SAT Math ยท Self-Study Workbook
Geometry
Master Notes
10 Core Topics ยท 10 Key Problems

โœ๏ธ Write your name: ___________________________
๐Ÿ“… Date started: ___________________________

โš ๏ธ Don't peek at answers!
TOPIC 01 ๐Ÿ“ Angles & Lines

Vertical angles are EQUAL. Supplementary angles add up to 180ยฐ. Complementary angles add up to 90ยฐ.

๐Ÿ”‘ KEY WORDS: Vertical = Equal / Supplement = 180 / Complement = 90
Two lines intersect. One angle is 3x + 10. Its vertical angle is 5x โˆ’ 20. Find x.
๐Ÿ’ก Vertical angles are equal!
\(3x + 10 = 5x - 20\)
\(30 = 2x\)
\(x = 15\)
Q1
EASY
In the figure, two straight lines intersect. One angle measures \((4x + 5)ยฐ\) and the angle adjacent to it (supplementary) measures \((2x + 15)ยฐ\). What is the value of x? โš ๏ธ Tricky: adjacent โ‰  vertical!
(A) 20
(B) 25
(C) 26.67
(D) 30
My Work & Answer
๐Ÿชค TRAP: Don't set them equal โ€” they're supplementary!
TOPIC 02 ๐Ÿ”บ Triangle Angle Sum

The three angles of ANY triangle always sum to 180ยฐ. An exterior angle equals the sum of the two non-adjacent interior angles.

๐Ÿ”‘ KEY WORDS: Interior Sum = 180 / Exterior = Remote Interior ร— 2
A triangle has angles \(x\), \(2x\), and \(3x\). Find each angle.
\(x + 2x + 3x = 180\)
\(6x = 180 \Rightarrow x = 30\)
Angles: 30ยฐ, 60ยฐ, 90ยฐ โ€” it's a right triangle!
Q2
EASY
In triangle ABC, \(\angle A = 2x\), \(\angle B = x + 30\), and \(\angle C = 3x - 10\). An exterior angle at C is formed by extending side BC. What is the measure of that exterior angle? โš ๏ธ Tricky: find interior first!
(A) 110ยฐ
(B) 120ยฐ
(C) 130ยฐ
(D) 140ยฐ
My Work & Answer
๐Ÿชค TRAP: Exterior angle = 180ยฐ โˆ’ interior angle at C (not 180ยฐ โˆ’ 90ยฐ)
TOPIC 03 ๐Ÿ“ Pythagorean Theorem

In a right triangle: \(a^2 + b^2 = c^2\) where c is the hypotenuse (longest side, opposite the right angle).

๐Ÿ”‘ MEMORIZE TRIPLES: 3-4-5 / 5-12-13 / 8-15-17 / 7-24-25
A right triangle has legs 6 and 8. Find the hypotenuse.
\(6^2 + 8^2 = c^2\)
\(36 + 64 = 100\)
\(c = \sqrt{100} = 10\) โœ” (Recognize: 6-8-10 is a 3-4-5 triple ร— 2!)
Q3
EASY
A rectangle has width 5 and length 12. What is the length of its diagonal? โš ๏ธ Tricky: diagonal splits into right triangles!
(A) 11
(B) 13
(C) 14
(D) 17
My Work & Answer
๐Ÿชค TRAP: 5 + 12 = 17 is WRONG. You need \(\sqrt{5^2+12^2}\)
TOPIC 04 โœจ Special Right Triangles

45-45-90: sides in ratio 1 : 1 : โˆš2    30-60-90: sides in ratio 1 : โˆš3 : 2

๐Ÿ”‘ 45-45-90 โ†’ multiply leg ร— โˆš2 to get hypotenuse
๐Ÿ”‘ 30-60-90 โ†’ short leg ร— 2 = hypotenuse; short leg ร— โˆš3 = long leg
Quick Formula Reference
\[ \underbrace{45\text{-}45\text{-}90}_{\text{isoceles right}} \quad \text{legs} = x, \quad \text{hyp} = x\sqrt{2} \] \[ \underbrace{30\text{-}60\text{-}90}_{\text{half equilateral}} \quad \text{short}=x, \quad \text{long}=x\sqrt{3}, \quad \text{hyp}=2x \]
Q4
EASY
In a 30-60-90 triangle, the hypotenuse is 10. What is the length of the longer leg? โš ๏ธ Tricky: which is the "long leg"?
(A) \(5\)
(B) \(5\sqrt{2}\)
(C) \(5\sqrt{3}\)
(D) \(10\sqrt{3}\)
My Work & Answer
๐Ÿชค TRAP: hyp = 2x โ†’ x = 5. Long leg = 5โˆš3, NOT 5โˆš2!
TOPIC 05 ๐Ÿ”ต Circles โ€” Arc, Chord, Sector

Central angle = Arc measure. Inscribed angle = ยฝ ร— Arc measure. Arc length = \(\dfrac{\theta}{360} \times 2\pi r\). Sector area = \(\dfrac{\theta}{360} \times \pi r^2\).

๐Ÿ”‘ INSCRIBED ANGLE = HALF the intercepted arc
๐Ÿ”‘ Arc Length โ†’ think "fraction of circumference"
๐Ÿ”‘ Sector Area โ†’ think "fraction of total area"
A circle has radius 6. A sector has a central angle of 90ยฐ. Find its area.
\(\text{Sector Area} = \dfrac{90}{360} \times \pi(6)^2 = \dfrac{1}{4} \times 36\pi = 9\pi\)
Q5
EASY
An inscribed angle in a circle intercepts an arc of 110ยฐ. A central angle intercepts the same arc. What is the difference between the central angle and the inscribed angle? โš ๏ธ Tricky: central angle โ‰  inscribed angle!
(A) 55ยฐ
(B) 65ยฐ
(C) 70ยฐ
(D) 110ยฐ
My Work & Answer
๐Ÿชค TRAP: Inscribed = 110/2 = 55ยฐ. Central = 110ยฐ. Difference = 55ยฐ.
TOPIC 06 ๐Ÿ“ฆ Area & Perimeter of Polygons

Triangle area: \(\dfrac{1}{2}bh\)  ยท  Trapezoid: \(\dfrac{1}{2}(b_1+b_2)h\)  ยท  Regular polygon: use apothem!

๐Ÿ”‘ TRAP(ezoid) has TWO bases โ†’ average them, then ร— height
๐Ÿ”‘ "Height" is ALWAYS perpendicular to the base
Q6
EASY
A trapezoid has parallel sides of length 8 and 14, and a height of 6. A triangle with the same base of 14 and same height of 6 is drawn beside it. How much greater is the trapezoid's area than the triangle's area? โš ๏ธ Tricky: don't confuse the formulas!
(A) 24
(B) 33
(C) 42
(D) 48
My Work & Answer
๐Ÿชค TRAP: Trapezoid = ยฝ(8+14)ร—6 = 66. Triangle = ยฝร—14ร—6 = 42. Diff = 24!
TOPIC 07 ๐Ÿ“ Similar Triangles & Proportions

If two triangles are similar (AA, SAS, SSS), corresponding sides are proportional. If the sides ratio is \(k\), then areas ratio is \(k^2\)!

๐Ÿ”‘ SIDES ratio = k  โ†’  AREA ratio = kยฒ  โ†’  VOLUME ratio = kยณ
๐Ÿ”‘ AA (Angle-Angle) is the easiest similarity proof
Two similar triangles have corresponding sides 4 and 6. The area of the smaller triangle is 20. Find the larger triangle's area.
Ratio of sides: \(\dfrac{6}{4} = \dfrac{3}{2}\)
Ratio of areas: \(\left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\)
Larger area: \(20 \times \dfrac{9}{4} = 45\)
Q7
EASY
In the figure, a small triangle has sides 3, 4, 5. A similar triangle has hypotenuse 15. What is the perimeter of the larger triangle? โš ๏ธ Tricky: find the scale factor first!
(A) 30
(B) 36
(C) 40
(D) 45
My Work & Answer
๐Ÿชค TRAP: Scale = 15/5 = 3. Small perimeter = 12. Large = 12 ร— 3 = 36!
TOPIC 08 ๐ŸงŠ Volume & Surface Area

Cylinder: \(V = \pi r^2 h\). Cone: \(V = \dfrac{1}{3}\pi r^2 h\). Sphere: \(V = \dfrac{4}{3}\pi r^3\). SA of sphere: \(4\pi r^2\).

๐Ÿ”‘ Cone/Pyramid = โ…“ ร— (base shape) ร— height
๐Ÿ”‘ "Surface Area" = SUM of all faces
๐Ÿ”‘ SPHERE: "4 great circles" โ†’ SA = 4ฯ€rยฒ
Q8
EASY
A cylinder has radius 3 and height 8. A cone has the same radius and height as the cylinder. What is the ratio of the cone's volume to the cylinder's volume? โš ๏ธ Tricky: this is a ratio โ€” you don't need exact values!
(A) \(\dfrac{1}{4}\)
(B) \(\dfrac{1}{3}\)
(C) \(\dfrac{1}{2}\)
(D) \(\dfrac{2}{3}\)
My Work & Answer
๐Ÿชค TRAP: Cone = โ…“ ร— Cylinder ALWAYS. Ratio = 1:3 โ†’ cone/cylinder = 1/3
TOPIC 09 ๐Ÿ“ Coordinate Geometry

Midpoint: \(\left(\dfrac{x_1+x_2}{2},\, \dfrac{y_1+y_2}{2}\right)\). Distance: \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). Slope: \(\dfrac{y_2-y_1}{x_2-x_1}\).

๐Ÿ”‘ MIDPOINT = average the coordinates
๐Ÿ”‘ DISTANCE = diagonal of a right triangle โ†’ Pythagorean theorem!
๐Ÿ”‘ Perpendicular slopes โ†’ product = โˆ’1 (negative reciprocal)
Q9
EASY
Point M is the midpoint of \(\overline{AB}\). If \(A = (2, -3)\) and \(M = (5, 1)\), what are the coordinates of point B? โš ๏ธ Tricky: work backwards from midpoint!
(A) \((3.5,\,-1)\)
(B) \((7,\, 4)\)
(C) \((8,\, 5)\)
(D) \((10,\, 2)\)
My Work & Answer
๐Ÿชค TRAP: B = 2M โˆ’ A โ†’ x: 2(5)โˆ’2=8, y: 2(1)โˆ’(โˆ’3)=5 โ†’ B=(8,5)
TOPIC 10 ๐Ÿ”ฒ Circle Equation in Coordinate Plane

Standard form of a circle: \((x-h)^2 + (y-k)^2 = r^2\)   where (h, k) is the center and r is the radius.

๐Ÿ”‘ CENTER: read (h, k) directly โ€” but WATCH THE SIGNS!
๐Ÿ”‘ RADIUS: take โˆš of the right side
๐Ÿ”‘ Must COMPLETE THE SQUARE if given general form axยฒ+bx+...
What is the center and radius of \((x-3)^2 + (y+4)^2 = 25\)?
Center: \((3,\,-4)\)   โ€” Watch! \(y+4\) means \(k = -4\)
Radius: \(r = \sqrt{25} = 5\)
Q10
EASY
A circle in the coordinate plane has equation
\(x^2 + y^2 - 6x + 8y + 9 = 0\)
What is the radius of this circle? โš ๏ธ Tricky: complete the square first!
(A) 3
(B) 4
(C) 5
(D) \(\sqrt{34}\)
My Work & Answer
๐Ÿชค TRAP: Complete the square โ†’ (xโˆ’3)ยฒ+(y+4)ยฒ=16 โ†’ r=4, not rยฒ=4!
Quick Cheat Sheet
Angles
Vertical = Equal
Supplement + angle = 180ยฐ
Triangle sum = 180ยฐ
Exterior = sum of remote interior
Triangles
Pyth: \(a^2+b^2=c^2\)
45-45-90: \(x, x, x\sqrt{2}\)
30-60-90: \(x, x\sqrt{3}, 2x\)
Similar: sides ratio \(k\), area \(k^2\)
Circles
Circumference: \(2\pi r\)
Area: \(\pi r^2\)
Arc length: \(\frac{\theta}{360}\cdot2\pi r\)
Inscribed angle: \(\frac{1}{2}\) arc
Volume
Cylinder: \(\pi r^2 h\)
Cone: \(\frac{1}{3}\pi r^2 h\)
Sphere: \(\frac{4}{3}\pi r^3\)
Prism/Box: \(l \times w \times h\)

๐Ÿง  The #1 Most Common Mistake: Confusing "radius" and "diameter." Always double-check: does the problem give you r or d? If d, divide by 2 first!

๐Ÿ”‘ DIAMETER = 2 ร— RADIUS. Always. Every time.
~ Keep going. You've got this! ๐Ÿ’ช ~
SAT GEOMETRY NOTES ยท 10 TOPICS ยท 10 PROBLEMS