Circles

Circle questions ask for circumference and area, the length of an arc or the area of a sector, the measures of angles in circles (central and inscribed), and the equation of a circle in the $xy$-plane. Angles may be given in degrees or in radians.

• Everything about a circle comes from its radius. If you are given a diameter, area, or circumference, find $r$ first.
• An arc or sector is a fraction of the whole circle: the central angle over $360^\circ$ (or over $2\pi$ radians).
• A central angle equals the measure of its arc. An inscribed angle (vertex on the circle) is half its intercepted arc.
• A radius drawn to a point of tangency is perpendicular to the tangent line.
• The equation $(x - h)^2 + (y - k)^2 = r^2$ describes a circle with center $(h, k)$ and radius $r$. If the equation is expanded, complete the square to get back to this form.

$C = 2\pi r \qquad A = \pi r^2 \qquad (x - h)^2 + (y - k)^2 = r^2$

• Inscribed angle $= \frac{1}{2} \times$ intercepted arc.
• Completing the square: $x^2 + 10x = (x + 5)^2 - 25$. Take half the coefficient, square it.

A circle has radius $9$. Find the length of the arc and the area of the sector cut off by an $80^\circ$ central angle.

1. Fraction of the circle: $\frac{80}{360} = \frac{2}{9}$.
2. Arc length: $\frac{2}{9} \cdot 2\pi(9) = \frac{2}{9} \cdot 18\pi = 4\pi$.
3. Sector area: $\frac{2}{9} \cdot \pi(9)^2 = \frac{2}{9} \cdot 81\pi = 18\pi$.

Check with radians: $80^\circ = \frac{4\pi}{9}$, and $r\theta = 9 \cdot \frac{4\pi}{9} = 4\pi$. Correct.

A circle in the $xy$-plane has equation $x^2 + y^2 + 10x - 4y = 7$. What are its center and radius?

1. Group the terms: $(x^2 + 10x) + (y^2 - 4y) = 7$.
2. Complete each square: $x^2 + 10x = (x + 5)^2 - 25$ and $y^2 - 4y = (y - 2)^2 - 4$.
3. Substitute: $(x + 5)^2 - 25 + (y - 2)^2 - 4 = 7$, so $(x + 5)^2 + (y - 2)^2 = 36$.
4. The center is $(-5, 2)$ and the radius is $\sqrt{36} = 6$.

Check: the point $(1, 2)$ is $6$ units right of the center, so it should be on the circle: $1 + 4 + 10 - 8 = 7$. Correct.

• Forgetting the square root. In $(x - 3)^2 + (y + 1)^2 = 64$, the radius is $8$, not $64$.
• Center sign errors. $(x + 5)^2$ means $h = -5$. The center has the opposite signs of the numbers in the parentheses.
• Not adding the completed squares to both sides. When you add $25$ and $4$ on the left, add them on the right too: $7 + 25 + 4 = 36$.
• Using the diameter in $\pi r^2$. A diameter of $14$ gives an area of $49\pi$.
• Treating an inscribed angle like a central angle. An inscribed angle intercepting a $140^\circ$ arc measures $70^\circ$.
• Mixing units. Use $\frac{\theta}{360}$ with degrees and $r\theta$ with radians, never a mix.