Free video lesson

How to Read a Circle Equation on the SAT

A circle equation is just two things — a center and a radius — hiding in plain sight. Learn to read them straight off (watch the sign flip!), draw the circle, and handle arc length and sector area with one fraction.

Math · Circles2:10Published July 6, 2026

On YouTube: SAT Circles: The Arc-Angle Connection

What this lesson covers

  • (x−h)² + (y−k)² = r² → center (h, k), radius r
  • The sign flip: (x−2) means h = +2
  • r² vs r: take the square root
  • Desmos: type it, the circle appears
  • Practice: arc & sector of a 90° slice, radius 8

Questions worked in the video

  1. 0:29(x−2)² + (y+1)² = 25 — find the center and radius.
  2. 1:09A 90° sector of a circle with radius 8 — find the arc length and sector area.

Worked examples

The questions the video works, written out: the setup, each step, the answer and the trap.

0:29Example 1

The equation (x−2)2+(y+1)2=25(x - 2)^2 + (y + 1)^2 = 25 describes a circle. What are its center and radius?

  1. Match the equation to the template (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where the center is (h, k) and the radius is r.
  2. The x-part is (x−2)2(x - 2)^2, so h = 2. The y-part is (y+1)2(y + 1)^2, which is (y−(−1))2(y - (-1))^2, so k = -1. The sign flips: a plus inside the parentheses means a negative coordinate.
  3. The right side is r2=25r^2 = 25, so r=25=5\displaystyle r = \sqrt{25} = 5. The center is (2, -1) and the radius is 5.
  4. Check with a point that should sit on the circle. Five units right of the center is (7, -1): (7−2)2+(−1+1)2=25+0=25(7 - 2)^2 + (-1 + 1)^2 = 25 + 0 = 25.

Answer: center (2, -1), radius 5

The template puts the center coordinates behind minus signs, so (x - 2) reads as h = 2 while (y + 1) reads as k = -1. The two traps are the sign flip, which produces the wrong center (-2, 1), and reporting 25 as the radius when 25 is r squared; the radius is 5.

1:09Example 2

A sector of a circle with radius 8 has a central angle of 90 degrees. What are the arc length and the area of the sector?

  1. A 90 degree slice is 90360=14\displaystyle \frac{90}{360} = \frac{1}{4} of the whole circle. That fraction multiplies both the circumference and the area.
  2. Circumference: 2πr=2π(8)=16π2\pi r = 2\pi(8) = 16\pi. Arc length: 14⋅16π=4π\displaystyle \frac{1}{4} \cdot 16\pi = 4\pi.
  3. Area: πr2=π(82)=64π\pi r^2 = \pi(8^2) = 64\pi. Sector area: 14⋅64π=16π\displaystyle \frac{1}{4} \cdot 64\pi = 16\pi.
  4. Check: four such arcs, 4⋅4π=16π4 \cdot 4\pi = 16\pi, rebuild the full circumference, and four such sectors, 4⋅16π=64π4 \cdot 16\pi = 64\pi, rebuild the full area.

Answer: arc length 4π4\pi, sector area 16π16\pi

An arc is a fraction of the circumference and a sector is the same fraction of the area, so both answers are one quarter of the whole-circle value. The trap is forgetting the fraction and answering 16π16\pi for the arc, which is the full circumference, or mixing the two formulas and squaring the radius inside the arc.

One more, same method

The equation (x+3)2+(y−4)2=36(x + 3)^2 + (y - 4)^2 = 36 describes a circle. What are its center and radius?

  1. Match to (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. The x-part (x+3)2(x + 3)^2 is (x−(−3))2(x - (-3))^2, so h = -3.
  2. The y-part (y−4)2(y - 4)^2 already has a minus sign, so k = 4. The center is (-3, 4).
  3. r2=36r^2 = 36, so r = 6.
  4. Check: six units right of the center is (3, 4): (3+3)2+(4−4)2=36+0=36(3 + 3)^2 + (4 - 4)^2 = 36 + 0 = 36.

Answer: center (-3, 4), radius 6

Both signs flip this time, in opposite directions: the plus in (x + 3) gives a negative h, and the minus in (y - 4) gives a positive k. Reading the signs as written gives the wrong center (3, -4), and reading 36 as the radius gives a circle six times too big.

Chapters

Lesson transcript

The narration of the video, word for word, under its chapter headings.

0:00A circle = center + radius

Welcome to SAT Climb. A circle equation looks scary, but it's just two things: a center and a radius. Read them off, and you can draw the whole circle.

0:16The setup: an equation

Here's one: (x−2)² + (y+1)² = 25. Where's the center? How big is the radius?

0:29Read center & radius

Match the template: (x−h)² + (y−k)² = r². So h = 2, k = −1 — watch the sign flip — center (2, −1). And r² = 25, so the radius is 5. Draw it.

0:49Confirm in Desmos

Not sure? Type it into Desmos. The circle appears, centered at (2, −1), radius 5. Exactly what you read.

1:09Your turn (arc & sector)

Your turn. A 90° slice of a circle, radius 8. Arc = (θ/360)·2πr; sector = (θ/360)·πr². Find both.

1:35Three traps

Three traps. The sign flip: (x−2) means h = +2. r vs r²: 25 = r², so take the square root. And forgetting the θ/360 fraction for arcs and sectors.

1:56Recap

Circles: solved. Center, radius, and a slice. Start free at satclimb.com.

Read the written version: the Circles strategy guide, then try 25 hard Circles questions with full explanations. Both are free, no account needed.

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