Circles

Read the center from the subtracted coordinates

The equation $(x-2{)}^2+(y+3{)}^2=25$ describes a circle centered at $(2,-3)$ with radius 5. Standard form is $(x-h{)}^2+(y-k{)}^2=r^2$, so the right side gives the squared radius.

The signs inside the parentheses are opposite the center coordinates. To verify the center, substitute it: both squared terms become zero. To verify a point on the circle, their sum must equal 25.

Find area and circumference from the radius

A circle with diameter 12 has radius 6. Its area is $A=\pi r^2=36\pi$, and its circumference is $C=2\pi r=12\pi$. Area and circumference answer different questions and have different units.

Doubling the radius doubles circumference but quadruples area. If an area is multiplied by 9, the radius is multiplied by 3, not 9.

Use a fraction of a turn for arcs and sectors

A 60-degree central angle is $\frac{60}{360}=\frac{1}{6}$ of a full circle. For radius 6, its arc length is $\frac{1}{6}(12\pi )=2\pi$, and its sector area is $\frac{1}{6}(36\pi )=6\pi$.

The arc uses circumference; the sector uses area. A central angle has its vertex at the center. An inscribed angle intercepting the same arc is half the central angle, so identify the vertex before using the number.

Use radians directly when they are given

With an angle $\theta$ in radians, arc length is $s=r\theta$ and sector area is $A=\frac{1}{2}{r}^2\theta$. For $r=6$ and $\theta =\frac{\pi }{3}$, those formulas again give $2\pi$ and $6\pi$.

Do not insert a degree measure into these radian formulas. Convert degrees using ${\theta }_{\text{radians}}={\theta }_{\text{degrees}}\cdot \frac{\pi }{180}$, or use the fraction-of-360 method instead.

Check your understanding

Sample circles questions

Practice this skill

1. Label radius, diameter, and the angle vertex before calculating.
2. Choose circumference for arcs and area for sectors.
3. Check a coordinate-circle answer by substituting a known point.

Print the circles worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.