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intermediate20 min

Quantitative Reasoning: Geometry

Applying core geometry formulas correctly, and why GRE figures are usually not drawn to scale.

What you'll learn

  • Apply core geometry formulas -- area, perimeter, volume, and the Pythagorean theorem -- accurately
  • Recognize that GRE figures are generally not drawn to scale and must not be estimated visually
  • Apply triangle and circle properties to original practice problems

Explanation

Geometry questions cover lines and angles, triangles (including the Pythagorean theorem and special right triangles), circles, quadrilaterals, three-dimensional figures (surface area and volume), and coordinate geometry. Formal proofs are not required -- the skill being tested is correctly applying known formulas and relationships to the specific figure or values given.

The single most important GRE-specific habit is remembering that geometric figures are generally not drawn to scale unless a question explicitly states otherwise. You cannot estimate a length or angle by how it visually appears in the figure -- every measurement must come from the values, relationships, or labels explicitly given in the problem. This trips up test-takers who are used to trusting diagrams at face value in other contexts.

Worked original example: a right triangle has legs of length 6 and 8. Using the Pythagorean theorem, a^2 + b^2 = c^2, the hypotenuse is sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. This is a scaled-up version of the well-known 3-4-5 right triangle (multiplied by 2), which is worth recognizing on sight to save calculation time.

A second worked example: a circle has a radius of 5. Its area is pi * r^2 = pi * 25 = 25*pi, and its circumference is 2 * pi * r = 10*pi. Mixing up area and circumference formulas under time pressure is a common, avoidable error -- area uses the radius squared, circumference does not.

Common mistakes

  • Estimating a length or angle by how a figure visually looks, instead of remembering that GRE figures are generally not drawn to scale unless stated.
  • Mixing up the formulas for a circle's area (pi r^2) and its circumference (2 pi r) under time pressure.
  • Assuming the Pythagorean theorem applies to any triangle, when it only applies directly to right triangles.

Knowledge check

Knowledge check

1. On the GRE, if a geometric figure is not explicitly labeled as drawn to scale, what should you assume?
2. A right triangle has legs of length 6 and 8. What is the length of its hypotenuse?
3. A circle has a radius of 5. What is its area, in terms of pi?
4. The Pythagorean theorem, a^2 + b^2 = c^2, applies directly to:

Takeaway

Never estimate a length or angle from how a GRE figure looks -- calculate strictly from given values, since figures are generally not drawn to scale.

Summary

Geometry questions test correct application of standard formulas (Pythagorean theorem, area, circumference) to given values, with the GRE-specific caution that figures are usually not drawn to scale.

References

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