GRE Triangles
Triangles are the most heavily tested topic in GRE geometry, and the right triangle is at the center of it. Master a handful of ratios and you can solve most of them without touching trigonometry. Drag the tool below to see the Pythagorean theorem hold as the triangle changes.
The Pythagorean theorem
In a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). This one relationship unlocks distances, diagonals, and half of coordinate geometry. Drag the legs and watch the hypotenuse and the equation update.
Pythagorean theorem
A right triangle has legs of length 3 and 4. How long is the hypotenuse (the longest side)?
The two special right triangles
These fixed side ratios appear constantly. Memorize both and you skip the algebra:
- 45-45-90: sides in ratio x : x : x√2. The isosceles right triangle; also the diagonal of a square.
- 30-60-90: sides in ratio x : x√3 : 2x, with x opposite the 30° angle. Also half of an equilateral triangle.
Facts that show up again and again
- Angles sum to 180°. Two given angles fix the third.
- Area = ½ × base × height, with height perpendicular to the base.
- Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples.
- Third-side rule: |a − b| < c < a + b for any triangle.
- Bigger angle, longer opposite side. The order of angles matches the order of sides.
Frequently asked questions
What are the special right triangles on the GRE?
Two of them. The 45-45-90 triangle has sides in ratio x : x : x√2. The 30-60-90 triangle has sides in ratio x : x√3 : 2x, where x is opposite the 30° angle. Memorizing these two ratios saves you from ever needing trigonometry.
What Pythagorean triples should I memorize?
The most common are 3-4-5, 5-12-13, 8-15-17, and 7-24-25, plus any multiple of them (like 6-8-10). Spotting a triple lets you find the third side instantly instead of squaring and taking a root.
What is the third-side rule?
In any triangle, the length of one side must be greater than the difference of the other two and less than their sum: |a − b| < c < a + b. GRE questions love asking which lengths could form a triangle.
How do you find the area of a triangle?
Area = ½ × base × height, where the height is perpendicular to the base. For a right triangle the two legs are the base and height, so the area is ½ × leg × leg.
Practice GRE geometry, adaptively
Grezi drills you on triangles, circles, and the whole geometry set with real GRE questions and an AI tutor that finds your blind spots.
More GRE Quant: all topics, coordinate geometry.