Inscribed Shaped

Here is your no-BS, test-ready cheat sheet. This is exactly what you want to memorize and review the night before.

🔥 SAT Circle → Square / Triangle Cheat Sheet

🚨 STEP 0: Recognize the situation

If you see:

• messy equation like (x^2 + y^2 + ...)

• and words like circle, inscribed, square, triangle

👉 You are doing:
Complete the square → get (r^2) → plug into formula

⚡ STEP 1: Clean the equation FAST

Always do this first:

• Factor out number in front (usually 4, 9, etc.)

Example:
[
4x^2-32x+4y^2+24y-80=0
]

Divide by 4:
[
x^2-8x+y^2+6y-20=0
]

⚡ STEP 2: Complete the square

Pattern:
[
x^2 + bx → (x + \tfrac{b}{2})^2
]

Example:

[
x^2 - 8x → (x-4)^2 \quad (+16)
]
[
y^2 + 6y → (y+3)^2 \quad (+9)
]

Add both to the right side.

Final form:
[
(x-h)^2+(y-k)^2=r^2
]

👉 YOU ONLY CARE ABOUT (r^2)

⚡ STEP 3: DO NOT FIND r (unless needed)

Most of the time:
👉 You can use (r^2) directly

🔲 CASE 1: Circle inscribed in a square

Key fact:

• square side = diameter = 2r

Area:

[
A = (2r)^2 = 4r^2
]

🔥 Shortcut:

[
\boxed{\text{Square area} = 4r^2}
]

Example

[
(x-4)^2+(y+3)^2=45
]

[
r^2=45
]

[
\text{Area}=4(45)=180
]

🔺 CASE 2: Equilateral triangle INSCRIBED in circle

Key formula (MEMORIZE):

[
\boxed{A=\frac{3\sqrt3}{4}r^2}
]

Example

[
(x-2)^2+(y+1)^2=12
]

[
r^2=12
]

[
A=\frac{3\sqrt3}{4}(12)=9\sqrt3
]

🔺 CASE 3: Right triangle in a circle

Key fact:

[
\boxed{\text{Hypotenuse = diameter = 2r}}
]

Then:

• use Pythagorean theorem or area formula depending on what they give you

⚠ COMMON TRAPS (this is where people lose points)

❌ Mistake 1:

Using (2r^2) instead of (4r^2)

👉 Always square the diameter:
[
(2r)^2 = 4r^2
]

❌ Mistake 2:

Actually finding (r) when not needed

👉 Stay in (r^2) whenever possible

❌ Mistake 3:

Messing up completing the square

👉 ALWAYS:

• half the number

• square it

❌ Mistake 4:

Forgetting to divide first

👉 If you see (4x^2), divide everything by 4 immediately

🧠 ULTRA FAST SAT STRATEGY

When you finish completing the square and get:

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

Immediately ask:

👉 “What shape are they asking about?”

🔁 FULL SPEED EXAMPLE

[
9x^2-54x+9y^2+72y-144=0
]

Step 1:

Divide by 9

[
x^2-6x+y^2+8y-16=0
]

Step 2:

Complete square

[
(x-3)^2+(y+4)^2=41
]

Step 3:

[
r^2=41
]

Step 4 (square):

[
\text{Area}=4(41)=\boxed{164}
]

🎯 WHAT TO MEMORIZE (night before)

You only need this:

1.

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

2.

[
\boxed{\text{Square area} = 4r^2}
]

3.

[
\boxed{\text{Equilateral triangle area} = \frac{3\sqrt3}{4}r^2}
]

4.

[
\boxed{\text{Right triangle: hypotenuse = 2r}}
]

💥 FINAL ADVICE

• This is a guaranteed SAT question type

• You are not solving geometry — you are extracting (r^2)

• Speed comes from:

  • dividing early

  • completing square clean

  • using formulas instantly

If you want, I can give you:

• a 10-question drill set

• or a timed version to simulate SAT pressure