SAT Math · Geometry and Trigonometry

Equation of a circle

Read the center and radius from (x − h)² + (y − k)² = r², and complete the square when the equation is expanded.

Try Aristotle free

Updated

(x − 2)² + (y + 1)² = 9: center (2, −1), radius 3.−4−268−6−4−2246r = 3(2, −1)

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

(x−2)2+(y+1)2=9(x - 2)^2 + (y + 1)^2 = 9: center (2,−1)(2, -1), radius 3.

What is the equation of a circle?

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

A circle is every point exactly rr away from a center (h,k)(h, k). The equation is that sentence in algebra: a point (x,y)(x, y) is on the circle precisely when its distance to the center works out to rr.

Every point on the circle is exactly 3 from (1, 2).−224−246r = 3(1, 2)
Every point on the circle is exactly 3 from (1, 2).

It is the Pythagorean theorem, for every point at once

(x−h)2⏟across2+(y−k)2⏟up2=r2⏟radius2\underbrace{(x-h)^2}_{\text{across}^2} + \underbrace{(y-k)^2}_{\text{up}^2} = \underbrace{r^2}_{\text{radius}^2}

From the center to any point on the circle, go across x−hx - h and up y−ky - k. Those are the legs of a right triangle whose hypotenuse is the radius. Square the legs, add, and you have the equation.

3² + 4² = 5², so (3, 4) is on the radius-5 circle.−6−4−246−6−4−2246345(3, 4)
3² + 4² = 5², so (3, 4) is on the radius-5 circle.

Read the center backwards, and root the radius

(x+1)2+(y−2)2=16(x + 1)^2 + (y - 2)^2 = 16⇒  center (−1,2), r=4\Rightarrow\; \text{center } (-1, 2),\ r = 4

The equation subtracts the center, so a plus inside means a negative coordinate: x+1x + 1 is x−(−1)x - (-1). The right side is the radius squared, so 16 means a radius of 4, not 16.

x + 1 puts the center at x = −1, and 16 means r = 4.−6−4−224−2246r = 4(−1, 2)
x + 1 puts the center at x = −1, and 16 means r = 4.

Why the equation looks like this

Answer each question in your head, then scroll for the answer.

(x − 2)² + (y + 1)² = 9: center (2, −1), radius 3.−4−268−6−4−2246r = 3(2, −1)

Think it through

  1. Question 1

    A point (x,y)(x, y) is on a circle centered at (h,k)(h, k). What are the horizontal and vertical gaps to the center?

  2. Answer

    x−hx - h across and y−ky - k up.

  3. Question 2

    Those gaps form a right triangle. What is its hypotenuse?

  4. Answer

    The radius rr, so (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.

  5. Question 3

    Center (2,−1)(2, -1). What goes inside the parentheses?

  6. Answer

    (x−2)(x - 2) and (y+1)(y + 1). The signs flip.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

A circle with center (−3, 4) and radius 5.−10−8−6−4−224−4−226810(−3, 4)

Question 1

What are the center and radius of (x+3)2+(y−4)2=25(x + 3)^2 + (y - 4)^2 = 25?

The circle x² + y² − 4x + 2y − 4 = 0, centered at (2, −1).−4−268−6−4−2246(2, −1)

Question 2

What is the radius of the circle x2+y2−4x+2y−4=0x^2 + y^2 - 4x + 2y - 4 = 0?

A circle centered at (4, 2) passing through (7, 6).−2246810−4−22468(4, 2)(7, 6)

Question 3

A circle has center (4,2)(4, 2) and passes through (7,6)(7, 6). What is the equation of the circle?

A circle of radius 5 centered at the origin, with (3, 4) on it and (4, −4) outside it.−6−4−2246−6−4−2246(3, 4)(4, −4)

Question 4

Is the point (4,−4)(4, -4) inside, on, or outside the circle x2+y2=25x^2 + y^2 = 25?

Stuck on a question like this?

Talk it through with Aristotle. It asks guiding questions instead of giving the answer. Intro is free with three sessions.

Try Aristotle free

The formulas

Standard form

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Center (h,k)(h, k), radius rr. A center at the origin gives x2+y2=r2x^2 + y^2 = r^2.

Expanded form

x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0

The same circle multiplied out. Complete the square to read it.

Completing the square

x2+Dx=(x+D2)2−(D2)2x^2 + Dx = \left(x + \tfrac{D}{2}\right)^2 - \left(\tfrac{D}{2}\right)^2

Halve the xx coefficient, square it, and add it to both sides. Do the same for yy.

Circle facts, for reference

C=2πrA=πr2C = 2\pi r \qquad A = \pi r^2

Both on the reference sheet. A tangent is perpendicular to the radius.

Try it

Move the circle

Move the center and resize. Watch the signs flip.

A circle with center (2, -1) and radius 3.−8−48−8−448(2, −1)
2
−1
3

Standard form(x − 2)² + (y + 1)² = 9

Expandedx² + y² − 4x + 2y − 4 = 0

The signs inside flip: center (2, −1) shows up as (x − 2)².

Common mistakes

Keeping the signs from the equation

(x−2)2+(y+1)2=9(x - 2)^2 + (y + 1)^2 = 9 has center (2,−1)(2, -1), not (−2,1)(-2, 1).

Reading r² as the radius

If the right side is 9, the radius is 3. The number on the right is the radius squared.

Adding to only one side when completing the square

Whatever you add to complete the square, add to the right side too.

Using the diameter as the radius

From a diameter's endpoints, the center is the midpoint and rr is half the length.

How it shows up on the SAT

Geometry and Trigonometry is about 15% of SAT Math. Expect to read the center or radius, complete the square, or test whether a point is on the circle.

What Aristotle tracks in circles

7 SAT Math skills, including:

  • Solving problems using circle properties

  • Using radians and trigonometric ratios in the unit circle

  • Writing equations for circles in the xy-plane

  • Relating circle equations to circle graphs

Questions students ask

A few things you might wonder before getting started.

What is the equation of a circle?

A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r². If the center is the origin, it simplifies to x² + y² = r².

How do you find the center and radius from the equation?

How do you complete the square for a circle equation?

Is the equation of a circle on the SAT reference sheet?

How do you tell if a point is on a circle?

How can Aristotle help with SAT geometry?

More SAT concepts

The Pythagorean theorem

a² + b² = c² for any right triangle, plus the triples and special triangles that save time on SAT Math.

The unit circle

Sine, cosine, and radians from one circle, with the values SAT Math expects you to know.

The quadratic formula

Solve any ax² + bx + c = 0, and use the discriminant to count solutions without solving.

Tutoring for this topic

SAT tutor

SAT Math and Reading and Writing, organized by official domain.

Geometry tutor

Proofs, transformations, trigonometry, and everything between.

Precalculus tutor

Precalculus and AP Precalculus, the runway into calculus.

A learner having an idea, surrounded by maps and equations

Practice it out loud

Talk it through with Aristotle

Work a question like these out loud. Aristotle guides you one step at a time, so you can do the next one on your own.

Try Aristotle today