🎯⭐ INTERACTIVE LESSON

Circles

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Circles - Complete Interactive Lesson

Part 1: Circle Equations

⭕ Circle Equations

Part 1 of 7 — Standard Form, Center, Radius & Completing the Square

Circle equations appear on almost every SAT. The standard form is the key:

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

Component	Meaning
$(h, k)$	Center of the circle
$r$	Radius
$r^2$	The number on the right side

Important: The signs in $(x - h)$ and $(y - k)$ are subtractions, so if the equation has $(x + 3)^2$ , that means .

Reading Center & Radius — Worked Examples

Example 1: Find the center and radius of $(x - 4)^2 + (y + 1)^2 = 25$ .

Practice — Reading Circle Equations 🔍

Completing the Square — Key SAT Skill

Many SAT problems give the circle in general form and ask you to find the center or radius. You must complete the square.

Example: Find the center and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$ .

Complete the square practice. 🧮

Consider the circle $x^2 + y^2 + 8x - 2y - 8 = 0$ .

Match each equation to its center. 🔍

SAT-Style Questions 📋

Part 2: Arc Length & Sectors

📐 Arc Length & Sector Area

Part 2 of 7 — Formulas, Degree ↔ Radian Conversion

When a central angle $\theta$ (in radians) intercepts an arc of a circle with radius $r$ :

Quantity	Formula (radians)	Formula (degrees)
Arc length	$s = r\theta$

Part 3: Right Triangle Trig

📏 Right Triangle Trigonometry

Part 3 of 7 — SOH-CAH-TOA, Finding Sides & Angles, Special Triangles

For a right triangle with an acute angle $\theta$ :

Ratio	Formula	Mnemonic
$\sin \theta$	$\frac{\text{opposite}}{\text{hypotenuse}}$

Part 4: Unit Circle Basics

🔵 Unit Circle Basics

Part 4 of 7 — Radian Measure, Coordinates at Key Angles, Reference Angles

The unit circle is a circle with radius $1$ centered at the origin. Any point on the unit circle has coordinates:

$(\cos \theta, \sin \theta)$

This means:

$\cos \theta =$ the -coordinate

Part 5: Trig Applications

🎯 Trig on the SAT

Part 5 of 7 — Complementary Angles, Pythagorean Identity & SAT Favorites

Two high-frequency SAT trig concepts:

1. Complementary Angle Relationship:

If $A + B = 90°$ , then:

$\sin A = \cos B \qquad \text{and} \qquad \cos A = \sin B$

Part 6: Problem-Solving Workshop

🔶 Circle Theorems & Tangent Lines

Part 6 of 7 — Inscribed Angles, Central Angles, Tangent-Radius Perpendicularity

Key circle theorems tested on the SAT:

Theorem	Statement
Central angle	Central angle $=$ intercepted arc
Inscribed angle	Inscribed angle $= \frac{1}{2}$ intercepted arc

Part 7: Review & Applications

🏁 Review & Mixed Practice

Part 7 of 7 — Formula Cheat Sheet, Mixed Problems & Exam Strategies

Formula Cheat Sheet

Topic	Formula
Circle equation	$(x-h)^2 + (y-k)^2 = r^2$

Compare to $(x - h)^2 + (y - k)^2 = r^2$ .
$h = 4$ , $k = -1$ (note the $+1$ means $k = -1$ ).
$r^2 = 25$ , so $r = 5$ .
Center: $(4, -1)$ , Radius: $5$ .

Example 2: Write the equation of a circle with center $(-2, 7)$ and radius $3$ .

$(x - (-2))^2 + (y - 7)^2 = 3^2$ $(x + 2)^2 + (y - 7)^2 = 9$

SAT Tip: If you see $r^2 = 36$ , the radius is $6$ , not $36$ . The SAT loves including $r^2$ as a wrong answer for the radius.

Group $x$ and $y$ terms: $(x^2 - 6x) + (y^2 + 4y) = 12$ .
Complete the square for $x$ : half of $-6$ is $-3$ ; $(-3)^2 = 9$ .
Complete the square for $y$ : half of $4$ is $2$ ; $(2)^2 = 4$ .
Add both to each side: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$ .
Factor: $(x - 3)^2 + (y + 2)^2 = 25$ .
Center: $(3, -2)$ , Radius: $5$ .

The pattern: For $x^2 + bx$ , add $\left(\frac{b}{2}\right)^2$ to both sides.

What is the $x$ -coordinate of the center?
What is the $y$ -coordinate of the center?
What is the radius?

s = \frac{θ}{360} \cdot 2 π r

Degree ↔ Radian Conversion:

\text{radians} = \text{degrees} \times \frac{\pi}{180} \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}

Degrees	Radians
$30°$	$\frac{\pi}{6}$
$45°$	$\frac{\pi}{4}$
$60°$	$\frac{\pi}{3}$
$90°$	$\frac{\pi}{2}$
$180°$	$\pi$
$360°$	$2\pi$

Worked Examples

Example 1 — Arc Length: A circle has radius $10$ cm. Find the length of the arc intercepted by a central angle of $\frac{\pi}{3}$ radians.

$s = r\theta = 10 \cdot \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47 \text{ cm}$

Example 2 — Sector Area: A pizza slice has radius $8$ inches and a central angle of $45°$ .

Convert: $45° = 45 \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.

Example 3 — Degree Method: A sector has radius $6$ and central angle $120°$ . $A = \frac{120}{360} \cdot \pi(6)^2 = \frac{1}{3} \cdot 36\pi = 12\pi$

SAT Tip: Always check whether the angle is given in degrees or radians. Using the wrong formula is the #1 mistake on these problems.

Practice — Arc Length & Sector Area 🔍

Conversion Practice — Degrees ↔ Radians

Example: Convert $150°$ to radians.

$150 \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$

Example: Convert $\frac{3\pi}{4}$ radians to degrees.

$\frac{3\pi}{4} \times \frac{180}{\pi} = \frac{3 \times 180}{4} = 135°$

Shortcut: To convert degrees → radians, divide by 180 and multiply by $\pi$ . To go back, divide by $\pi$ and multiply by 180.

Compute each value (give exact answers as integers or simplified fractions). 🧮

Convert $270°$ to radians. Enter the numerator when written as $\frac{?\pi}{2}$ .
A circle has radius $4$ and arc length $6\pi$ . What is the central angle in degrees?
The area of a full circle with radius $10$ is $100\pi$ . What fraction of the area is a sector with central angle $72°$ ? Enter as a fraction (e.g. 1/5).

Match each degree measure to its radian equivalent. 🔍

SAT-Style Questions 📋

Also: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

SAT Tip: The SAT provides the reference formulas for special triangles on the formula sheet, but memorizing them saves precious time.

Special Right Triangles

45-45-90 Triangle:

$\text{sides} = 1 : 1 : \sqrt{2}$

If a leg is $a$ , the hypotenuse is $a\sqrt{2}$ .

$\sin 45° = \cos 45° = \frac{\sqrt{2}}{2} \qquad \tan 45° = 1$

30-60-90 Triangle:

$\text{sides} = 1 : \sqrt{3} : 2$

Short leg (opposite $30°$ ) $= x$
Long leg (opposite $60°$ ) $= x\sqrt{3}$

Angle	$\sin$	$\cos$	$\tan$
$30°$	$\frac{1}{2}$

Worked Examples

Example 1: In a right triangle, $\sin A = \frac{5}{13}$ . Find $\cos A$ .

Opposite $= 5$ , hypotenuse $= 13$ .
Adjacent $= \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ .
$\cos A = \frac{12}{13}$ .

Example 2: A ladder leans against a wall, making a $60°$ angle with the ground. If the foot of the ladder is $4$ meters from the wall, how long is the ladder?

The side adjacent to $60°$ is $4$ m, and we want the hypotenuse.
$\cos 60° = \frac{4}{\text{hyp}} \Rightarrow \frac{1}{2} = \frac{4}{\text{hyp}} \Rightarrow \text{hyp} = 8$ m.

Example 3: A square has side length $6$ . What is the length of its diagonal?

The diagonal creates a $45\text{-}45\text{-}90$ triangle: $d = 6\sqrt{2}$ .

Practice — Right Triangle Trig 🔍

Compute each value. 🧮

In a right triangle, the legs are $9$ and $12$ . What is the hypotenuse?
$\sin 30° + \cos 60° = ?$ (enter a decimal or fraction)
A $45\text{-}45\text{-}90$ triangle has hypotenuse $10$ . What is the length of each leg? Give the exact decimal (rounded to 2 places).

Match each trig expression to its value. 🔍

SAT-Style Questions 📋

\sin \theta =

the

y

-coordinate

Key angles and their coordinates:

$\theta$ (rad)	$\theta$ (deg)	$(\cos \theta,\; \sin \theta)$
$0$	$0°$	$(1, 0)$
$\frac{\pi}{6}$	$30°$	$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$\frac{\pi}{4}$	$45°$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
$\frac{\pi}{3}$	$60°$	$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
$\frac{\pi}{2}$	$90°$	$(0, 1)$

Reference Angles

A reference angle is the acute angle formed between the terminal side and the $x$ -axis.

Quadrant	Reference angle formula	$\sin$	$\cos$
I ( $0°$ – $90°$ )	$\theta_{\text{ref}} = \theta$	$+$	$+$
II ( $90°$ – $180°$ )	$\theta_{\text{ref}} = 180° - \theta$
III ( $180°$ – $270°$ )	$\theta_{\text{ref}} = \theta - 180°$
IV ( $270°$ – $360°$ )	$\theta_{\text{ref}} = 360° - \theta$

Worked Example: Find $\cos 150°$ .

$150°$ is in Quadrant II; reference angle $= 180° - 150° = 30°$ .
$\cos 30° = \frac{\sqrt{3}}{2}$ .

Sign Patterns — "All Students Take Calculus"

A handy mnemonic for which trig functions are positive:

Quadrant	Positive functions	Mnemonic
I	All ( $\sin$ , $\cos$ , $\tan$ )	All
II	$\sin$ only	Students
III	$\tan$ only	Take
IV	$\cos$ only	Calculus

Example: $\sin 210°$

QIII; reference angle $= 210° - 180° = 30°$ .
$\sin 30° = \frac{1}{2}$ .

Practice — Unit Circle 🔍

Find each value. Enter as a simplified fraction or integer. 🧮

$\cos 0° = ?$
$\sin 270° = ?$
What is the reference angle for $315°$ (in degrees)?

Match each angle to its sine value. 🔍

SAT-Style Questions 📋

This comes from the fact that in a right triangle, the two acute angles add to $90°$ , and one angle's opposite side is the other angle's adjacent side.

2. Pythagorean Identity:

$\sin^2 \theta + \cos^2 \theta = 1$

This is always true — for every angle. Two useful rearrangements:

$\sin^2 \theta = 1 - \cos^2 \theta \qquad \cos^2 \theta = 1 - \sin^2 \theta$

Complementary Angles — Worked Examples

Example 1: If $\sin 32° = 0.53$ , what is $\cos 58°$ ?

Since $32° + 58° = 90°$ , they are complementary.

$\cos 58° = \sin 32° = 0.53$

Example 2: In right triangle $PQR$ with a right angle at $Q$ , $\sin P = \frac{4}{5}$ . What is ?

Angles $P$ and $R$ are complementary ( $P + R = 90°$ ), so: $\cos R = \sin P = \frac{4}{5}$ .

SAT Tip: If a question says " $\sin x° = \cos y°$ ," immediately conclude $x + y = 90$ (assuming $0 < x, y < 90$ ).

Pythagorean Identity — Worked Examples

Example 1: If $\cos \theta = \frac{3}{5}$ and $\theta$ is in Quadrant I, find $\sin \theta$ .

$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}$

Example 2: Simplify $\frac{\sin^2 x + \cos^2 x}{\cos x}$ .

$= \frac{1}{\cos x} = \sec x$

Example 3: If $\sin \theta = 0.6$ , what is $\sin^2 \theta + \cos^2 \theta$ ?

$= 1 \quad \text{(always, by identity)}$

The SAT might try to distract you — don't compute; just recognize the identity.

Practice — SAT Trig Concepts 🔍

Compute each value. 🧮

If $\sin(2x)° = \cos(x + 15)°$ and both angles are acute, what is $x$ ?
If $\cos \theta = 0.8$ , what is $\sin^2 \theta$ ? (Enter as a decimal.)
$\sin^2 40° + \cos^2 40° = ?$

Match each expression to its equivalent. 🔍

SAT-Style Questions 📋

Central Angle vs. Inscribed Angle:

If a central angle and an inscribed angle intercept the same arc, the central angle is twice the inscribed angle.

$\text{Central angle} = 2 \times \text{Inscribed angle}$

Inscribed Angles — Worked Examples

Example 1: A central angle measures $80°$ . An inscribed angle intercepts the same arc. What is the inscribed angle?

$\text{Inscribed} = \frac{80°}{2} = 40°$

Example 2: An inscribed angle in a semicircle?

Any angle inscribed in a semicircle intercepts a $180°$ arc.

$\text{Inscribed angle} = \frac{180°}{2} = 90°$

This is Thales' theorem: an angle inscribed in a semicircle is always a right angle.

Example 3: Two inscribed angles intercept the same arc. What can you say?

They are equal — inscribed angles that intercept the same arc are congruent.

SAT Tip: If you see a triangle inscribed in a circle with one side as a diameter, immediately mark a $90°$ angle.

Tangent Lines — Key Properties

A tangent touches the circle at exactly one point and is perpendicular to the radius at that point.

Worked Example: Point $P$ is outside a circle with center $O$ and radius $5$ . A tangent from $P$ touches the circle at $T$ , and $OP = 13$ . Find $PT$ .

$OT \perp PT$ (tangent-radius), so triangle $OTP$ is a right triangle.
$OT = 5$ (radius), $OP = 13$ (given).

Two tangents from one point: If two tangent segments are drawn from the same external point, they have equal length.

So if $PA$ and $PB$ are tangent to a circle at $A$ and $B$ , then $PA = PB$ .

Practice — Circle Theorems 🔍

Solve each problem. 🧮

A central angle measures $110°$ . What is the minor arc it intercepts (in degrees)?
An inscribed angle measures $55°$ . What is the intercepted arc (in degrees)?
Triangle $ABC$ is inscribed in a circle with $BC$ as a diameter. What is angle $A$ (in degrees)?

Match each scenario to the correct conclusion. 🔍

SAT-Style Questions 📋

Special triangles:

$45\text{-}45\text{-}90$	$1 : 1 : \sqrt{2}$
$30\text{-}60\text{-}90$	$1 : \sqrt{3} : 2$

SAT Exam Strategies for Circles & Trig

1. Read the question twice. Common traps:

Asking for $r^2$ vs. $r$
Giving degrees when you need radians (or vice-versa)
Asking for an angle when you solve for a side

2. Draw and label. Sketch the circle or triangle directly on the test — label known sides, angles, and what you're solving for.

3. Look for right triangles. Tangent-radius? That's a right angle. Diameter as a chord? Thales gives you $90°$ .

4. Use process of elimination. Trig ratios are between $-1$ and $1$ (for $\sin$ and $\cos$ ). If an answer is $> 1$ , it can't be a sine or cosine value.

5. Pythagorean triples save time: $3\text{-}4\text{-}5$ , $5\text{-}12\text{-}13$ , $8\text{-}15\text{-}17$ , $7\text{-}24\text{-}25$ and their multiples.

Mixed Practice Set 1 🔍

Mixed Practice Set 2 🔍

Final mixed problems — enter your answers. 🧮

What is the area of a circle with equation $(x+1)^2 + (y-3)^2 = 49$ ? Enter in terms of $\pi$ — give just the coefficient (e.g., if the answer is $25\pi$ , enter 49).
A tangent to a circle of radius $7$ is drawn from a point $25$ units from the center. How long is the tangent segment?
$\cos^2 72° + \sin^2 72° = ?$

Quick-fire review — choose the correct value. 🔍

📋 What You've Mastered

Across all 7 parts you've covered:

✅ Circle equations — standard form, completing the square, center & radius
✅ Arc length & sector area with degree and radian formulas
✅ Right triangle trig — SOH-CAH-TOA and special triangles
✅ The unit circle — coordinates, reference angles, sign patterns
✅ SAT trig favorites — complementary angles and Pythagorean identity
✅ Circle theorems — inscribed angles, central angles, tangent-radius
✅ Mixed practice and exam strategies

Next steps:

Time yourself: aim for each circle/trig problem in under 90 seconds.
On test day, sketch diagrams — even rough drawings reveal right triangles and relationships.
If stuck, try plugging in the answer choices. For trig questions, use special values ( $30°$ , $45°$ , $60°$ ) to test.

Good luck on the SAT! 🚀

Area:

A = \frac{1}{2}(8)^2 \cdot \frac{\pi}{4} = \frac{1}{2}(64)\cdot \frac{\pi}{4} = 8\pi \approx 25.13

sq in.

Cosine is negative in QII:

\cos 150° = -\frac{\sqrt{3}}{2}

Sine is negative in QIII:

\sin 210° = -\frac{1}{2}

\sin \theta = \frac{4}{5} \quad (\text{positive in QI})

By the Pythagorean theorem:

PT = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12