How to run this lesson. Send the pre-work a day ahead. Open with goals (5 min), then angles (12), right triangles (16), circles & volume (12), trig & similarity (8), the Vocabulary Lab (7), and a wrap that assigns the next-two-days work. The dark navy notes are for the instructor only.
The frame for today
One line to open: “The SAT hands you the formulas — your job is recognition, not memorization.” Today is about spotting the triangle type, the right formula, and the hidden 3-4-5.
00 Before you begin
Pre-work handout
Five minutes of warm-up makes the lesson land faster.
Geometry & Trig — Pre-work
Complete on paper before our session · about 10 minutes
Warm-up
1. The angles of a triangle add to ______ degrees.
2. A right triangle has legs 3 and 4. The hypotenuse is ______.
3. A circle has radius 2. Its area is ______.
4. Two angles are complementary. One is 30°. The other is ______.
Answer key (instructor)
1 · 180 2 · 5 3 · 4π 4 · 60°
02 Angles
The angle rules that repeat
A triangle’s angles add to 180°. A straight line is 180°. Complementary angles add to 90°; supplementary add to 180°; vertical angles are equal.
Triangle sum1
Two angles of a triangle measure 50° and 60°. What is the third angle?
Why B
All three add to 180°: 180 − (50 + 60) = 70°.Complementary2
Angles A and B are complementary. If A = 35°, what is B?
Why B
Complementary means they add to 90°: 90 − 35 = 55°.03 Right triangles
Three triangle tools
For right triangles, reach for the Pythagorean theorem, the common triples, or the two special triangles.
Pick the fastest tool
Pythagorean
a² + b² = c² always works for a right triangle.
Triples
Recognize 3-4-5, 5-12-13, 8-15-17 and their multiples — instant answer.
Special right
45-45-90 sides are x, x, x√2. 30-60-90 sides are x, x√3, 2x.
Triple3
A right triangle has legs of length 9 and 12. What is the hypotenuse?
Why B
9-12-15 is the 3-4-5 triangle tripled → 15. (Check: 81 + 144 = 225 = 15².)Solve it another way
True shortcut If the legs are multiples of 3 and 4, the hypotenuse is the matching multiple of 5.
Special right4
In a 45-45-90 right triangle, each leg is 5. What is the hypotenuse?
Why B
In a 45-45-90 triangle the hypotenuse is a leg times √2 → 5√2.04 Circles & volume
Area, circumference, volume
Circle area is πr²; circumference is 2πr. For solids, volume multiplies the base area by the height.
Circle area5
A circle has radius 4. What is its area?
Why B
Area = πr² = π(4)² = 16π.Cylinder volume6
A cylinder has radius 3 and height 5. What is its volume?
Why B
Volume = πr²h = π(9)(5) = 45π.05 Trig & similarity
SOH-CAH-TOA, and scaling
Sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. Similar triangles keep the same shape — every side scales by the same factor.
Sine7
In a right triangle, the side opposite a 30° angle is 6 and the hypotenuse is 12. What is sin(30°)?
Why A
sin = opposite / hypotenuse = 6 / 12 = 0.5.Similar triangles8
A 3-4-5 triangle is enlarged so its shortest side is 9. How long is its longest side?
Why B
The shortest side went from 3 to 9, a scale factor of 3. The longest side: 5 × 3 = 15.Recognition over memorization
The reference sheet gives the formulas — train the student to label what they have (legs? radius? angle?) and pick the matching tool. Drill the triples and the two special triangles until they’re instant; that’s where the time savings live.
06 Vocabulary Lab
This week’s words — think in synonyms and antonyms
Tap a card to flip it. Learn each word next to its opposite — that’s how the test frames them.
Quick check
Synonym9
Which word is closest in meaning to intricate?
Why B
Intricate = complicated and detailed, like complex.Antonym10
Which word is most nearly opposite to diminish?
Why C
Diminish = make smaller; its opposite is increase.Word in context11
The two highways ______ at the city center before splitting off again.
Why A
Converge = come together from different directions.07 Wrap-up
Three things to carry out the door
Say them out loud — that’s how they stick.
Angles sum to 180°
In every triangle — your most-used rule.
Know the triples
3-4-5, 5-12-13, and the 45-45-90 / 30-60-90 specials.
πr² and 2πr
Circle area and circumference — memorize the pair.
Close the loop
Have the student set one target for the mid-week. Confirm the two lesson days, point them to the workbook, and preview that Lesson 5 is Problem-Solving & Data Analysis — the last math domain.
08 Keep the fire lit
Next-two-days handout
Short, daily, cumulative.
Geometry & Trig — Work for the Next Two Days
About 30 minutes a day · sketch the figure every time
Day 1 · Angles, triangles, circles
Reach for the right tool
1. Triangle angles are 40° and 75°. The third? ______
2. Right triangle legs 8 and 15. Hypotenuse? ______
3. Circle radius 5. Area? ______
4. Two complementary angles; one is 28°. The other? ______
5. 45-45-90 triangle with legs 7. Hypotenuse? ______
Day 2 · Volume, trig, similarity
Scale and solve
1. Cylinder radius 2, height 10. Volume? ______
2. Right triangle: opposite = 3, hypotenuse = 6. sin θ? ______
3. Rectangle area 60, length 12. Width? ______
4. Similar triangles, scale factor 4. A side of 3 becomes? ______
Answer key (instructor)
Day 1: 1 · 65° 2 · 17 3 · 25π 4 · 62° 5 · 7√2
Day 2: 1 · 40π 2 · 0.5 3 · 5 4 · 12
Day 2: 1 · 40π 2 · 0.5 3 · 5 4 · 12
This sets up Lesson 5. Data and word problems lean on ratios and arithmetic — the same careful setup you used to label triangles.