Digital SAT · Math · Unit 4 · Geometry & Trigonometry

Lesson 4.4 · Circles

On the test: 1–3 questions a test — most often the equation of a circle (complete the square!), then arcs and sectors, a tangent, or a circle around a square

Big idea. Everything about a circle comes from ONE number, the radius, and ONE centre. Lengths and areas are fractions of 2πr and πr². Angles are halves of arcs (on the circle, inside, or outside). A tangent meets the radius at 90°. And on the xy-plane the equation (x − h)² + (y − k)² = r² is just Pythagoras: every point is r away from the centre.

🎛 Move it · the words of a circle

Tap each word: its part lights up.

tangent: a line that touches the circle at ONE point — always at 90° to the radius there.

🧑‍🏫 Rule 1 · Circumference and areastep 1 of 4

1
the two formulas
$C = 2 π r = π d, A = π r^{2}$
Both on the reference sheet. Area needs the RADIUS.

🎛 Move it · arc and sector

Turn the central angle, change the radius: the arc and the sector are always the same fraction θ/360 of the circle.

central angle θ72°radius r5

The whole circle: C = 2πr = πd, A = πr². A central angle θ cuts off the SAME fraction θ/360 of both: arc = (θ/360)·2πr, sector = (θ/360)·πr².

In radians the fraction is θ/2π, so arc = rθ and sector = ½r²θ. Going backwards: arc ÷ circumference = angle ÷ 360 — so a 20° arc of 4 inches means the whole circle is 4 × 18 = 72.

Double the radius → circumference ×2, area ×4 (area grows with r²).

🧑‍🏫 Rule 2 · An arc or a sector is a fraction of the circlestep 1 of 5

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the fraction
$\frac{arc}{2 π r} = \frac{sector}{π r ^{2}} = \frac{θ}{360}$
One fraction, three places.

🧑‍🏫 Rule 3 · Diameters make equal (vertical) arcsstep 1 of 4

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📋 The question · SAT · May 2024 · Math 800 Book Q14
A circle has diameters $\overline{A C}$ and $\overline{B D}$ . The circumference of the circle is $84 π$ , and the length of arc $D A$ is 2 times the length of arc $A B$ . What is the length of arc $B C$ ?
1. A) $2 π$
2. B) $14 π$
3. C) $21 π$
4. D) $28 π$
half circle
$A B + D A = \frac{84 π}{2} = 42 π$
May 2024: arcs on one side of a diameter make a half circle.

🎛 Move it · the unit circle

Big angles in radians: drop whole turns (2π) until the point is easy to read.

angle θ225°extra full turns0

On a circle of radius 1, the point at angle θ IS (cos θ, sin θ): cos = how far across, sin = how far up. The blue and red legs and the radius make a right triangle with hypotenuse 1 — so sin² + cos² = 1 (Pythagoras).

Quadrant 3: Tan positive (sin, cos negative). The size comes from the reference angle 45° (the angle to the nearest x-axis); the quadrant only decides the signs.

Add full turns (2π each) — the point does not move: 5π/4 lands where 5π/4 does. To simplify a huge angle, subtract 2π until it is between 0 and 2π.

🧑‍🏫 Rule 4 · Angles on the unit circlestep 1 of 4

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📋 The question · SAT · May 2024 · Math 800 Book Q15
Point $N$ lies on a unit circle in the $x y$ -plane and has coordinates $(1, 0)$ . Point $O$ is the center of the circle and has coordinates $(0, 0)$ . Point $M$ also lies on the unit circle, and the measure of angle $N O M$ is $\frac{272 π}{3}$ radians. If the coordinates of point $M$ are $(a, b)$ , where $a$ and $b$ are constants, what is the value of $a$ ?
1. A) $\frac{1}{2}$
2. B) $\frac{3}{2}$
3. C) $- \frac{1}{2}$
4. D) $- \frac{3}{2}$
drop whole turns
$\frac{272 π}{3} = 90 π + \frac{2 π}{3}$
May 2024: $90 π$ is 45 full turns.

🎛 Move it · central and inscribed angles

Move A, B, P and Q. Then try the diameter and the cyclic quadrilateral tabs.

move A-30°move B70°move P200°move Q250°

Central angle = its arc. Inscribed angle = HALF its arc — 50 = ½ × 100.

Move P and Q anywhere on the same side: every inscribed angle on the same arc is equal. Cross over to the other side and the angle sees the other arc (now 360 − arc).

🧑‍🏫 Rule 5 · Central = arc, inscribed = half the arcstep 1 of 4

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central
$∠ A O B = arc A B$
Corner at the centre.

🧑‍🏫 Rule 6 · A diameter makes a right anglestep 1 of 4

1
why
$\frac{1}{2} \cdot 180° = 90°$
The arc of a diameter is a half circle.

🧑‍🏫 Rule 7 · Cyclic quadrilateralstep 1 of 1

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📋 The question · Gen Z Study · cyclic quadrilateral
Quadrilateral $A B C D$ is inscribed in a circle. If $∠ A = 72°$ , what is $∠ C$ , in degrees?
(type the answer)
opposite angles
$∠ A + ∠ C = 180 \Rightarrow 108$
The two opposite angles open onto arcs that make the whole 360°.

🎛 Move it · two chords crossing

Move the four points. The angle is half the two red arcs; the two products stay equal.

move point 1150°move point 2200°move point 3330°move point 440°

Two chords crossing inside: the angle = ½(sum of the two arcs it opens onto) — the arc in front of it and the arc behind its vertical partner. (A central angle is the special case where both arcs are equal.)

The pieces multiply equally: AE · EB = CE · ED. (The two triangles AEC and DEB are similar.) If one chord is a diameter at right angles to the other, it cuts that chord in half — so BD² = AD · DC.

🧑‍🏫 Rule 8 · The angle between crossing chordsstep 1 of 1

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📋 The question · Gen Z Study · crossing chords (angle)
Two chords intersect inside a circle. The angle they form opens onto an arc of $70°$ , and its vertical angle opens onto an arc of $50°$ . What is the measure of the angle, in degrees?
(type the answer)
half the sum
$\frac{70 + 50}{2} = 60$
The arc in front of the angle and the arc in front of its vertical partner.

🧑‍🏫 Rule 9 · The pieces of crossing chordsstep 1 of 6

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📋 The question · Gen Z Study · crossing chords (products)
Chords $\overline{A B}$ and $\overline{C D}$ of a circle intersect at $E$ . If $A E = 4$ , $E B = 6$ , and $C E = 3$ , what is $E D$ ?
(type the answer)
products
$A E \cdot E B = C E \cdot E D \Rightarrow 4 \cdot 6 = 3 \cdot E D$
$E D = 8$ .

🧑‍🏫 Rule 10 · The perpendicular from the centre halves a chordstep 1 of 1

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📋 The question · Gen Z Study · perpendicular bisects a chord
A chord is 5 units from the center of a circle of radius 13. What is the length of the chord?
(type the answer)
right triangle
$(\frac{c}{2})^{2} + 5^{2} = 1 3^{2} \Rightarrow \frac{c}{2} = 12$
Chord = 24. Switch the lab to “perpendicular from the centre” to see it.

🎛 Move it · lines from a point outside

Two secants, a secant and a tangent, two tangents. Move P and the lines.

P's distance11upper line12°lower line18°

Angle at a point OUTSIDE = ½(far arc − near arc). (Inside it was ½ of the SUM; on the circle it is ½ of one arc; at the centre, the whole arc.)

A tangent is perpendicular to the radius at the point it touches. Two tangents from one point are equal in length — and each is √(PO² − r²) by Pythagoras.

Outside piece × whole line is the same for every line from P: PA·PB = PC·PD, and for a tangent PT² = PC·PD. (Always PO² − r² = 96.)

🧑‍🏫 Rule 11 · A tangent is perpendicular to the radiusstep 1 of 2

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📋 The question · Gen Z Study · tangent length
Point $P$ is 13 units from the center of a circle of radius 5. A line through $P$ is tangent to the circle at $T$ . What is $P T$ ?
(type the answer)
right triangle
$P T = 1 3^{2} - 5^{2} = 12$
Radius, tangent and $P O$ make a right triangle — $P O$ is the hypotenuse.

🧑‍🏫 Rule 12 · Angles outside, and the tangent–chord anglestep 1 of 3

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the family
$centre: arc ∣ on: \frac{1}{2} arc ∣ inside: \frac{1}{2} (sum) ∣ outside: \frac{1}{2} (difference)$
Where the corner is decides the rule.

🧑‍🏫 Rule 13 · Outside piece × whole linestep 1 of 2

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the rule
$P A \cdot P B = P C \cdot P D = P T^{2}$
The same for every line from $P$ .

🎛 Move it · the circle on the grid

Move the centre and the radius; watch both forms of the equation. Test a point. Turn the tangent point round the circle.

centre h2centre k-1radius r3test point x4test point y1tangent at45°

(x − h)² + (y − k)² = r²: centre (h, k) — read the signs BACKWARDS — and the right side is r², not r. Expand it and you get the general form x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k; to go back, complete the square in x and in y.

A point: put it in the left side — equal to r²: on the circle, less: inside, more: outside. Moving the circle changes only (h, k); changing the size changes only r².

The tangent is perpendicular to the radius, so its slope is the negative reciprocal of the radius' slope. The circle touches the x-axis when |k| = r and the y-axis when |h| = r.

Rule · Why this is the equation

Every point (x, y) on the circle is r away from the centre (h, k). The distance formula — Pythagoras — says (x − h)² + (y − k)² = r².

🧑‍🏫 Rule 14 · Read the centre and the radiusstep 1 of 5

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📋 The question · SAT · March 2024 · Math 800 Book Q10
What is the center of the circle in the $x y$ -plane defined by the equation $(x - 9)^{2} + (y + 6)^{2} = 81$ ?
1. A) $(- 9, - 6)$
2. B) $(- 9, 6)$
3. C) $(9, - 6)$
4. D) $(9, 6)$
signs backwards
$(x - 9)^{2} + (y + 6)^{2} \Rightarrow (9, - 6)$
March 2024: C.

🧑‍🏫 Rule 15 · Expanded form ↔ standard formstep 1 of 6

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📋 The question · SAT · August US 2023 · Math 800 Book Q5
A circle in the $x y$ -plane has its center at $(2, - 5)$ and has a radius of 3. An equation of this circle is $x^{2} + y^{2} + a x + b y + c = 0$ , where $a$ , $b$ , and $c$ are constants. What is the value of $c$ ?
(type the answer)
expand
$(x - 2)^{2} + (y + 5)^{2} = 9$
August US 2023.

🧑‍🏫 Rule 16 · Is a point on the circle?step 1 of 6

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📋 The question · SAT · June 2023 · Math 800 Book Q3
$(x + 3)^{2} + (y - 4)^{2} = 25$ In the $x y$ -plane, the graph of the given equation is a circle. Which point lies on this circle?
1. A) $(- 3, 4)$
2. B) $(3, - 4)$
3. C) $(11 + 3, 14 - 4)$
4. D) $(11 - 3, 14 + 4)$
substitute
$(11 - 3 + 3)^{2} + (14 + 4 - 4)^{2} = 11 + 14 = 25$
June 2023: equal to $r^{2}$ → on it. D.

🧑‍🏫 Rule 17 · Moving and resizing a circlestep 1 of 6

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📋 The question · SAT · August US 2023 · Math 800 Book Q4
Circle A has equation $(x - 3)^{2} + (y + 1)^{2} = 16$ . In the $x y$ -plane, circle B is obtained by translating circle A to the right 2 units. Which equation represents circle B?
1. A) $(x - 5)^{2} + (y + 1)^{2} = 16$
2. B) $(x - 3)^{2} + (y - 1)^{2} = 16$
3. C) $(x - 3)^{2} + (y + 3)^{2} = 16$
4. D) $(x - 1)^{2} + (y + 1)^{2} = 16$
move right 2
$(3, - 1) \to (5, - 1)$
August US 2023: only the centre changes. A.

🧑‍🏫 Rule 18 · The centre is the midpoint of a diameterstep 1 of 2

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📋 The question · SAT · August US 2024 · Math 800 Book Q22
A circle in the $x y$ -plane has a diameter with endpoints $(a, 11)$ and $(a, d)$ , where $a$ and $d$ are constants. An equation of this circle is $(x - a)^{2} + (y - 17)^{2} = r^{2}$ , where $r$ is a positive constant. What is the value of $d$ ?
1. A)5
2. B)12
3. C)17
4. D)23
midpoint
$\frac{11 + d}{2} = 17 \Rightarrow d = 23$
August US 2024: D.

🧑‍🏫 Rule 19 · Tangent slope, and touching an axisstep 1 of 6

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📋 The question · SAT · November 2023 · Math 800 Book Q8
A circle in the $x y$ -plane has its center at $(- 3, - 8)$ . Line $k$ is tangent to this circle at the point $(- 5, - 9)$ . What is the slope of line $k$ ?
1. A) $- 2$
2. B) $- \frac{1}{2}$
3. C) $\frac{1}{2}$
4. D)2
radius slope
$\frac{- 9 - ( - 8 )}{- 5 - ( - 3 )} = \frac{1}{2}$
November 2023: centre to the point of tangency.

🎛 Move it · squares and circles

Switch between the circle inside the square and the square inside the circle.

radius r9

A circle inscribed in a square touches all four sides: side = diameter = 2r, perimeter 8r. (So: complete the square → r → perimeter 8r.)

🧑‍🏫 Rule 20 · Circle in a square: side = diameterstep 1 of 5

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📋 The question · SAT · March US 2023 · Math 800 Book Q2
$x^{2} - 10 x + y^{2} - 6 y - 47 = 0$ In the $x y$ -plane, the graph of the given equation is a circle. If this circle is inscribed in a square, what is the perimeter of the square?
1. A)18
2. B)36
3. C)72
4. D)188
r → side
$r = 9 \Rightarrow s = 18 \Rightarrow P = 72$
March US 2023: C.

🧑‍🏫 Rule 21 · Square in a circle: diagonal = diameterstep 1 of 2

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📋 The question · College Board · Practice Test 9 · Module 2 Q24
A square is inscribed in a circle. The radius of the circle is $\frac{20 2}{2}$ inches. What is the side length, in inches, of the square?
1. A)20
2. B) $\frac{20 2}{2}$
3. C) $202$
4. D)40
side
$d = 202 \Rightarrow s = \frac{20 2}{2} = 20$
Practice Test 9: A.

🧑‍🏫 Rule 22 · Triangles in circles, and scalingstep 1 of 4

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📋 The question · SAT · August US 2023 · Math 800 Book Q6
$figure from the test$
In the figure, $\overline{A C}$ is a diameter of the circle with center $O$ , and $A C = 10$ . Triangle $A C B$ is equilateral. The length of $\overline{B P}$ is $n$ times the length of $\overline{A O}$ . What is the value of $n$ ?
1. A) $5$
2. B) $3$
3. C) $3 - 1$
4. D) $5 - 1$
the height
$B O = \frac{3}{2} \cdot 10 = 53$
August US 2023: $B O$ is the height of the equilateral triangle.

Real question 1SAT · December 2024 · Math 800 Book Q34type the answer

$figure from the test$

In the figure shown, points $A$ , $B$ , $C$ , and $E$ lie on the circle, and $A B < B C$ . Segment $A C$ is perpendicular to segment $B E$ at point $D$ , and $B D = 346$ . The diameter of the circle is 175. If $\frac{C D}{A D} = r$ , what is the value of $r$ ?

Take with you

▸C = 2πr = πd, A = πr². Arc and sector are the fraction θ/360 of them; in radians arc = rθ, sector = ½r²θ. Vertical central angles cut equal arcs.
▸Central angle = arc. Inscribed angle = ½ arc; same arc → equal; a diameter → 90°. Cyclic quadrilateral: opposite angles add to 180°.
▸Crossing chords: angle = ½(sum of arcs), AE·EB = CE·ED. The perpendicular from the centre halves a chord: (½c)² + d² = r².
▸Tangent ⟂ radius; two tangents from a point are equal; outside angle = ½(far − near); tangent–chord = ½ arc; PA·PB = PT².
▸(x − h)² + (y − k)² = r²: signs backwards, r² on the right. Complete the square to read it. Point in: equal → on, less → inside. Diameter endpoints → midpoint is the centre. Tangent slope = −1 / radius slope.
▸Circle in a square: side = 2r. Square in a circle: diagonal = 2r. Scale lengths by k → areas by k².

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