SAT Math: Geometry & Trigonometry
Geometry and Trigonometry makes up roughly 15% of the Digital SAT Math section, the smallest of the four domains, but it draws on a wide range of formulas and rules that can feel unfamiliar if you haven't reviewed them recently.
This domain covers area and volume, properties of lines, angles, and triangles, similarity and congruence, right-triangle trigonometry, and circles — much of it accessible once a small set of relationships and formulas are fresh in your memory.
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Area and volume
Area and volume questions typically give you enough information to plug directly into a formula, but sometimes require you to first find a missing dimension using another geometric relationship. The Digital SAT's reference sheet includes formulas for circles, rectangles, triangles, and several solids, but knowing the most common ones (rectangle area, triangle area, circle area, and volume of a rectangular prism and cylinder) by heart saves valuable time.
Worked example
A cylinder has a radius of 3 and a height of 10. What is its volume, in terms of π?
- Use the formula V = πr²h
- Substitute: V = π(3²)(10) = π(9)(10)
- Multiply: V = 90π
Answer: 90π cubic units
Lines, angles, and triangles
When two parallel lines are cut by a transversal, the resulting angles follow predictable rules: corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles sum to 180°. In any triangle, the interior angles always sum to 180°, and the exterior angle at any vertex equals the sum of the two non-adjacent interior angles.
The triangle inequality theorem states that the sum of any two side lengths must exceed the third side; this shows up in questions asking which of several possible side lengths could complete a triangle.
Similarity and congruence
Similar triangles have equal corresponding angles and proportional corresponding sides; congruent triangles have both equal angles and equal side lengths. When two triangles are established as similar (often through an Angle-Angle argument, since two equal angles guarantee similarity), you can set up a proportion between corresponding sides to solve for an unknown length.
Worked example
Triangle ABC is similar to triangle DEF. Side AB = 6, side BC = 9, and the corresponding side DE = 8. Find EF.
- Because the triangles are similar, corresponding sides are proportional: AB/DE = BC/EF
- Substitute known values: 6/8 = 9/EF
- Cross-multiply: 6 × EF = 8 × 9 = 72
- Divide: EF = 72/6 = 12
Answer: EF = 12
Right triangles and SOHCAHTOA
For a right triangle, the three basic trigonometric ratios relate an angle to the triangle's sides: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — commonly remembered as SOHCAHTOA. A related and frequently tested fact is that sine and cosine of complementary angles are equal: sin(θ) = cos(90° − θ), because the 'opposite' side for one acute angle in a right triangle is the 'adjacent' side for the other.
Worked example
In a right triangle, angle A and angle B are complementary. If sin(A) = 0.6, what is cos(B)?
- Complementary angles satisfy sin(A) = cos(B) when A + B = 90°.
- Since sin(A) = 0.6, cos(B) must also equal 0.6.
Answer: cos(B) = 0.6
Special right triangles
Two triangle shapes recur often enough to be worth memorizing directly: the 45-45-90 triangle has sides in the ratio 1 : 1 : √2, and the 30-60-90 triangle has sides in the ratio 1 : √3 : 2 (opposite the 30°, 60°, and 90° angles respectively). Recognizing these ratios lets you find all three side lengths instantly from just one known side, without needing the Pythagorean theorem or trigonometric ratios.
Radians and degrees
The Digital SAT expects familiarity with radian measure, particularly the conversion between degrees and radians: multiply degrees by π/180 to convert to radians, or multiply radians by 180/π to convert to degrees. A full circle is 360° or 2π radians, so a half circle is π radians and a quarter circle is π/2 radians — memorizing these common benchmark angles speeds up arc and sector problems considerably.
Circles: equations, arcs, and sectors
The standard equation of a circle is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius; if an equation is given in expanded form, complete the square on both the x and y terms to identify the center and radius. Arc length and sector area are both fractions of the full circle's circumference or area, based on the central angle: arc length = (θ/360°) × 2πr, and sector area = (θ/360°) × πr², using the same fraction of 360° (or 2π radians) that the central angle represents.
Worked example
A circle has radius 6. What is the arc length of a sector with a central angle of 60°?
- Find the fraction of the circle: 60°/360° = 1/6
- Find the full circumference: C = 2πr = 2π(6) = 12π
- Multiply the fraction by the circumference: (1/6)(12π) = 2π
Answer: 2π
Common mistakes to avoid
- Mixing up which side is 'opposite' versus 'adjacent' relative to the reference angle in SOHCAHTOA problems.
- Forgetting to complete the square when a circle's equation is given in expanded form, making it impossible to identify the center and radius directly.
- Using the wrong ratio for a 30-60-90 or 45-45-90 triangle, such as pairing the wrong side with √3 versus √2.
- Applying the arc length formula with degrees when the given angle is already in radians (or vice versa) without converting first.
- Assuming two triangles are similar or congruent without confirming the correspondence of angles, leading to a mismatched proportion.
- Forgetting that the exterior angle of a triangle equals the sum of the two remote interior angles, and instead assuming it equals 180° minus only one interior angle incorrectly.
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