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Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry Syllabus

The Quantitative Reasoning: Geometry syllabus for Graduate Record Examinations (GRE) in full: 4 chapters, 12 topics and 21 sub-topics, every one listed below, plus 50 Quantitative Reasoning: Geometry flashcards with answers.

4Chapters
12Topics
21Sub-topics
50Flashcards

Quantitative Reasoning: Geometry syllabus: all 4 chapters

Expand any chapter to see its topics and sub-topics.

  1. Lines, Angles and Triangles

     3 topics
    • Angle relationships
      • Parallel lines and transversals
      • Vertical and supplementary angles
    • Triangle properties
      • Angle-sum and triangle inequality
      • Similar and congruent triangles
    • Right triangles
      • Pythagorean theorem
      • Special right triangles (45-45-90, 30-60-90)

  2. Polygons and Circles

     3 topics
    • Quadrilaterals and polygons
      • Area and perimeter formulas
      • Interior and exterior angle sums
    • Circle properties
      • Circumference and area
      • Arcs, sectors and chords
      • Inscribed and central angles
    • Composite and shaded-region figures
      • Decomposing complex shapes
      • Finding shaded areas

  3. Three-Dimensional Geometry

     3 topics
    • Surface area and volume of solids
      • Rectangular solids and cubes
      • Cylinders
    • Diagonals and 3D distance
      • Space diagonal of a box
    • Visualising and unfolding solids
      • Nets of common solids

  4. Coordinate Geometry

     3 topics
    • Distance and midpoint
      • Distance formula
      • Midpoint formula
    • Equations of circles and lines
      • Standard circle equation
    • Slope, intercepts and graph interpretation
      • Identifying coordinates from graphs

Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry flashcards — 50 questions with answers

  1. When two lines intersect, what is the relationship between the two pairs of opposite (vertical) angles?

    Vertical angles are equal (congruent). The two angles directly across from each other at the intersection have the same measure.

  2. What is the sum of two angles that form a linear pair (supplementary angles on a straight line)?

    $180^{\circ}$. Angles that combine to a straight line are supplementary, so they sum to $180^{\circ}$.

  3. Define complementary angles and give their sum.

    Two angles are complementary if their measures add to $90^{\circ}$.

  4. When a transversal crosses two parallel lines, what is true of alternate interior angles and of corresponding angles?

    Both pairs are equal. Alternate interior angles are congruent and corresponding angles are congruent; co-interior (same-side interior) angles are supplementary, summing to $180^{\circ}$.

  5. What is the sum of the interior angles of any triangle?

    $180^{\circ}$.

  6. State the Exterior Angle Theorem for a triangle.

    An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

  7. In any triangle, how does the length of a side relate to the angle opposite it?

    The largest angle lies opposite the longest side and the smallest angle lies opposite the shortest side; equal sides lie opposite equal angles.

  8. State the Triangle Inequality Theorem.

    The sum of the lengths of any two sides of a triangle must be greater than the length of the third side: $a + b > c$ for all sides.

  9. What are the side and angle properties of an equilateral triangle?

    All three sides are equal and all three interior angles equal $60^{\circ}$.

  10. What defines an isosceles triangle and what follows about its angles?

    It has at least two equal sides; the angles opposite those equal sides (the base angles) are also equal.

  11. What is the formula for the area of a triangle with base $b$ and height $h$?

    $$A = \frac{1}{2}bh$$

  12. State the Pythagorean Theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$.

    $$a^{2} + b^{2} = c^{2}$$

  13. List three common Pythagorean triples used to identify right triangles quickly.

    $(3,4,5)$, $(5,12,13)$, and $(8,15,17)$, along with their multiples such as $(6,8,10)$.

  14. What are the side ratios of a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle?

    The sides are in ratio $1 : 1 : \sqrt{2}$ (leg : leg : hypotenuse).

  15. What are the side ratios of a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle?

    The sides are in ratio $1 : \sqrt{3} : 2$ (side opposite $30^{\circ}$ : side opposite $60^{\circ}$ : hypotenuse).

  16. In a right triangle, what is the relationship between the altitude to the hypotenuse and the two segments it creates?

    The altitude is the geometric mean of the two hypotenuse segments: $h = \sqrt{p \cdot q}$, where $p$ and $q$ are the segment lengths.

  17. What is the sum of the interior angles of any quadrilateral?

    $360^{\circ}$.

  18. List the defining properties of a parallelogram.

    Opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other.

  19. What additional properties does a rectangle have beyond those of a parallelogram?

    All four angles are $90^{\circ}$ and the diagonals are equal in length (and still bisect each other).

  20. What distinguishes a rhombus from a general parallelogram?

    All four sides are equal, and its diagonals are perpendicular bisectors of each other and bisect the vertex angles.

Show the other 30 Quantitative Reasoning: Geometry flashcards

  1. What is the area of a trapezoid with parallel sides $b_1$ and $b_2$ and height $h$?

    $$A = \frac{1}{2}(b_1 + b_2)h$$

  2. What is the formula for the sum of interior angles of a polygon with $n$ sides?

    $$S = (n - 2)\times 180^{\circ}$$

  3. What is the measure of each interior angle of a regular polygon with $n$ sides?

    $$\text{Interior angle} = \frac{(n-2)\times 180^{\circ}}{n}$$

  4. What is the sum of the exterior angles of any convex polygon, one per vertex?

    $360^{\circ}$, regardless of the number of sides.

  5. What is the area of a rhombus in terms of its diagonals $d_1$ and $d_2$?

    $$A = \frac{1}{2}d_1 d_2$$

  6. Give the formulas for the circumference and area of a circle of radius $r$.

    Circumference $C = 2\pi r$ and area $A = \pi r^{2}$.

  7. What is the relationship between a central angle and the inscribed angle subtending the same arc?

    The central angle is twice the inscribed angle that subtends the same arc: $\theta_{\text{central}} = 2\,\theta_{\text{inscribed}}$.

  8. What is the measure of an inscribed angle that subtends a diameter (semicircle)?

    $90^{\circ}$. An angle inscribed in a semicircle is always a right angle (Thales' theorem).

  9. How do you find the arc length of a sector with central angle $\theta$ (in degrees) in a circle of radius $r$?

    $$\text{Arc length} = \frac{\theta}{360^{\circ}}\times 2\pi r$$

  10. What is the area of a sector with central angle $\theta$ (in degrees) and radius $r$?

    $$A = \frac{\theta}{360^{\circ}}\times \pi r^{2}$$

  11. What is the relationship between a radius drawn to the point of tangency and the tangent line?

    The radius is perpendicular to the tangent line at the point of tangency, forming a $90^{\circ}$ angle.

  12. What is the general strategy for finding the area of a shaded region in a composite figure?

    Find the area of the larger/outer figure and subtract the area of the unshaded inner figure(s): Shaded $=$ Total area $-$ Unshaded area.

  13. How do you find the area of a composite figure made of several simple shapes?

    Decompose it into non-overlapping basic shapes (rectangles, triangles, semicircles), compute each area, then add them together.

  14. What is the surface area of a rectangular solid (box) with dimensions $l$, $w$, and $h$?

    $$SA = 2(lw + lh + wh)$$

  15. What is the volume of a rectangular solid with dimensions $l$, $w$, and $h$?

    $$V = lwh$$

  16. Give the formulas for the volume and surface area of a sphere of radius $r$.

    Volume $V = \frac{4}{3}\pi r^{3}$ and surface area $SA = 4\pi r^{2}$.

  17. What are the volume and lateral (curved) surface area of a right circular cylinder with radius $r$ and height $h$?

    Volume $V = \pi r^{2}h$; lateral surface area $= 2\pi r h$; total surface area $= 2\pi r h + 2\pi r^{2}$.

  18. What are the volume and lateral surface area of a right circular cone with radius $r$, height $h$, and slant height $l$?

    Volume $V = \frac{1}{3}\pi r^{2}h$; lateral surface area $= \pi r l$, where $l = \sqrt{r^{2}+h^{2}}$.

  19. What is the formula for the length of the space diagonal of a rectangular box with dimensions $l$, $w$, $h$?

    $$d = \sqrt{l^{2} + w^{2} + h^{2}}$$

  20. What is the length of the space diagonal of a cube with edge length $s$?

    $$d = s\sqrt{3}$$

  21. What is the length of the face diagonal of a cube with edge length $s$?

    $$d = s\sqrt{2}$$

  22. When a cube is unfolded into a net, how many faces, edges, and vertices does the cube have?

    A cube has 6 (square) faces, 12 edges, and 8 vertices. A valid net shows all 6 squares connected so they fold without overlap.

  23. What does Euler's formula state for the faces, vertices, and edges of a convex polyhedron?

    $$V - E + F = 2$$ where $V$ = vertices, $E$ = edges, $F$ = faces.

  24. State the distance formula between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane.

    $$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$$

  25. State the midpoint formula for the segment joining $(x_1, y_1)$ and $(x_2, y_2)$.

    $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

  26. What is the standard equation of a circle with center $(h, k)$ and radius $r$?

    $$(x - h)^{2} + (y - k)^{2} = r^{2}$$

  27. How do you find the slope of the line through points $(x_1, y_1)$ and $(x_2, y_2)$?

    $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

  28. What is the slope-intercept form of a line, and what does each variable represent?

    $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept (the value of $y$ where the line crosses the $y$-axis).

  29. What is the relationship between the slopes of two parallel lines versus two perpendicular lines?

    Parallel lines have equal slopes ($m_1 = m_2$); perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).

  30. How do you find the $x$-intercept and $y$-intercept of a line from its equation?

    For the $x$-intercept, set $y = 0$ and solve for $x$; for the $y$-intercept, set $x = 0$ and solve for $y$.

Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry FAQ

What is in the Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry syllabus?

Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry is split into 4 chapters: Lines, Angles and Triangles, Polygons and Circles, Three-Dimensional Geometry and Coordinate Geometry. Together they contain 12 topics and 21 sub-topics.

How many chapters are in Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry?

4 chapters, containing 12 topics and 21 sub-topics. The largest chapter is Lines, Angles and Triangles, with 3 topics.

Are there flashcards for Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry?

Yes — a 50-card Quantitative Reasoning: Geometry deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.