What this category really tests
Geometry & Trigonometry is the smallest Math domain on the digital SAT, but it's loaded with formulas you can read straight from the reference sheet (the SAT provides one in Bluebook). Your job is to recognize the shape and pick the right tool.
Triangles: the workhorses
The big three triangle facts
- Angles of any triangle sum to 180°.
- Pythagorean theorem (right triangles): a² + b² = c².
- Triangle inequality: any side is less than the sum of the other two.
Special right triangles, memorize these
- 45-45-90: sides in ratio 1 : 1 : √2. The legs are equal; the hypotenuse is leg × √2.
- 30-60-90: sides in ratio 1 : √3 : 2. Short leg is opposite 30°; long leg is opposite 60°; hypotenuse is opposite 90°.
The SAT also loves Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and their multiples. If you see two of these numbers, the third is probably the answer.
Similar triangles
Two triangles are similar if their angles match. Their sides are proportional. The SAT loves "two triangles inside one figure" questions: identify the similar pair, set up a ratio, solve.
Right-triangle trigonometry: SOH CAH TOA
For a given acute angle θ in a right triangle:
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent
The SAT's favorite trig identity to test: sin θ = cos (90° - θ). Sine and cosine are complementary. If sin x = 0.6 and you're asked for cos (90° - x), the answer is 0.6.
Practical tip: when given side lengths and an angle, draw the triangle, label O, A, H, and pick the ratio that contains your knowns and unknowns. Trig becomes a translation exercise.
Circles: the small set you need
- Circumference: C = 2πr.
- Area: A = πr².
- Arc length: (θ / 360°) · 2πr (where θ is the central angle in degrees).
- Sector area: (θ / 360°) · πr².
- Inscribed angle theorem: an inscribed angle is half the central angle that subtends the same arc.
- Equation of a circle: (x - h)² + (y - k)² = r², centered at (h, k) with radius r. Be ready to complete the square to convert standard form to this.
Radians and degrees: for SAT purposes, you'll usually convert by π radians = 180°. So 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6.
Polygons and angles
Sum of interior angles of a polygon with n sides = (n - 2) · 180°. So a hexagon has (6 - 2) · 180° = 720°.
For regular polygons, divide by n to get each interior angle. A regular pentagon has 540° / 5 = 108° per angle.
Parallel lines cut by a transversal create equal alternate interior angles, equal corresponding angles, and supplementary same-side interior angles. The SAT mostly tests this in figure-style problems where you must chase angles through a diagram.
Area, surface area, and volume
The Bluebook reference sheet gives you the formulas for circle area, triangle area, and the volumes of rectangular box, cylinder, sphere, cone, and pyramid. You don't need to memorize, but you should:
- Know which formula to grab without scanning the whole sheet.
- Be careful with units. A cube of side 4 cm has volume 64 cm³, not 64 cm².
- Watch for "scale factor" questions. If you scale a 3D figure by k, lengths scale by k, areas by k², volumes by k³.
A worked example
"In the figure (not shown), triangle ABC is inscribed in a circle of radius 5, with AB as a diameter. If the length of AC is 6, what is the length of BC?"
Key insight: any triangle inscribed in a circle with a side as diameter is a right triangle, with the right angle opposite the diameter. So angle C = 90°, AB = diameter = 10.
By the Pythagorean theorem: AC² + BC² = AB². So 6² + BC² = 10². So BC² = 64, so BC = 8.
The 6-8-10 triangle is just a scaled 3-4-5. Recognize it and you save algebra.
How to practice geometry efficiently
- Draw every figure, even when one is shown. Re-drawing forces you to label sides and angles, which often makes the answer obvious.
- Memorize the special right triangles and the Pythagorean triples. They appear constantly.
- Practice converting circle equations between standard and (x - h)² + (y - k)² = r² form using completing the square.
- For trig, label O, A, H first. Then pick the ratio. Don't try to solve trig in your head.