These are 5 hard SAT geometry and trigonometry questions from our question bank, all at the top of our difficulty scale. Geometry and Trigonometry is about 15% of the Math section, 5 to 7 questions, but the hard ones lean on a few ideas again and again: how area scales, similar triangles, and the link between sine and cosine. Try each one, then open the solution.
Remember that Bluebook gives you a reference sheet with the area and volume formulas, so you do not need to memorize them. What you need is to know when to use them.
Question 1 (Hard, Area, Perimeter, and Volume)
- A. The area of PQRS would be quadrupled.
- B. The area of PQRS would be increased by 75%.
- C. The area of PQRS would be decreased by 75%.
- D. The area of PQRS would be unaffected
Show answer and solution
Answer: D. The area of PQRS would be unaffected
Trapezoid area is (base 1 + base 2) / 2 × height. Decreasing both bases by 75% multiplies them by 0.25. Quadrupling the height multiplies it by 4. 0.25 × 4 = 1, so the area is unchanged.
Tip: For “how does the area change” questions, multiply the scale factors. You never need the actual lengths.
Question 2 (Hard, Area, Perimeter, and Volume)
What is the area, in square units, of the square?
- A. 20
- B. 25
- C. 50
- D. 100
Show answer and solution
Answer: C. 50
The vertices sit on the axes at 5 and −5, so each diagonal is 10 long. For a square (or any rhombus), area = d1 × d2 / 2 = 10 × 10 / 2 = 50.
Tip: When you know the diagonals of a square, area = diagonal2 / 2. No side length needed.
Question 3 (Hard, Right-Triangle Trigonometry)
Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angle. The length of AB is 2.4 times the length of DE.
If , what is the value of sin D?
- A. 0.35
- B. 0.55
- C. 0.63
- D. 0.72
Show answer and solution
Answer: D. 0.72
tan A = 21/20 means opposite 21 and adjacent 20, so the hypotenuse is √(212 + 202) = 29. Similar triangles have equal corresponding angles, so sin D = sin A = 21/29 ≈ 0.72. The 2.4 scale factor does not matter.
Tip: Trig ratios do not change when a triangle is scaled. Ignore the scale factor and work in one triangle.
Want more at this level?
Targeted Practice builds sets from the skills you miss.
Question 4 (Hard, Right-Triangle Trigonometry)
- A. 5
- B. 19
- C. 23
- D. 29
Show answer and solution
Answer: A. 5
cos(Q) = sin(R) for acute angles means Q and R are complementary: Q + R = 90. So (x + 61) + (4x + 4) = 90, 5x = 25, and x = 5.
Tip: sin(x) = cos(90 − x). Whenever a sine equals a cosine, the two angles add to 90°.
Question 5 (Hard, Triangles)
- A. 16.8
- B. 17.6
- C. 18.4
- D. 19.6
Show answer and solution
Answer: B. 17.6
Parallel lines cut two transversals in the same ratio, so AB / BC = FE / ED. That gives 9 / 18.5 = 8.55 / ED, so ED = 8.55 × 18.5 / 9 = 17.575, which rounds to 17.6.
Tip: Three or more parallel lines crossing two lines create matching ratios. Set up the proportion first, then solve.
Review: Triangles notes
How to get better
- Learn the idea behind each solution. Each tip above is a pattern that shows up on many hard questions.
- Read the notes linked in each solution, then drill the skill in Targeted Practice.
- Watch it explained in the free Advanced Math video series with Mr. O.
- Test yourself under time with a free full-length practice test.
Want more? Try these hard SAT math questions and hard SAT algebra questions.
Frequently asked questions
How many geometry questions are on the SAT?
Geometry and Trigonometry is about 15% of the Math section, about 5 to 7 of the 44 questions.
Do you get a formula sheet on the SAT?
Yes. Bluebook has a reference sheet with common area, volume and triangle formulas.
What trig is on the SAT?
Mostly right-triangle trig (sine, cosine, tangent), the fact that sin(x) = cos(90 − x), and radians.
Hard SAT Math practice
This guide is part of Digital SAT Vocabulary and Advanced Math: Resource Hub.
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Sources
- College Board, The Math Section: Overview.
Frequently Asked Questions
How many geometry questions are on the SAT?
Geometry and Trigonometry is about 15% of the Math section, about 5 to 7 of the 44 questions.
Do you get a formula sheet on the SAT?
Yes. Bluebook has a reference sheet with common area, volume and triangle formulas.
What trig is on the SAT?
Mostly right-triangle trig (sine, cosine, tangent), the fact that sin(x) = cos(90 - x), and radians.