GRE Solids and Volume

GRE Geometry tests three-dimensional solids inside one content area with lines, circles, triangles, quadrilaterals, other polygons and the Pythagorean theorem, per ETS's Quantitative Reasoning content-area list. The GRE Math Review's Geometry chapter lists six basic solids, including rectangular solids, cubes, cylinders, spheres, pyramids and cones, and it works through volume and surface area formulas for only two of them. The two with worked formulas are the rectangular solid and the right circular cylinder. The Math Review does not work through the sphere, pyramid and cone formulas, so know them yourself rather than relying on the question to supply them.

This guide lists volume and surface area formulas for the six solids ETS names, walks through composite and inscribed-solid problems, and covers the unit-conversion trap which turns a correct setup into a wrong answer. Worked examples run through each formula and setup. Practice the topic with our free Solids and Volume drills at three difficulty levels, no signup required.

Where Solids and Volume Sit in GRE Geometry

GRE Quantitative Reasoning organizes content into four areas: arithmetic, algebra, geometry and data analysis. Read the full breakdown in our GRE Quantitative Reasoning overview. The Geometry content area covers three-dimensional figures, area, perimeter and volume, alongside lines, circles, triangles, quadrilaterals, other polygons and the Pythagorean theorem.

Our guide to circles, quadrilaterals and polygons covers the flat-shape half of GRE Geometry. ETS's Math Review lists rectangular solids, cubes, cylinders, spheres, pyramids and cones as basic three-dimensional figures, and the chapter works through volume and surface area formulas for only two: the rectangular solid and the right circular cylinder. Solids appear in section 3.6, Three-Dimensional Figures, and they reuse the area and circumference results from the flat shapes. A cylinder's volume, for example, is the area of its circular base multiplied by its height.

Volume and Surface Area Formulas for the Basic Solids

The table below lists volume and surface area for the six solids the Math Review names. The rectangular solid and cylinder rows come directly from ETS, and the cube rows follow from the rectangular solid, since a cube has equal length, width and height. The cone, sphere and pyramid rows follow standard geometry and are worth memorizing, since the review chapter does not work through them. Each letter keeps one meaning across the table: l, w and h are length, width and height, s is a cube's side length, and r is the radius of a circular base. A cone's slant height is the straight-line distance from the tip to the edge of the base, which is longer than the vertical height h.

Rectangular Solids and Cubes

A rectangular solid's volume equals length times width times height. ETS gives the formula as V = lwh, with surface area A = 2(lw + lh + wh), the sum of the areas of the three pairs of matching faces, doubled. A cube is a rectangular solid with equal length, width and height, so its volume simplifies to the side length cubed, and its surface area simplifies to six times the side length squared.

Take a rectangular solid with length 6, width 4 and height 3. Its volume is 6 times 4 times 3, or 72. Its surface area is 2 times the sum of (6 times 4), (6 times 3) and (4 times 3), or 2 times (24 + 18 + 12), which totals 108.

Right Circular Cylinders

ETS gives a right circular cylinder's volume as V = πr²h and its surface area as A = 2πr² + 2πrh, where r is the radius of the circular base and h is the height. The two circular ends contribute 2πr² to the surface area, and the curved side contributes 2πrh, the circumference of the base times the height. Picture the curved side unrolled flat: it becomes a rectangle with width 2πr and height h. Keep the radius and the diameter apart, because a figure labeled with a diameter of 6 has a radius of 3.

Take a cylinder with radius 3 and height 10. Its volume is π times 3 squared times 10, or 90π. Its surface area is 2π times 9, plus 2π times 3 times 10, or 18π + 60π, which totals 78π.

Cones, Spheres and Pyramids

ETS's Math Review names spheres, pyramids and cones as basic three-dimensional figures without working through their formulas in the chapter. Know the standard formulas below rather than relying on the question to supply them. Each one is short, and a single practice session is enough to fix all three in memory. For the official wording and the rest of the Geometry chapter, read ETS's Math Review PDF directly.

A cone's volume is one third times pi times the radius squared times the height: V = (1/3)πr²h. A sphere's volume is four thirds times pi times the radius cubed: V = (4/3)πr³, with surface area A = 4πr². A pyramid with a rectangular base has volume one third times the base's length times width times height: V = (1/3)lwh.

Take a sphere with radius 6. Its volume is four thirds times pi times 6 cubed, or four thirds times pi times 216, which is 288π. Take a cone with radius 3 and height 8. Its volume is one third times pi times 9 times 8, or 24π.

Composite and Inscribed-Solid Problems

Composite solids combine two or more basic shapes into one figure. Inscribed solids fit one shape exactly inside another, sharing a measurement such as a radius or side length. Both problem types build on the formulas above, so identify the shapes first, then apply the right formula to each.

Composite Solids: Add or Subtract Volume

A composite solid's volume is the sum of its parts, or the whole minus a missing piece. Look at the figure's outline before picking a route. A solid built from stacked or attached shapes needs addition. A solid with a piece removed, such as a hole or notch, needs subtraction.

• Identify each basic solid inside the figure.
• Find the volume of each solid using the formulas above.
• Add the volumes for a combined figure, or subtract a removed piece from a whole figure.

Take a solid made from a rectangular box, 8 by 5 by 4, with a cylinder of radius 2 and height 4 attached on top. The box's volume is 8 times 5 times 4, or 160. The cylinder's volume is π times 2 squared times 4, or 16π. The combined solid's volume is 160 + 16π.

Take a cube with side 10, with a cylindrical hole of radius 1 and height 10 drilled straight through. The cube's volume is 10 cubed, or 1,000. The cylindrical hole's volume is π times 1 squared times 10, or 10π. The remaining solid's volume is 1,000 minus 10π.

Inscribed Solids: Read the Shared Measurement

An inscribed solid touches the solid around it at every possible point, which fixes one measurement between the two shapes. A sphere inscribed in a cube touches each face at its center, so the sphere's diameter equals the cube's side length. A cylinder inscribed in a rectangular box, standing upright, has a diameter equal to the box's shorter base side and a height equal to the box's height.

Take a cube with side 8. A sphere inscribed inside it has diameter 8, so its radius is 4. Its volume is four thirds times pi times 4 cubed, or four thirds times pi times 64, which is (256/3)π.

Unit-Conversion Traps in Volume Problems

Volume problems often mix units, such as feet and inches, or meters and centimeters. Converting a linear measurement is a single multiplication, but converting a volume measurement cubes the conversion factor, since volume has three dimensions. The safer route is to convert every dimension into one unit first, then apply the volume formula once. A setup with a radius in centimeters and a height in meters returns a wrong answer even when the formula is right.

Cubing the Conversion Factor

One foot equals 12 inches, so converting a length from feet to inches multiplies by 12. Converting a volume from cubic feet to cubic inches multiplies by 12 cubed, or 1,728, not by 12. A 2 cubic foot volume equals 2 times 1,728, or 3,456 cubic inches.

The same rule applies to any unit pair. Converting square units squares the conversion factor, and converting cubic units cubes it. Write out the conversion factor once, then raise it to the power matching the dimension before multiplying.

Common Conversion Mistakes

Three setup errors turn a correct formula into a wrong answer. Each one appears before any arithmetic happens, in the line where you write the measurements down. Check the units on every dimension against the unit the question asks for, then multiply.

• Multiplying by the linear conversion factor instead of raising it to the third power.
• Converting only one dimension of a solid, such as height, and leaving length and width in the original unit.
• Mixing units inside one formula, such as a radius in inches and a height in feet.

We offer three difficulty levels of drills for Geometry: Solids and Volume on our GRE practice page, along with full-length practice tests and Quant section practice. Every drill set is free, with no signup required. Start at the easy level, then move up once you set up composite and inscribed problems without hesitation.

Sources

• ETS: GRE General Test, Quantitative Reasoning content preparation
• ETS: GRE Math Review, Chapter 3: Geometry (PDF)