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Quantitative Reasoning: Geometry

Every geometry fact ETS lists — lines, triangles, circles, polygons, solids — plus what a GRE figure is and is not allowed to tell you.

21 topics · 92 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Figures Not Drawn to Scale, and What You May Assume

  1. How it worksHow does the ETS rule split a geometric figure into things you may assume and things you may not?
  2. How it worksWalk me through what stays true in a GRE figure: straight lines, point order, relative position.
  3. Common mistakeWhy doesn't an angle that looks like 90 degrees in a GRE figure count as a right angle?
  4. Going furtherWhat changes when the picture is a coordinate plane or a bar graph instead of a bare triangle?
Your turnhard1 of 3
A figure shows triangle PQR with PQ = 7 and QR = 14, and the angle at Q looks like a right angle. Quantity A is the length of PR. Quantity B is √(72 + 142).

Why the Construction of Proofs Is Not Tested

  1. How it worksHow does a GRE geometry question turn a theorem into a number instead of an argument?
  2. Common mistakeWhy isn't memorizing two-column proof steps any use to you on GRE test day?
  3. Worked exampleWalk me through getting an inscribed angle from a 140 degree arc without writing a proof step.
  4. Going furtherWhat still has to be justified in your head even when no proof gets written down?
Your turneasy1 of 3
A student memorizes 10 two-column proof formats before a GRE. How much of that will the Quant section ask for?

Parallel and Perpendicular Lines and the Transversal

  1. How it worksHow does sliding one intersection along the transversal onto the other match all the angles up?
  2. How it worksWalk me through vertical, corresponding, alternate interior and same-side interior angles at a glance.
  3. Worked exampleFind all eight angles when parallel lines are cut by a transversal making one angle of 65 degrees.
  4. Common mistakeWhy isn't the alternate interior angles rule true when the two lines are not parallel?
  5. Going furtherWhat happens to the eight angles as one of the two lines rotates 5 degrees out of parallel?
Your turneasy1 of 3
Two parallel lines are cut by a transversal making one angle of 112 degrees. Find the measure of the same-side interior angle next to it, in degrees.

The Triangle Angle Sum and Exterior Angles

  1. How it worksHow does tearing off a triangle's three corners and laying them side by side make 180 degrees?
  2. How it worksHow does an exterior angle end up equal to the two interior angles it does not touch?
  3. Worked exampleFind the third angle and the exterior angle at that vertex when a triangle has angles 47 and 68.
  4. Common mistakeWhy isn't a triangle able to hold two obtuse angles at the same time?
Your turneasy1 of 3
A triangle has interior angles of 58 and 66 degrees. Find the exterior angle at the third vertex, in degrees.

The Triangle Inequality

  1. How it worksHow does the third side get squeezed between the difference and the sum of the other two?
  2. Worked exampleFind every possible integer length for the third side of a triangle with sides 7 and 11.
  3. Common mistakeWhy isn't a third side of exactly 5 allowed when the other two sides are 2 and 3?
  4. Going furtherWhat happens to the range of the third side as the two known sides get closer in length?
Your turnmedium1 of 3
A triangle has sides of length 10 and 18. Find the number of possible integer lengths for the third side.

Isosceles and Equilateral Triangles

  1. How it worksHow does folding an isosceles triangle along its axis land one half exactly on the other?
  2. Worked exampleFind the base angles of an isosceles triangle whose apex angle measures 104 degrees.
  3. Worked exampleFind the height and the area of an equilateral triangle with side length 10.
  4. Common mistakeWhy isn't every triangle with two equal angles automatically equilateral?
Your turneasy1 of 3
Find the measure of a base angle, in degrees, of an isosceles triangle whose apex angle is 114 degrees.

The Pythagorean Theorem and Common Triples

  1. How it worksWalk me through the rearrangement where four copies of a right triangle tile a larger square.
  2. Worked exampleFind the hypotenuse of a right triangle with legs 9 and 12, and the triple it scales up from.
  3. Worked exampleFind the missing leg of a right triangle whose hypotenuse is 26 and whose other leg is 10.
  4. Common mistakeWhy doesn't a 5-12-13 triangle stay a right triangle once you add 1 to every side?
  5. Going furtherWhat happens to a squared plus b squared when the angle between the legs opens past 90 degrees?
Your turneasy1 of 3
A right triangle has hypotenuse 33 and one leg 12. Find the other leg.

45-45-90 and 30-60-90 Right Triangles

  1. How it worksHow does a 30-60-90 triangle end up with sides in the ratio 1 to root 3 to 2?
  2. How it worksHow does a square's diagonal turn every 45-45-90 triangle into 1 to 1 to root 2?
  3. Worked exampleFind the other two sides of a 30-60-90 triangle whose shortest side has length 6.
  4. Worked exampleFind the diagonal of a square whose area is 50.
  5. Common mistakeWhy isn't the side opposite the 60 degree angle twice the side opposite the 30 degree angle?
Your turnmedium1 of 3
A 30-60-90 triangle has its shortest side equal to 15. Find the length of the side opposite the 60 degree angle.

Triangle Area and the Altitude

  1. How it worksHow does half of base times height come straight out of a rectangle cut along its diagonal?
  2. How it worksHow does an altitude land outside the triangle entirely when one of the angles is obtuse?
  3. Worked exampleFind the area of a triangle with base 15 whose opposite vertex sits 6 units off that base.
  4. Going furtherWhat happens to the area as the top vertex slides along a line parallel to the base?
Your turneasy1 of 3
Find the area of a triangle with base 23 whose opposite vertex sits 14 units off that base.

Congruent Figures and the Congruence Criteria

  1. How it worksHow do side-side-side, side-angle-side and angle-side-angle each pin a triangle down completely?
  2. Common mistakeWhy isn't side-side-angle enough to force two triangles to be congruent?
  3. Worked exampleFind which congruence rule settles two triangles with sides 8 and 5 and a 40 degree angle between them.
  4. Going furtherWhat changes about congruence when one triangle is reflected instead of just slid and turned?
Your turnhard1 of 3
Two triangles share two sides of lengths 8 and 14 and a 58 degree angle that is NOT between those sides. Are they congruent?

Similar Figures and the Scale Factor

  1. How it worksHow does one triangle grow into a similar one with every angle pinned and every side times k?
  2. Worked exampleFind x when a triangle with sides 6, 9 and 12 is similar to one with sides 10, 15 and x.
  3. Worked exampleFind a flagpole's height from its 24 foot shadow when a 6 foot person casts a 4 foot shadow.
  4. Common mistakeWhy isn't a 2 by 4 rectangle similar to a 4 by 6 rectangle even though both sides grew by 2?
Your turneasy1 of 3
A triangle with sides 7, 11 and 20 is similar to a triangle whose shortest side is 26. Find the side of the second triangle corresponding to 20.

How Area and Volume Scale with Length

  1. How it worksHow does a scale factor of k become k squared for area and k cubed for volume?
  2. Worked exampleFind the area of a triangle similar to one of area 18 when its sides are 5 times as long.
  3. Common mistakeWhy doesn't a circle with twice the circumference enclose twice the area?
  4. Going furtherWhat happens to the surface-area-to-volume ratio of a cube as the cube gets bigger?
  5. Ties togetherHow does the k cubed rule connect a scaled-up cylinder to a scaled-up rectangular box?
Your turneasy1 of 3
A triangle has area 31 square centimeters. A similar triangle is built with every side k times as long. Write, in terms of k, the area of the larger triangle in square centimeters.

Quadrilaterals: Square, Rectangle, Parallelogram, Trapezoid, Rhombus

  1. How it worksHow do square, rectangle, rhombus, parallelogram and trapezoid nest inside one another?
  2. How it worksHow does a trapezoid's area formula come from averaging its two parallel sides?
  3. Worked exampleFind the area of a trapezoid whose parallel sides are 9 and 15 and whose height is 6.
  4. Worked exampleFind the area of a rhombus whose diagonals measure 10 and 24.
  5. Common mistakeWhy doesn't a parallelogram's area equal the product of its two side lengths?
Your turneasy1 of 3
A trapezoid has parallel sides of length 10 and 20 and a height of 11. Find its area.

Polygon Angle Sums and Regular Polygons

  1. How it worksHow does fanning diagonals from one vertex cut an n-sided polygon into n minus 2 triangles?
  2. How it worksHow do the exterior angles of any convex polygon always come out to 360 degrees in total?
  3. Worked exampleFind each interior angle and each exterior angle of a regular decagon.
  4. Common mistakeWhy isn't a polygon's interior angle sum just 180 times its number of sides?
Your turneasy1 of 3
Find the sum, in degrees, of the interior angles of a convex polygon with 15 sides.

Circles: Radius, Diameter, Chord, Circumference and Area

  1. How it worksHow does unrolling a circle's edge into a straight segment show circumference equals 2 π r?
  2. How it worksHow does slicing a circle into thin wedges and interleaving them build the π r squared area?
  3. Worked exampleFind the circumference and the area of a circle whose radius is 5.2.
  4. Common mistakeWhy doesn't a chord that seems to pass near the middle have to be a diameter?
Your turnhard1 of 3
A circle has circumference 45. Find its area.

Arcs, Sectors and Central Angles

  1. How it worksHow does the fraction of 360 degrees set both the arc length and the sector area at once?
  2. Worked exampleFind the arc length and sector area for a 50 degree central angle in a circle of radius 5.
  3. Worked exampleFind the central angle of a sector whose area is 6 π inside a circle of radius 6.
  4. Common mistakeWhy doesn't a 60 degree arc have a length of 60 in a circle of radius 10?
  5. Going furtherWhat happens to arc length and sector area when the radius triples but the angle stays at 40 degrees?
Your turneasy1 of 3
A circle has radius 10. Find the length of the arc cut off by a central angle of 108 degrees.

Inscribed Angles and the Angle in a Semicircle

  1. How it worksHow does an inscribed angle end up exactly half of the central angle on the same arc?
  2. How it worksHow does a triangle with one side lying on the diameter always come out with a right angle?
  3. Worked exampleFind the inscribed angle standing on a 140 degree arc, and the central angle on that same arc.
  4. Common mistakeWhy isn't an inscribed angle the same number of degrees as the arc it stands on?
  5. Going furtherWhat happens to the inscribed angle when its vertex crosses over to the other arc of the circle?
Your turneasy1 of 3
An inscribed angle of a circle stands on an arc measuring 154 degrees. Find the measure of that inscribed angle in degrees.

Tangent Lines and the Point of Tangency

  1. How it worksHow does a secant cutting a circle twice become a tangent as its two crossing points merge?
  2. Worked exampleFind the tangent length from a point 13 units from the center of a circle of radius 5.
  3. Common mistakeWhy isn't a tangent line perpendicular to every radius rather than just one of them?
  4. Going furtherWhat happens to the two tangent lines from an outside point as that point moves toward the circle?
Your turnmedium1 of 3
A point P lies 32 units from the center of a circle of radius 10. Find the length of the tangent segment from P to the circle.

Rectangular Solids: Volume and Surface Area

  1. How it worksHow does unfolding a box into six flat rectangles turn surface area into 2lw plus 2lh plus 2wh?
  2. Worked exampleFind the volume and the surface area of a rectangular solid measuring 8.5 by 5 by 10.
  3. Going furtherWhat happens to volume and to surface area when every edge of a cube is tripled?
  4. Common mistakeWhy doesn't the box holding the most volume also have the largest surface area?
Your turneasy1 of 3
A rectangular solid measures 12 by 7 by 11. Find its surface area.

The Diagonal of a Rectangular Box

  1. How it worksWalk me through stacking two Pythagorean theorems to reach the space diagonal of a box.
  2. Worked exampleFind the length of the space diagonal of a rectangular box measuring 3 by 4 by 12.
  3. Common mistakeWhy isn't a box's space diagonal simply the longest of its three face diagonals?
  4. Ties togetherHow does a cube's space diagonal tie the 45-45-90 triangle to a factor of root 3?
Your turneasy1 of 3
A rectangular box measures 7 by 9 by 15. Find the length of its space diagonal.

Right Circular Cylinders

  1. How it worksHow does unrolling a can's label into a flat rectangle give a lateral surface of 2 π r h?
  2. Worked exampleFind the volume and surface area of a right circular cylinder with radius 3 and height 6.5.
  3. Worked exampleFind the height of a cylinder whose volume is 100 π and whose radius is 5.
  4. Common mistakeWhy doesn't doubling a cylinder's radius double the amount the cylinder holds?
  5. Going furtherWhat happens to the volume when a cylinder's height is halved and its radius is doubled?
Your turneasy1 of 3
A right circular cylinder has radius 7 and height 11. Find its volume.

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