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

Total area: $A = \pi r^2 = \pi(9)^2 = 81\pi$

Sector area: $\frac{1}{9} \times 81\pi = 9\pi$

Answer: $9\pi$

SAT Tip: Sector = "pizza slice." Find what fraction of the whole circle it is!

Step 2: Calculate area: $A = \pi r^2 = \pi(7)^2 = 49\pi \approx 153.94 \text{ cm}^2$

Answer: $49\pi$ cm² (approximately 153.94 cm²)

Common mistake: Using the diameter (14) instead of the radius (7) in the formula.

Step 2: Calculate area: $A = \pi r^2 = \pi(7)^2 = 49\pi \approx 153.94 \text{ cm}^2$

Answer: $49\pi$ cm² (approximately 153.94 cm²)

Common mistake: Using the diameter (14) instead of the radius (7) in the formula.

$A = \frac{72}{360} \times \pi(10)^2 = \frac{1}{5} \times 100\pi = 20\pi \approx 62.83$

Answer: $20\pi$ square units (approximately 62.83)

Check: 72° is $\frac{1}{5}$ of 360°, so the sector is $\frac{1}{5}$ of the full circle area. $\frac{1}{5} \times 100\pi = 20\pi$ ✓

$A = \frac{72}{360} \times \pi(10)^2 = \frac{1}{5} \times 100\pi = 20\pi \approx 62.83$

Answer: $20\pi$ square units (approximately 62.83)

Check: 72° is $\frac{1}{5}$ of 360°, so the sector is $\frac{1}{5}$ of the full circle area. $\frac{1}{5} \times 100\pi = 20\pi$ ✓

Step 2: The distance from center $(2, 3)$ to the line $y = -1$ is: $r = |3 - (-1)| = |4| = 4$

Step 3: Write the equation: $(x - 2)^2 + (y - 3)^2 = 16$

Check: The closest point on the circle to $y = -1$ is directly below the center: $(2, 3-4) = (2, -1)$. This point is on the line $y = -1$ ✓

Answer: $(x - 2)^2 + (y - 3)^2 = 16$

Step 2: The distance from center $(2, 3)$ to the line $y = -1$ is: $r = |3 - (-1)| = |4| = 4$

Step 3: Write the equation: $(x - 2)^2 + (y - 3)^2 = 16$

Check: The closest point on the circle to $y = -1$ is directly below the center: $(2, 3-4) = (2, -1)$. This point is on the line $y = -1$ ✓

Answer: $(x - 2)^2 + (y - 3)^2 = 16$