Equation of a Circle

Standard Form

$(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$
• Radius: $r$

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

To convert to standard form: complete the square for both $x$ and $y$.

Example: $x^2 + y^2 - 6x + 4y - 12 = 0$ $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$ $(x - 3)^2 + (y + 2)^2 = 25$

Center: $(3, -2)$, Radius: $5$

Arc Length and Sector Area

An arc is a portion of the circumference. A sector is the "pie slice" region.

For a central angle of $\theta$ degrees:

$\text{Arc length} = \frac{\theta}{360} \times 2\pi r$

$\text{Sector area} = \frac{\theta}{360} \times \pi r^2$

Key insight: The fraction $\frac{\theta}{360}$ represents what fraction of the full circle is captured.

Central and Inscribed Angles

• Central angle: vertex at the center of the circle
• Inscribed angle: vertex on the circle

An inscribed angle is half the central angle that subtends the same arc.

If a central angle is 80°, the inscribed angle on the same arc is 40°.

Special case: Inscribed angle on a semicircle

If an inscribed angle subtends a diameter (semicircle), the angle is always 90°.

Tangent Lines

A tangent line touches the circle at exactly one point and is perpendicular to the radius at that point.

If a tangent and a radius meet at the point of tangency, they form a 90° angle.

SAT Question Types

Type 1: Find the Area or Circumference

Plug into the formulas. Watch for diameter vs. radius!

Type 2: Equation of a Circle

Given center and radius → write equation, or given equation → find center and radius.

Type 3: Complete the Square

Convert general form to standard form.

Type 4: Arc Length / Sector Area

Use the fraction of the circle based on the central angle.

Type 5: Inscribed/Central Angles

Use the 2:1 relationship between central and inscribed angles.

Common SAT Mistakes

1. Confusing radius and diameter — the formula uses radius, but the problem may give diameter
2. Forgetting to square $r$ in the circle equation — it's $r^2$, not $r$
3. Sign errors in the circle equation — $(x-3)^2$ means center $x = 3$ (positive), not $-3$
4. Using 360 instead of $2\pi$ when the angle is in radians
5. Not completing the square properly when converting circle equations

Area formula: $A = \pi r^2$ $A = \pi(6)^2 = 36\pi$

Answer: $36\pi$

SAT Tip: Always convert diameter to radius first! Area uses radius, not diameter.

Compare to given equation: $(x - 4)^2 + (y - (-1))^2 = 3^2$

Center: $(h, k) = (4, -1)$ Radius: $r = \sqrt{9} = 3$

Answer: Center: $(4, -1)$, Radius: $3$

SAT Tip: Watch the signs! $(y + 1)$ means $k = -1$, not $+1$

Compare with $(x + 3)^2 + (y - 5)^2 = 36$:

$(x + 3)^2 = (x - (-3))^2$, so $h = -3$ $(y - 5)^2$, so $k = 5$ $r^2 = 36$, so $r = 6$

Center: $(-3, 5)$ Radius: $6$

Key: Watch the signs! $(x + 3)^2$ means the center $x$-coordinate is $-3$, not $+3$.

Compare with $(x + 3)^2 + (y - 5)^2 = 36$:

$(x + 3)^2 = (x - (-3))^2$, so $h = -3$ $(y - 5)^2$, so $k = 5$ $r^2 = 36$, so $r = 6$

Center: $(-3, 5)$ Radius: $6$

Key: Watch the signs! $(x + 3)^2$ means the center $x$-coordinate is $-3$, not $+3$.

Step 2: Complete the square for $x$: Half of 8 = 4, square it = 16 $(x^2 + 8x + 16)$

Step 3: Complete the square for $y$: Half of $-10$ = $-5$, square it = 25 $(y^2 - 10y + 25)$

Step 4: Add the same values to the right side: $(x^2 + 8x + 16) + (y^2 - 10y + 25) = -16 + 16 + 25$ $(x + 4)^2 + (y - 5)^2 = 25$

Answer: Center $(-4, 5)$, Radius $= \sqrt{25} = 5$

Step 2: Complete the square for $x$: Half of 8 = 4, square it = 16 $(x^2 + 8x + 16)$

Step 3: Complete the square for $y$: Half of $-10$ = $-5$, square it = 25 $(y^2 - 10y + 25)$

Step 4: Add the same values to the right side: $(x^2 + 8x + 16) + (y^2 - 10y + 25) = -16 + 16 + 25$ $(x + 4)^2 + (y - 5)^2 = 25$

Answer: Center $(-4, 5)$, Radius $= \sqrt{25} = 5$