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Educerie · SAT · Math

Geometry and Trigonometry · GT.4 Circles

Where it is examined
one question on most forms, and it is usually either the circle equation or an arc-and-sector calculation.
The question this unit answers
the equation has two squared brackets, or the diagram has a slice cut out of a circle. What is being asked, and which fraction do you need?
Before you start
completing the square, from Advanced Math — it is how a circle equation in expanded form is turned back into something readable.

What you must be able to do

You must be able toWhat it looks like on the test
Read centre and radius from an equation(x − h)² + (y − k)² = r²
Complete the square to get thereWhen the equation arrives expanded
Find arc length and sector areaA fraction of the whole circle
Use central and inscribed anglesThe inscribed angle is half the central one
Work in radiansArc = rθ, sector = ½r²θ

1The idea in one paragraph

A circle is a centre and a radius, and everything in this unit is either extracting those two things from an equation or taking a fraction of the whole circle. The fraction is always the angle over 360° — or over 2π in radians — and it applies to the circumference for an arc and to the area for a sector. There is nothing else to learn here.

In (x − h)² + (y − k)² = r², the centre is (h, k) with the signs flipped, and the right-hand side is r squared, not r.

2The equation of a circle

(x − 3)² + (y + 2)² = 25 has centre (3, −2) and radius 5.

Two sign traps, both of which the test uses:

  • The centre coordinates are the opposite of the numbers written: (x + 2) means the centre is at −2.
  • The right side is r², so a 25 means a radius of 5 — not 25 and not 12.5.

3When the equation is expanded

x2 + y2 − 6x + 4y − 12 = 0
x2 − 6x → (x − 3)2 − 9halve −6, square it
y2 + 4y → (y + 2)2 − 4halve 4, square it
(x − 3)2 + (y + 2)2 = 12 + 9 + 4
(x − 3)2 + (y + 2)2 = 25
centre (3, −2), radius 5

Halve the coefficient, square it, add and subtract — the same procedure as in Advanced Math, done twice.

4Arcs and sectors

For a central angle of θ degrees:

  • Arc length = (θ/360) × 2πr
  • Sector area = (θ/360) × πr²
60/360 = 1/6the fraction of the whole circle
arc = (1/6)(2π × 12) = 4π
sector = (1/6)(π × 122) = 24π

In radians the same facts are shorter: arc = rθ and sector area = ½r²θ. Radian questions on this test usually want these forms.

5Central and inscribed angles

A central angle has its vertex at the centre; an inscribed angle has its vertex on the circle. An inscribed angle is half the central angle standing on the same arc.

The most useful consequence: an angle inscribed in a semicircle is 90°, because the central angle is a straight 180°.

6Tangents and chords

A tangent touches at one point and is perpendicular to the radius at that point, which turns many circle questions into right-triangle questions — and right triangles you already know how to handle.

Where points are lost

  • Reading the centre with the wrong signs.
  • Taking the right-hand side as the radius rather than its square.
  • Forgetting to add back what completing the square removed.
  • Using the diameter in an arc or sector formula.
  • Mixing degrees and radians in the same calculation.
  • Halving the wrong angle between central and inscribed.

Work it right

  1. Get the equation into (x − h)² + (y − k)² = r² form, completing the square if needed.
  2. Read the centre with flipped signs, and take the square root for the radius.
  3. For an arc or sector, write the fraction θ/360 first.
  4. Choose circumference for an arc, area for a sector.
  5. Check whether the angle is in degrees or radians before substituting.

Try it

Q1. A circle has equation (x + 4)² + (y − 1)² = 36. What are its centre and radius?

A) (4, −1), r = 6 B) (−4, 1), r = 6 C) (−4, 1), r = 36 D) (4, −1), r = 36

Q2. A circle has equation x² + y² − 10x + 6y + 18 = 0. What is its radius? (Type your answer.)

Q3. A circle has radius 9. What is the length of an arc subtended by a central angle of 40°?

A) 2π B) 4π C) 6π D) 9π

Q4. A sector of a circle of radius 6 has a central angle of 120°. What is its area?

A) 6π B) 12π C) 18π D) 36π

Q5. An inscribed angle in a circle measures 35°. What is the measure of the central angle standing on the same arc, in degrees? (Type your answer.)

In one breath

A circle is a centre and a radius, and the equation gives you both with the signs flipped and the right-hand side squared. When it arrives expanded, complete the square on each variable and remember to add back what you removed. Everything else is a fraction of the whole circle — the angle over 360°, applied to the circumference for an arc and to the area for a sector — and an inscribed angle is always half the central angle on the same arc.

Answers

Q1. B — (−4, 1), r = 6. flip the signs, square-root the right side A keeps the signs as written. C and D report 36 as the radius.

Q2. 4. complete the square twice

x2 + y2 − 10x + 6y + 18 = 0
(x − 5)2 − 25 + (y + 3)2 − 9 + 18 = 0
(x − 5)2 + (y + 3)2 = 16
r = 4

Answering 16 stops one step early: the right-hand side is r², not r.

Q3. A — 2π. the fraction is 40/360 = 1/9

circumference = 2π(9) = 18π
arc = (40/360)(18π)
arc = (1/9)(18π) = 2π

B and C use the wrong fraction.

Q4. B — 12π. a third of the area

area = π(62) = 36π
sector = (120/360)(36π)
sector = (1/3)(36π) = 12π

A uses the arc formula; D is the whole circle.

Q5. 70. the central angle is twice the inscribed angle 2 × 35 = 70. Answering 17.5 halves it the wrong way.

Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Geometry and Trigonometry, skill/knowledge testing point "Circles"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.

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