The fastest way to get good at geometry is to memorize a small set of patterns that show up over and over. Here are 10 mental shortcuts every student should have on speed-dial. Each one saves 30 seconds to several minutes per problem.
If you see a right triangle with two of (3, 4, 5), the third is the missing one. Same for (5, 12, 13). And (8, 15, 17), (7, 24, 25). Recognizing these saves you from running a² + b² = c² every time.
Tip: any multiple works too. (6, 8, 10), (9, 12, 15), (10, 24, 26) are scaled triples.
If you ever see a 30-60-90 triangle, the sides are always in the ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). Memorize this once, never run trig again for these.
Example: hypotenuse = 10. Short leg = 5. Long leg = 5√3 ≈ 8.66.
An isosceles right triangle. Legs are equal; hypotenuse = leg × √2.
Example: legs = 7. Hypotenuse = 7√2 ≈ 9.90.
Combined with #2, these two “special right triangles” cover most exam problems involving exact non-decimal answers.
Direct application of #3. A square with side 5 has diagonal 5√2 ≈ 7.07.
For a cube, the SPACE diagonal (corner to opposite corner through the body) is side × √3.
If two parallel lines are crossed by a transversal:
This means in any parallel-line setup, you only need to know ONE angle to know all 8. Use this in proofs to invoke ASA or AA-similarity — see our Parallel Lines and Transversals Calculator.
For any triangle: each side must be LESS than the sum of the other two. So sides 3, 4, 8 cannot form a triangle (3 + 4 = 7 < 8).
Use this to quickly reject impossible-looking triangle problems before trying to solve them.
If two similar shapes have a linear ratio of k, their areas are in ratio k². If you double a shape (k=2), its area quadruples (4×). Triple it (k=3), area is 9×.
For 3D solids, volume ratios follow k³. Doubling all dimensions of a cube increases volume 8×.
Even non-right triangles. Pick any side as the base, drop a perpendicular from the opposite vertex to that side, measure the perpendicular’s length, plug in. Works for obtuse triangles too (where you might need to extend the base externally to drop the perpendicular).
For when you don’t have the height but have all 3 sides, use Heron’s formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. (See our Heron’s Formula Calculator.)
An angle inscribed in a circle (vertex on the circle, sides being chords) equals exactly HALF the central angle subtending the same arc.
Special case: any angle inscribed in a semicircle is exactly 90°. This is Thales’ Theorem — incredibly useful for proving right angles in circle problems.
For any n-gon: triangle (n=3) has 180° total, quadrilateral (n=4) has 360°, pentagon (n=5) has 540°, hexagon (n=6) has 720°.
For REGULAR polygons, divide by n to get each interior angle: regular hexagon = 720°/6 = 120° per angle.
Bonus: SUM of EXTERIOR angles is always 360° regardless of n. Each exterior angle of a regular n-gon = 360°/n.
| π (pi) | ≈ 3.14159 | circle: C/d |
| √2 | ≈ 1.414 | diagonal of unit square |
| √3 | ≈ 1.732 | long leg of 30-60-90 |
| √5 | ≈ 2.236 | golden ratio φ = (1+√5)/2 |
| π/180 | ≈ 0.01745 | degrees → radians multiplier |
Test these tricks against the actual calculations on our 60+ calculators. The point of memorizing these patterns is so that you can sanity-check the calculator’s output instantly — and skip the calculator entirely on simpler problems.
How do I memorize all of these? Don’t try in one sitting. Pick 2 per week, write them on a sticky note, work through 5 problems using each. After 5 weeks you’ll have all 10 cemented.
Are these tricks taught in schools? Mostly yes, but scattered across multiple chapters. The advantage of seeing them collected here is recognizing them as a SET — the same 10 patterns will solve 80%+ of routine geometry problems.
What about trigonometry — should I memorize sin/cos values? The 30-60-90 and 45-45-90 ratios above effectively give you sin/cos of 30°, 45°, 60° without trig. For other angles, use a calculator. For exam settings without calculators, you only need: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2 (and cos values are the mirror of these).