Triangles
Triangles are the single most important shape on the GRE. The angle-sum rule, the Pythagorean theorem and the two special right triangles do most of the heavy lifting across the whole Geometry section, and they resurface inside quadrilateral, circle and coordinate-geometry problems too. The GRE deliberately does not test trigonometry — instead it reuses the 45-45-90 and 30-60-90 triangles so often that memorising their side ratios is worth more than any other geometry fact. Two habits pay off constantly: recognise the common Pythagorean triples on sight, and remember that GRE figures are not drawn to scale, so a triangle that looks equilateral or right-angled is only so if the problem says so.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Memorise the two special right triangles cold: 45-45-90 is 1 : 1 : √2, and 30-60-90 is 1 : √3 : 2 (short : long : hypotenuse). They replace all the trig the GRE deliberately leaves out.
- Learn the Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples so you can read the third side without squaring anything.
- Anchor every 30-60-90 to the short leg: hypotenuse = 2 × short, long leg = short × √3.
- The height of an equilateral triangle of side s is (s√3)/2 — it is just the long leg of a 30-60-90.
- Ratio of areas of similar triangles is the SQUARE of the side ratio; ratio of perimeters is the plain side ratio.
- Shadow, ramp and "reflection" word problems are almost always similar-triangle proportions — set up matching sides over matching sides.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using a slanted side as the "height" in the area formula. The height must be perpendicular to the chosen base.
- Assuming a triangle is right-angled, isosceles or equilateral because it looks that way. GRE figures are not drawn to scale.
- Squaring the side ratio to get perimeters (it stays linear) or forgetting to square it for areas (it must be squared).
- Mixing up the 30-60-90 sides — putting √3 opposite the 30° instead of the 60°. The short leg (×1) is always opposite the 30°.
- Forgetting the triangle inequality when choosing a third side, so an "impossible" triangle slips through.
- Calling the hypotenuse a leg. The hypotenuse is opposite the right angle and is the longest side.
📈 GRE exam insight & question patterns
Quantitative Comparison — a right triangle has legs 6 and 8. Quantity A: the length of the hypotenuse. Quantity B: 10. Which is greater?
Equal. 6² + 8² = 100, so the hypotenuse is exactly 10 (the 3-4-5 triple doubled).
A GRE figure shows a triangle that looks equilateral but only marks two 60° angles. Is the third side equal to the others?
Two 60° angles force the third to be 60°, so it truly is equilateral — but you conclude that from the angle sum, not from the drawing, which is not to scale.
Numeric Entry — an equilateral triangle has a side of 10. Enter its height in the form a√3.
Height = (s√3)/2 = (10√3)/2 = 5√3.
Two triangles are similar with side ratio 1 : 3. How many times larger is the bigger triangle’s area?
Area scales with the square of the side ratio: 3² = 9 times larger.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Triangles when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (6 topics) | 6/6 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 10 cards |
Formula Reference Sheet
Angles & sides
| Angle sum of a triangle | a + b + c = 180° |
|---|---|
| Exterior angle | = sum of the two remote interior angles |
| Triangle inequality | |a − b| < third side < a + b |
| Isosceles | equal sides ⇒ equal base angles |
Right triangles
| Pythagorean theorem | a² + b² = c² (c = hypotenuse) |
|---|---|
| Common triples | 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and multiples) |
| 45-45-90 sides | 1 : 1 : √2 (leg : leg : hypotenuse) |
| 30-60-90 sides | 1 : √3 : 2 (opposite 30° : 60° : 90°) |
Area
| Any triangle | Area = ½ × base × height |
|---|---|
| Equilateral, side s | Area = (√3 / 4) × s² |