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Circles: SAT Math notes
Read and write equations of circles in the xy-plane, and use the radius, arcs, central angles, tangent lines and radians to find lengths and angles.
Study it with worked examples and 8 practice questionsPractise Circles questions
What this topic tests
Circle questions come in two flavours. Some are about the equation of a circle in the xy-plane. Others treat the circle as a shape: how long an arc is, what fraction of the circle an angle cuts off, how a tangent line meets the radius, or how to measure angles in radians.
The equation (x-h)2+(y-k)2=r2 is the key to the first kind. Its center is (h, k) and its radius is r. When an equation comes expanded, you complete the square to get back to this form. This equation is not on the reference sheet, so learn it.
For the second kind, almost everything is a proportion. An arc or a central angle is a fraction of the whole circle, and that same fraction applies to the circumference 2πr, to the full 360°, and to the full 2π radians. The reference sheet gives C=2πr, A=πr2, and the fact that a circle has 360° or 2π radians of arc.
Key ideas
- Standard form: (x-h)2+(y-k)2=r2. The center is (h, k), so flip the signs you see in the brackets. The right side is r2, not r.
- To complete the square on x2+bx, add (b/2)2 to both sides. The center's x-coordinate is -b/2.
- Every point on the circle is exactly r from the center. A point closer than r is inside; farther is outside.
- Arc length is the central angle's share of the circumference: angle/360×2πr.
- A tangent line is perpendicular to the radius at the point where it touches the circle.
- 180°=π radians. Degrees to radians: multiply by (π)/180. Radians to degrees: multiply by 180/(π).
- Adding or taking away full turns (2π or 360°) doesn't change the sine or cosine of an angle.
The 4 kinds of Circles question
Sorted from every Circles question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. The circle equation in standard form
How to spot it. You see an equation like (x-3)2+(y+16)2=289, or you're asked to write one. The question asks for the center or radius, which points are on or inside the circle, or the new equation after the circle is moved or resized.
How to answer it:
- Read the center from the brackets, flipping both signs: (x+4) means h=-4, and (y-5) means k=5.
- Take the square root of the right side to get the radius.
- To write an equation, put the center in the brackets and r2 on the right. If you're given a point on the circle instead of the radius, the squared distance from the center to that point is r2.
- Moving a circle changes only the center. Resizing it changes only the right side, so square the new radius.
- To test a point, find its squared distance from the center and compare it with r2.
The trap. Two classic slips. Keeping the signs from the brackets: the center of (x-1)2+(y+7)2=1 is (1, -7), not (-1, 7). And treating the right side as the radius when it's the radius squared, which also means doubling the radius multiplies the right side by 4, not 2.
2. Completing the square to find the center and radius
How to spot it. The circle's equation is expanded, with terms like x2+20x and y2+16y, and you're asked for the center, the radius, or a constant in the equation.
How to answer it:
- Group the x terms and the y terms, and move the plain number to the right side. If x2 and y2 have a coefficient like 2, divide everything by it first.
- For x2+bx, add (b/2)2 to both sides and write the x part as (x+b/2)2. Do the same for y.
- Read the center and radius from the standard form you now have.
- Shortcut if you only need the center: for x2+bx+y2+dy, the center is (-b/2, -d/2).
The trap. Adding (b/2)2 on the left but forgetting to add it on the right, so the radius comes out wrong. Every number you add to complete a square has to be added to the other side too.
3. Arcs, central angles and tangents
How to spot it. The question names an arc, a central angle, a fraction of the circle, or a line tangent to the circle. It asks for an arc length, the circumference, the radius, or a length or slope involving a tangent.
How to answer it:
- Find the fraction of the circle: the central angle over 360°, or the arc length over the circumference.
- That one fraction links everything: (central angle)/360=(arc length)/(2πr). Fill in what you know and solve for the rest.
- Use the radii. They're all equal, so a triangle with two radii as sides is isosceles.
- For a tangent, draw the radius to the point where it touches. They meet at a right angle, so you can use the Pythagorean theorem, or perpendicular slopes (negative reciprocals).
The trap. Treating an arc's angle as if it were its length, or finding the circumference when the question asks for the radius. Write down what's being asked before you start.
4. Radians and the unit circle
How to spot it. An angle is given or asked for in radians (it has π in it), or you need the sine or cosine of an angle like (565π)/6.
How to answer it:
- Use 180°=π radians. Degrees to radians: multiply by (π)/180. Radians to degrees: multiply by 180/(π).
- For the sine or cosine of a big angle, take away full turns of 2π until the angle is between 0 and 2π.
- Change the angle that's left into degrees if that helps you recognise it: (π)/6=30°, (π)/4=45°, (π)/3=60°, (π)/2=90°.
- Get the value from the special right triangles: cos 30°=(√(3))/2, cos 45°=(√(2))/2, cos 60°=1/2.
The trap. Multiplying by the wrong fraction. A quick check: degree measures are bigger numbers than radian measures, so changing degrees to radians should make the number smaller. Also, one full turn is 2π, not π.
Practise Circles
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.