GRE® Geometry and Solid Geometry

Concepts Tested from Geometry in GRE

Properties of lines, rays, line sgement, parallel lines, angles, angle between intersecting lines, polygons, sum of interior angles of polygons, number of diagonals in a polygon, convex and concave polygons. properties of triangles, median, altitude, perpendicular bisector, angle bisector, circles, chords, tangent, quadrilaterals, tangent secant theorem, inscribed circle and circumscribing circles.

Concepts Tested in Mensuration (Solid Geometry)

Area of triangles, circles, quadrilaterals (including parallelogram, rhombus, rectangle, square, trapezium, and kite) and regular hexagon, surface area and volume of cube, cylinder, cone, sphere and hemisphere, length of the longest diagonal in a cube and cuboid (rectangular cube), volume of water displaced in a vessel when another object is dropped into it, and volume of solids formed by rotation of 2D shapes. A relatively easy topic. Key to success in this topic is to remember all the formulae.

1. A sector of a circle of radius 5 cm is recast into a right circular cone of height 4 cm. What is the volume of the resulting cone?

   1. 12 π cm³
   2. 100 π cm³
   3. 33 π cm³
   4. 32 π cm³
   5. 4 π cm³

   Correct AnswerChoice A
   12 π cm³

   Explanation Video Solution
   Approach to solve this Geometry Problem solving question

   Step 1: The radius of the sector will be the slant height of the cone.
   Step 2: The height of the cone, slant height of the cone and the radius form a right triangle in which the slant height is the hypotenuse.
   Step 3: Using data about slant height and height of the cone, compute radius of cone.
   Step 4: Compute volume of cone.

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2. A cylindrical vessel is filled with water up to some height. If a sphere of diameter 8 cm is dropped into the cylinder, the water level rises by half of the initial level. Instead, if a sphere of diameter 16 cm is dropped, the water level rises to a height h₂. What percentage of this new height h₂ is the initial level of water?

   1. 33\\frac{1}{3})%
   2. 40%
   3. 20%
   4. 25%
   5. 16\\frac{1}{6})%

   Correct AnswerChoice C
   20%

   Explanation Video Solution
   Approach to solve this Solid Geometry Problem

   Step 1: Assign variables for the initial height of water in the cylindrical vessel and the height to which water rises after sphere 1 is dropped.
   Step 2: Compute volume of sphere 1 and equate it to the increase in volume of water in the cylindrical vessel
   Step 3: Compute volume of sphere 2 and equate it to the increase in volume of water in the cylindrical vessel
   Step 4: Solve equations obtained in steps 2 and 3 to find the relation between h₂ and the initial level of water in the vessel.

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3. Chord AC at a distance of 7 cm from the center of a circle subtends an angle of 120 degrees at the center. What is the area of major segment?

   1. 410\\frac{1}{3})+ 7\\sqrt{3}) cm²
   2. 410\\frac{1}{3})+ 49\\sqrt{3}) cm²
   3. 1232 + 49\\sqrt{3}) cm²
   4. 1232 + 7\\sqrt{3}) cm²
   5. 410\\frac{2}{3})+ 49\\sqrt{3}) cm²

   Correct AnswerChoice E
   410\\frac{2}{3})+ 49√3 cm²

   Explanation Video Solution
   Steps to solve this GRE Geometry - Circles Problem

   Step 1: Because the angle subtended by the chord at the center of the circle is 120°, we have a 30 - 60 - 90 right triangle.
   Step 2: Using ratio of measure of sides of a 30 - 60 - 90 right triangle, compute hypotenuse of the triangle, which is the radius of the circle, and the base of the triangle.
   Step 3: Compute are of minor sector.
   Step 4: Compute are of triangle AOC and subtract it from area of minor sector to compute area of minor segment.
   Step 5: Area of major segment = Area of circle - area of minor segment

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4. The area for which of the following will necessarily be more than 50 square units.
   Indicate all such expressions

   1. Circle whose circumference is 22 units
   2. Parallelogram whose adjacent sides measure 20 units and 10 units.
   3. Rhombus whose perimeter is 52 units.
   4. Rectangle whose perimeter is 50 units.
   5. Square whose perimeter is 32 units.
   6. Right triangle whose hypotenuse measures 17 units.

   Correct AnswerChoice E

   Explanation Video Solution
   Approach to solve this GRE Select One or More Answer Choices Question

   Compute the area of each of the six 2D shapes given and mark all those which have an area greater than 50 square units.

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