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Circle questions on the SAT look like two topics: algebra about an equation, geometry about arcs and sectors. They run on two small ideas. The equation always wants the center and radius, and completing the square hands you both. Arc and sector questions always want a fraction of the whole circle, and the central angle names the fraction. Two moves, whole category.
Every SAT circle question wants one of two things. For a center or radius, put the equation in standard form (x-h)^2 + (y-k)^2 = r^2, completing the square first when it is given expanded. For an arc or a sector, take the central-angle fraction of the whole circle. Both moves lean only on formulas the test already prints, which the broader SAT math formula guide sorts into given versus memorize.
Key takeaways
- Standard form (x-h)^2 + (y-k)^2 = r^2 shows the center (h, k) and radius r directly.
- Given the expanded form, complete the square in x and y to recover it.
- Arc length and sector area are fractions: multiply the whole-circle value by \frac{\text{central angle}}{360^\circ}.
- Radians: degrees times \tfrac{\pi}{180}. A full circle is 2\pi radians.
Standard form: the equation that reads itself
A circle with center (h, k) and radius r has the equation (x-h)^2 + (y-k)^2 = r^2. The center hides behind minus signs, exactly like vertex form for a parabola: (x+2)^2 means h = -2, because x + 2 is x - (-2). And the right side is not the radius. It is the radius squared.
(x-3)^2 + (y+2)^2 = 25 describes a circle of radius 5, not 25. The equation ends in r^2, so take a square root before you answer. 25 is always in the choices, waiting.
Completing the square: expanded back to standard
The SAT rarely hands you standard form. It hands you the multiplied-out version, x^2 + y^2 - 6x + 4y - 12 = 0, and asks for the center or radius. Completing the square is the reverse gear:
- 1
Group by variable
Constant right, terms paired: (x^2 - 6x) + (y^2 + 4y) = 12.
- 2
Halve, square, add to both sides
Half of -6, squared: 9. Half of 4, squared: 4. So (x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4.
- 3
Factor and read
(x-3)^2 + (y+2)^2 = 25. Center (3, -2), radius \sqrt{25} = 5.
Try one
The equation x^2 + y^2 + 8x - 2y - 8 = 0 defines a circle in the xy-plane. What are its center and radius?
Arcs and sectors: everything is a fraction
A central angle carves out a slice, and the slice’s share of everything is \frac{\text{angle}}{360^\circ}. Arc length is that fraction of the circumference 2\pi r; sector area is the same fraction of \pi r^2. No separate arc formula to memorize, just a fraction times a formula that is on the reference sheet anyway. Its sibling category, area and volume, draws on the same reference sheet with the same division of labor.
| You want | Use | Watch for |
|---|---|---|
| Center and radius | Standard form (x-h)^2 + (y-k)^2 = r^2 | Signs flip; right side is r^2 |
| Standard form from expanded | Complete the square in x and y | Add the constants to both sides |
| Arc length | \frac{\text{angle}}{360^\circ} \times 2\pi r | Fraction of circumference |
| Sector area | \frac{\text{angle}}{360^\circ} \times \pi r^2 | Fraction of area, r squared |
| Radians | degrees \times \tfrac{\pi}{180} | Full circle = 2\pi |
A circle has radius 6. A sector of the circle has a central angle of 60^\circ. What is the sector’s area?
Degrees and radians without ceremony
Radians measure the same angles with 2\pi playing the role of 360^\circ. To convert, multiply degrees by \tfrac{\pi}{180}: 30^\circ = \tfrac{\pi}{6}, 60^\circ = \tfrac{\pi}{3}, 90^\circ = \tfrac{\pi}{2}. The fraction idea survives: an angle of \theta radians is \tfrac{\theta}{2\pi} of the circle, which is why arc length collapses to s = r\theta. If conversions feel shaky, real circle questions settle them faster than re-reading the rule.
Work equations, arcs, and sectors with a tutor that checks you took the square root.
Practice Circles →Frequently asked questions
- What is the standard form of a circle’s equation?
- (x minus h) squared plus (y minus k) squared equals r squared, with center (h, k) and radius r. The minus signs are built in, so a (x + 4) squared term means h is negative 4, and the right side is the radius SQUARED, not the radius.
- How do you find the center and radius of a circle from its equation?
- Complete the square: group the x-terms and y-terms, add (half the linear coefficient) squared for each variable to both sides, then factor. That recovers standard form, which shows the center directly and the radius as the root of the right side.
- How do you find arc length and sector area on the SAT?
- Both are fractions of a whole-circle quantity: multiply the circumference (2 pi r) or the area (pi r squared) by the central angle over 360. In radians the fraction is the angle over 2 pi, so arc length simplifies to radius times angle.
- How do you convert between degrees and radians?
- Multiply degrees by pi over 180 for radians, or radians by 180 over pi for degrees. Anchor on landmarks: 180 degrees is pi radians, 90 is pi over 2, 60 is pi over 3, 45 is pi over 4, and 30 is pi over 6.