an angle between them is not defined.

Table of Differences and Important Points

Property	External Tangency	Internal Tangency	Intersecting Circles
Angle	$0^\circ$	$180^\circ$	$0^\circ < \theta < 180^\circ$
Distance between centers	$d = r_1 + r_2$	$d =	r_1 - r_2
Nature of Intersection	Touch at one point	Touch at one point	Intersect at two points
Formula	Not applicable	Not applicable	$\cos(\theta) = \frac{r_1^2 + r_2^2 - d^2}{2r_1r_2}$

Examples

Example 1: External Tangency

Consider two circles with radii $r_1 = 5$ and $r_2 = 3$ that touch each other externally. The distance between their centers is $d = r_1 + r_2 = 5 + 3 = 8$. The angle between the two circles is $0^\circ$.

Example 2: Internal Tangency

Consider two circles with radii $r_1 = 7$ and $r_2 = 3$ that touch each other internally. The distance between their centers is $d = |r_1 - r_2| = |7 - 3| = 4$. The angle between the two circles is $180^\circ$.

Example 3: Intersecting Circles

Consider two circles with radii $r_1 = 6$ and $r_2 = 4$ whose centers are $8$ units apart. To find the angle between the circles, we use the formula:

$$ \cos(\theta) = \frac{6^2 + 4^2 - 8^2}{2 \cdot 6 \cdot 4} = \frac{36 + 16 - 64}{48} = \frac{-12}{48} = -\frac{1}{4} $$

Thus, $\theta = \cos^{-1}\left(-\frac{1}{4}\right)$, which can be calculated using a calculator to find the angle in degrees or radians.

In conclusion, the angle between two circles is a measure of how they intersect or touch each other. It is an important concept in geometry that helps us understand the spatial relationships between circles.