Circles

⭕ 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:

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

Reading Center & Radius — Worked Examples

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

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

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 $x^2 + y^2 - 6x + 4y - 12 = 0$.

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

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

Complete the square practice. 🧮

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

1. What is the $x$-coordinate of the center?

2. What is the $y$-coordinate of the center?

3. What is the radius?

Match each equation to its center. 🔍

SAT-Style Questions 📋

📐 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 $r$:

Degree ↔ Radian Conversion:

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 $45°$.

1. Convert: $45° = 45 \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.
2. 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.

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). 🧮

1. Convert $270°$ to radians. Enter the numerator when written as $\frac{?\pi}{2}$.

2. A circle has radius $4$ and arc length $6\pi$. What is the central angle in degrees?

3. 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 📋

📏 Right Triangle Trigonometry

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

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

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 $60°$) $= x\sqrt{3}$
• Hypotenuse $= 2x$

Worked Examples

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

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

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?

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

1. In a right triangle, the legs are $9$ and $12$. What is the hypotenuse?

2. $\sin 30° + \cos 60° = ?$ (enter a decimal or fraction)

3. A $45\text{-}45\text{-}90$ triangle has hypotenuse $10$. What is the length of each leg? Round to the nearest hundredth.

Match each trig expression to its value. 🔍

SAT-Style Questions 📋

🔵 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 $x$-coordinate
• $\sin \theta =$ the $y$-coordinate

Key angles and their coordinates:

Reference Angles

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

Worked Example: Find $\cos 150°$.

1. $150°$ is in Quadrant II; reference angle $= 180° - 150° = 30°$.
2. $\cos 30° = \frac{\sqrt{3}}{2}$.
3. Cosine is negative in QII: $\cos 150° = -\frac{\sqrt{3}}{2}$.

Sign Patterns — "All Students Take Calculus"

A handy mnemonic for which trig functions are positive:

Example: $\sin 210°$

1. QIII; reference angle $= 210° - 180° = 30°$.
2. $\sin 30° = \frac{1}{2}$.
3. Sine is negative in QIII: $\sin 210° = -\frac{1}{2}$.

Practice — Unit Circle 🔍

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

1. $\cos 0° = ?$

2. $\sin 270° = ?$

3. What is the reference angle for $315°$ (in degrees)?

Match each angle to its sine value. 🔍

SAT-Style Questions 📋

🎯 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:

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

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 $\cos R$?

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 $\sin \theta$.

$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}$ $\sin \theta = \frac{4}{5} \quad (\text{positive in QI})$

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

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

The numerator is exactly $1$ by the identity, so the whole expression collapses.

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. 🧮

1. If $\sin(2x)° = \cos(x + 15)°$ and both angles are acute, what is $x$?

2. If $\cos \theta = 0.8$, what is $\sin^2 \theta$? (Enter as a decimal.)

3. $\sin^2 40° + \cos^2 40° = ?$

Match each expression to its equivalent. 🔍

SAT-Style Questions 📋

🔶 Circle Theorems & Tangent Lines

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

Key circle theorems tested on the SAT:

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 $PT$.

1. $OT \perp PT$ (tangent-radius), so triangle $OTP$ is a right triangle.
2. $OT = 5$ (radius), $OP = 13$ (given).
3. By the Pythagorean theorem: $PT = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.

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. 🧮

1. A central angle measures $110°$. What is the minor arc it intercepts (in degrees)?

2. An inscribed angle measures $55°$. What is the intercepted arc (in degrees)?

3. 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 📋

🏁 Review & Mixed Practice

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

Formula Cheat Sheet

Special triangles:

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. 🧮

1. 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 25).

2. A tangent to a circle of radius $7$ is drawn from a point $25$ units from the center. How long is the tangent segment?

3. $\cos^2 72° + \sin^2 72° = ?$

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! 🚀