Practice Math — Geometry & Trigonometry questions for the Scholastic Assessment Test. Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Question 1medium
A right triangle has legs of length 5 and 12. The statement 'the length of the hypotenuse is 13 and the perimeter of the triangle is 30' is true.
Correct answer: A
Using the Pythagorean theorem: c² = 5² + 12² = 25 + 144 = 169, so c = 13. The perimeter is then 5 + 12 + 13 = 30. Both parts of the statement are correct, so the answer is True. This is a classic 5-12-13 Pythagorean triple, and combining the hypotenuse calculation with the perimeter sum tests two related skills in one step.
Question 2medium
In a circle with center O, a sector has a central angle of 120° and the circle has a radius of 6. The statement 'the arc length of the sector is 4π' is true.
Correct answer: A
Arc length = (central angle / 360°) × 2πr = (120/360) × 2π(6) = (1/3) × 12π = 4π. The statement is correct, so the answer is True. A common error is to use the sector area formula (πr² × θ/360) instead of the arc length formula, which would give 12π — that misconception makes this a meaningful medium-difficulty check.
Question 3medium
Triangle ABC is similar to triangle DEF with a scale factor of 3:5 (ABC to DEF). If the area of triangle ABC is 27 square units, the statement 'the area of triangle DEF is 75 square units' is true.
Correct answer: A
For similar figures with linear scale factor 3:5, the ratio of areas is (3:5)² = 9:25. Setting up the proportion: 27/Area(DEF) = 9/25, so Area(DEF) = 27 × 25/9 = 75 square units. The statement is True. A common mistake is to multiply by the linear scale factor (5/3) rather than its square (25/9), which incorrectly gives 45 — that error would lead a student to choose False.
Question 4medium
Line ℓ passes through the points (2, 5) and (6, 11). The statement 'a line perpendicular to ℓ has a slope of −2/3' is true.
Correct answer: A
The slope of line ℓ is (11 − 5)/(6 − 2) = 6/4 = 3/2. A perpendicular line has a slope equal to the negative reciprocal of 3/2, which is −2/3. The statement is True. Common errors include only taking the reciprocal (giving 2/3) or only negating (giving −3/2), both of which would cause a student to incorrectly choose False.
Question 5medium
In right triangle PQR, angle Q is the right angle, angle P measures 35°, and the hypotenuse PR has length 10. The statement 'the side QR opposite to angle P has length 10 sin(35°)' is true.
Correct answer: A
In right triangle PQR with the right angle at Q, side QR is opposite angle P and the hypotenuse is PR = 10. By SOH-CAH-TOA, sin(P) = opposite/hypotenuse = QR/PR, so QR = PR × sin(35°) = 10 sin(35°). The statement is True. A common error is to confuse which side is opposite versus adjacent, leading students to use cos(35°) instead, or to incorrectly identify PR as a leg rather than the hypotenuse.
Question 6medium
A cylinder has a radius of 4 and a height of 9. A cone has the same radius and the same height. The statement 'the volume of the cylinder is exactly three times the volume of the cone' is true.
Correct answer: A
The volume of the cylinder is πr²h = π(4²)(9) = 144π. The volume of the cone is (1/3)πr²h = (1/3)(144π) = 48π. Since 144π = 3 × 48π, the cylinder's volume is exactly three times the cone's volume. The statement is True. This relationship holds for any cylinder and cone sharing the same radius and height — the factor of 1/3 in the cone formula is precisely what creates this ratio. Students who misremember the cone formula (omitting the 1/3) would incorrectly choose False.
Question 7hard
In triangle PQR, angle Q = 90°, PQ = 7, and angle P = 60°. The triangle is inscribed in a rectangle such that vertices Q and R lie on the base of the rectangle, and vertex P lies on the top side directly above Q. What is the area of the rectangle?
Correct answer: B
In right triangle PQR with angle Q = 90° and angle P = 60°, angle R = 30°. Side PQ = 7 (leg adjacent to P), so tan(60°) = QR/PQ → QR = 7·tan(60°) = 7√3. The rectangle has P directly above Q, so the rectangle's height equals PQ = 7 and the rectangle's width equals QR = 7√3. Area = 7 × 7√3 = 49√3. Option A (7√3) is just the length of QR, not the area. Option D (14√3) results from using 2·PQ as one dimension. Option C (98√3) doubles the correct area, a common error when confusing the full rectangle with a triangle area.
Question 8hard
If sin(x°) = cos(3x − 10)°, and 0 < x < 90, what is the value of x?
Correct answer: C
Using the complementary angle identity, sin(x°) = cos(90° − x°). So cos(90 − x)° = cos(3x − 10)°. Setting the arguments equal: 90 − x = 3x − 10 → 100 = 4x → x = 25. Verify: sin(25°) = cos(65°) and cos(3·25−10)° = cos(65°) ✓. Option A (x=20) gives 3(20)−10 = 50, and 90−20 = 70 ≠ 50. Option B (x=30) gives 3(30)−10 = 80, and 90−30 = 60 ≠ 80. Option D (x=22.5) is a distractor from solving 90−x = 3x incorrectly without the −10 term: 90 = 4x → x = 22.5, ignoring the constant.
Question 9hard
In triangle ABC, angle C = 90°, BC = 5, and AC = 12. Point D is on hypotenuse AB such that CD is perpendicular to AB. What is the length of CD?
- A.5√2/2
- B.12/5
- C.65/12
- D.60/13✓
Correct answer: D
The hypotenuse AB = √(BC² + AC²) = √(25 + 144) = √169 = 13. The altitude from the right angle to the hypotenuse has length CD = (BC · AC)/AB = (5 · 12)/13 = 60/13. This is the geometric mean relationship in a right triangle: the altitude to the hypotenuse equals the product of the two legs divided by the hypotenuse. Distractor B(12/5) confuses one leg with the altitude formula. Distractor C(65/12) inverts the formula, using the hypotenuse squared over a leg. Distractor A suggests a 45-45-90 assumption that does not apply here.
Question 10hard
A solid is formed by attaching a cone of height 4 and base radius 3 on top of a cylinder of height 10 and radius 3. A second solid is a sphere. If the volume of the sphere equals the total volume of the composite solid, what is the radius of the sphere? (Use π where needed and simplify.)
- A.∛(306/4)✓
- B.∛(102/4)
- C.3∛(102/4)
- D.3
Correct answer: A
Volume of cylinder = π r² h = π(9)(10) = 90π. Volume of cone = (1/3)π r² h = (1/3)π(9)(4) = 12π. Total volume = 102π. Volume of sphere = (4/3)π R³ = 102π → R³ = 102π · 3/(4π) = 306/4 → R = ∛(306/4). Distractor B omits the factor of 3 when solving for R³. Distractor C incorrectly factors out 3 from the cube root. Distractor D results from guessing the radius equals the shared base radius without calculation.