GRE · Geometry · TOPIC LESSON
Build volume and exposed-surface calculations from dimensions, cross-sections and units.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
02 / UNDERSTAND & APPLY
A solid occupies three-dimensional space. An edge is a line segment where faces meet; a face is a surface region. Volume measures enclosed space in cubic units, while surface area measures the sum of exposed surface regions in square units. A drawing projects three dimensions onto a plane, so a rectangular face may look like a slanted parallelogram.
Use stated dimensions and the named solid’s properties. A rectangular prism has three mutually perpendicular edge directions even if its perspective sketch looks oblique. A height in a volume formula is the perpendicular distance between base planes, not any sloping edge that looks vertical on the page.
Mark the target before calculating: internal capacity, exterior paint, material for an open box, and a longest interior diagonal ask for different measurements. A hollow object may have both inner and outer dimensions; do not assume wall thickness is negligible unless the model says so.
03 / UNDERSTAND & APPLY
A prism has parallel congruent polygonal bases and a constant cross-sectional area parallel to those bases. If the base area is B and perpendicular height is h, volume is V = Bh. Think of stacking equal layers: changing the cross-sectional shape does not change the formula, but it changes how B is found.
A rectangular prism with dimensions l, w and h has volume lwh. A cube is the special case l = w = h = s and has volume s³. A triangular prism uses the area of its triangular base, not the product of all three triangle side lengths.
ORIGINAL WORKED EXAMPLE
A right triangular prism has a triangular base with perpendicular legs 6 cm and 8 cm. Its length between the triangular bases is 15 cm. Find its volume.
Answer 360 cm³.
04 / UNDERSTAND & APPLY
A closed rectangular prism has three pairs of congruent faces, with areas lw, lh and wh. Its total surface area is 2(lw + lh + wh). A cube has six square faces, so its surface area is 6s². These formulas count all outside faces once.
For an open-top box, subtract the missing top area lw from the closed-box total. If only the four side walls are painted, exclude both the top and bottom. A lid or an internal partition changes the surface count. Draw a face inventory rather than guessing whether to halve a total.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A rectangular box has length 12 cm, width 5 cm and height 4 cm. It has a bottom and four sides but no top. Ignoring thickness and overlap, find the material area.
Answer 196 cm².
05 / UNDERSTAND & APPLY
A right circular cylinder has radius r and perpendicular height h. Its circular base area is πr², so volume is πr²h. The diameter is 2r; entering the diameter where r belongs makes the volume four times too large.
Cutting the curved side along a vertical line and unrolling it produces a rectangle of width equal to the base circumference 2πr and height h. Its lateral area is 2πrh. A closed cylinder adds two circular bases, so total surface area is 2πrh + 2πr². An open container with a bottom adds only one base.
“Lateral area” means the curved side here, while “total surface area” depends on which bases are actually present. The end circles do not add any height or volume; they are boundary surfaces.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A closed right circular cylinder has radius 3 cm and height 10 cm. Find volume and total surface area.
Answer Volume 90π cm³; total surface area 78π cm².
06 / UNDERSTAND & APPLY
A pyramid or cone has volume one third of base area times perpendicular height: V = Bh/3. A right circular cone therefore has volume πr²h/3. The slant height of a cone runs along its side from the apex to the rim; it is not h. For a right cone, slant height s satisfies s² = r² + h², and lateral area is πrs. Add πr² for a closed base.
For a right square pyramid with base side a and vertical apex above the center, the slant height of one triangular face is √[h² + (a/2)²]. Its four triangular faces have total area 2a times that slant height. The apex-to-corner edge is a different length, so label the segment before substitution.
A sphere of radius r has surface area 4πr² and volume 4πr³/3. A hemisphere has half the sphere’s volume and half its curved surface area. If a flat circular base is included in total surface area, add πr². These formulas concern ideal solids; no calculus or trigonometry is needed for their use.
ORIGINAL WORKED EXAMPLE
A right circular cone has radius 6 cm and perpendicular height 8 cm. Find its volume and total surface area including the base.
Answer Volume 96π cm³; total surface area 96π cm².
07 / UNDERSTAND & APPLY
The diagonal across an l by w rectangular face is √(l² + w²). In a rectangular prism, that face diagonal and the perpendicular height h form another right triangle. The space diagonal joining opposite vertices is therefore √(l² + w² + h²). A cube’s face diagonal is s√2, but its space diagonal is s√3.
A cross-section is the region produced by a plane slicing a solid. Slices parallel to a prism’s bases have the same area as those bases. A cylinder cut parallel to its bases gives a circle of the same radius; a plane through the axis of a right cylinder gives a rectangle with width 2r and height h. A generic tilted slice is different, so specify the orientation.
ORIGINAL WORKED EXAMPLE
A rectangular prism has dimensions 3, 4 and 12 cm. Find its space diagonal.
Answer 13 cm.
08 / UNDERSTAND & APPLY
Volumes of nonoverlapping pieces add. A cavity removes volume, so subtract it from the outer solid, provided its dimensions describe the part actually removed. A cylinder drilled completely through a block removes πr² times the drilled length; a blind hole that stops inside the block has a smaller length.
Surface area behaves differently at a join. If two solid blocks are attached face to face, the touching faces become internal and are not exposed. Start with the sum of their separate surface areas and subtract both copies of the shared contact area. A new hole creates interior wall surface while removing part of the original exterior.
ORIGINAL WORKED EXAMPLE
Two solid cubes of side 4 cm are joined along one entire face. Find the combined volume and exposed surface area.
Answer Volume 128 cm³; exposed surface area 160 cm².
09 / UNDERSTAND & APPLY
A cubic unit contains three length factors. Since 1 m = 100 cm, 1 m³ = 1,000,000 cm³. Also, 1 L = 1,000 cm³ and 1 mL = 1 cm³. For capacity calculations, use interior dimensions; an outer measurement does not account for wall thickness.
A rectangular tank’s liquid volume is base area times liquid depth. If water is transferred without loss into another constant-cross-section container, the water volume remains fixed. Divide by the receiving base area to find the new depth, and check that the container is tall enough.
If all lengths of similar solids scale by k, surface area scales by k² and volume by k³. If only a cylinder’s radius doubles while height stays fixed, its volume is multiplied by four, not eight. Uniform similarity and independent dimension changes require different reasoning.
ORIGINAL WORKED EXAMPLE
An aquarium’s interior measures 50 cm by 30 cm, and water is 24 cm deep. Find the water volume in liters.
Answer 36 L.
10 / UNDERSTAND & APPLY
Numeric entries use the unit or coefficient requested. For QC use A, B, C or D as defined in the first section.
ORIGINAL PRACTICE · 1
140
Multiply the three perpendicular dimensions: 7×5×4 = 140 cm³.
ORIGINAL PRACTICE · 2
150
The positive cube root gives side 5 cm. Six faces contribute 6×5² = 150 cm².
ORIGINAL PRACTICE · 3
150
The radius is 5 cm. Volume is π(5²)(6) = 150π cm³, so k = 150.
ORIGINAL PRACTICE · 4
D. πr² + 2πrh
Count the one bottom disk and the curved rectangular side. Two base areas would describe a closed cylinder.
ORIGINAL PRACTICE · 5
75
One cubic meter is 1,000 L, so 0.075×1,000 = 75 L.
11 / UNDERSTAND & APPLY
For an insufficient-information answer, supply two valid solids rather than merely saying that a formula contains an unknown.
ORIGINAL PRACTICE · 1
D
A 4 by 4 by 4 cube has area 96, below 100. A 1 by 4 by 16 prism has the same volume and area 2(4 + 16 + 64) = 168, above 100.
ORIGINAL PRACTICE · 2
B and C
Area quantities use the square factor 2² = 4. Edge length uses 2 and volume uses 2³ = 8.
ORIGINAL PRACTICE · 3
7
The space diagonal is √(2² + 3² + 6²) = √49 = 7.
ORIGINAL PRACTICE · 4
120
Use one third of base area times perpendicular height: 45×8/3 = 120 cm³.
ORIGINAL PRACTICE · 5
444
Outer volume is 480. The removed prism has volume 2×3×6 = 36, so remaining volume is 444 cubic units.
ORIGINAL PRACTICE · 6
D
With the same radius, volume is proportional to height. Heights 2 and 5 make B larger; heights 5 and 2 make A larger. No height relation is given.
ORIGINAL PRACTICE · 7
27
The curved half-sphere area is (1/2)(4π×9) = 18π. Add the base area 9π to obtain 27π.
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