Right Triangle Trigonometry · Sub-skill drill
Special Right Triangles
Two special right triangles appear repeatedly on the SAT: the 30-60-90 triangle with sides in ratio 1:√3:2, and the 45-45-90 triangle with sides in ratio 1:1:√2. Memorizing these ratios saves time on every triangle question that uses them, because they let you read off side lengths without applying the trig functions. SAT questions sometimes hide these triangles inside larger figures; recognizing the special ratio is the entire skill.
How this sub-skill is tested on the SAT
Two special right triangles appear repeatedly on the SAT: the 30-60-90 triangle with sides in ratio 1:√3:2, and the 45-45-90 triangle with sides in ratio 1:1:√2. Memorizing these ratios saves time on every triangle question that uses them, because they let you read off side lengths without applying the trig functions. SAT questions sometimes hide these triangles inside larger figures; recognizing the special ratio is the entire skill.
This sub-skill sits inside the broader Right Triangle Trigonometry topic, which is part of the College Board's Additional Topics in Math content domain. Additional Topics in Math is the smallest official SAT Math domain by raw question count, but it carries outsized weight because the questions are concentrated at the harder end of each section. You will see roughly six of these per test, and they tend to separate students aiming for a 750 from students aiming for an 800. The domain covers right triangle trigonometry, circle theorems, volume formulas, complex number arithmetic, and the geometry of lines in the coordinate plane. Most of the formulas you need are listed at the start of the math section — but the test rewards students who have
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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In a right triangle with legs 5 and 12 and hypotenuse 13, what is the value of sin(A) where A is the angle opposite the leg of length 5?
- A 5/12
- B 5/13
- C 13/5
- D 12/13
Worked solution
Answer: B — 5/13
SOH-CAH-TOA: sin = opposite / hypotenuse. The leg opposite angle A has length 5; the hypotenuse is 13. So sin(A) = 5/13 = 5/13.
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In a right triangle with legs 18 and 24 and hypotenuse 30, what is the value of tan(B) where B is the angle opposite the leg of length 24?
- A 3/4
- B 4/3
- C 4/5
- D 3/5
Worked solution
Answer: B — 4/3
tan = opposite / adjacent. Opposite angle B is 24; the adjacent leg is 18. So tan(B) = 24/18 = 4/3.
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A right triangle has legs of length 21 and 72. What is the length of the hypotenuse?
- A 93
- B 74
- C 75
- D 76
Worked solution
Answer: C — 75
By the Pythagorean theorem, c^2 = a^2 + b^2 = 21^2 + 72^2 = 441 + 5184 = 5625. So c = √5625 = 75.
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In a right triangle with legs 20 and 48 and hypotenuse 52, what is the value of sin(A) where A is the angle opposite the leg of length 20?
- A 5/12
- B 5/13
- C 13/5
- D 12/13
Worked solution
Answer: B — 5/13
SOH-CAH-TOA: sin = opposite / hypotenuse. The leg opposite angle A has length 20; the hypotenuse is 52. So sin(A) = 20/52 = 5/13.
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In a right triangle with legs 36 and 48 and hypotenuse 60, what is the value of tan(B) where B is the angle opposite the leg of length 48?
- A 3/4
- B 4/3
- C 4/5
- D 3/5
Worked solution
Answer: B — 4/3
tan = opposite / adjacent. Opposite angle B is 48; the adjacent leg is 36. So tan(B) = 48/36 = 4/3.
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A right triangle has legs of length 21 and 72. What is the length of the hypotenuse?
- A 93
- B 74
- C 75
- D 76
Worked solution
Answer: C — 75
By the Pythagorean theorem, c^2 = a^2 + b^2 = 21^2 + 72^2 = 441 + 5184 = 5625. So c = √5625 = 75.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.