What is the surface area of a closed cylinder?

It is $2\pi r^2 + 2\pi r h$: the two circular ends contribute $2\pi r^2$, and the curved wall unrolls into a rectangle of area $2\pi r h$ (height $h$ times circumference $2\pi r$). If the cylinder is open at one or both ends, drop the corresponding $\pi r^2$ term.

Why does doubling an edge multiply a solid's volume by 8?

Volume is three-dimensional, so it scales with the cube of any change in a linear dimension. Doubling every edge multiplies by $2$ in all three directions, and $2 \times 2 \times 2 = 8$. Surface area, being two-dimensional, would scale by only $2^2 = 4$.

CramClub

Three-Dimensional Figures

Master GRE solid geometry — volume and surface area of boxes, cubes, and cylinders, cone and sphere formulas, the space diagonal of a box, and combining 2D facts inside 3D figures.

GREquantitative reasoninghard21 min read6 worked examples6 practice

Key Formulas

Box and cube volume

$V = \ell w h; \;\; V_{\text{cube}} = s^3$

A cube is a box with all edges equal.

Cylinder volume

$V = \pi r^2 h$

Base-circle area times height. Only the radius is squared.

Cone and sphere volume

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h; \;\; V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$

A cone is one-third of its matching cylinder.

Surface areas

$6s^2$ (cube); $2\pi r^2 + 2\pi r h$ (cylinder)

A closed cylinder is two circles plus an unrolled rectangle.

Space diagonal of a box

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

Two stacked right triangles: base diagonal, then the vertical edge.

Right circular cylinder

Note: Figure not drawn to scale.

Video Lesson

Explanation

From Flat Facts to Solid Figures

Three-dimensional problems on the GRE reward students who see a solid as a stack of familiar two-dimensional shapes. Master a short list of volume and surface-area formulas, learn the space-diagonal trick, and practice pulling a right triangle or a circle out of a 3D figure — and the "hard" solid-geometry questions become routine.

Volume: How Much Fits Inside

Volume is measured in cubic units. The three solids you must know cold:

Solid	Volume	Key idea
Rectangular box	$V = \ell w h$	multiply the three edge lengths
Cube (edge $s$ )	$V = s^3$	a box with all edges equal
Cylinder	$V = \pi r^2 h$	base-circle area times height

A cylinder is the cleanest case of the "stack a 2D shape" idea: its volume is simply the area of the circular base, $\pi r^2$ , multiplied by the height. Only the radius is squared — the height is a plain multiplier, and squaring it is a frequent slip.

Two more formulas appear less often but are worth memorizing so you are never stuck:

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3.$

A cone holds exactly one-third of the cylinder that shares its base and height — dropping that $\tfrac{1}{3}$ is one of the most common errors in the whole topic.

Surface Area: Wrapping the Outside

Surface area is the total area of every face, measured in square units — think of the paper needed to wrap the solid.

Solid	Surface area
Rectangular box	$2(\ell w + \ell h + w h)$
Cube (edge $s$ )	$6s^2$
Cylinder (closed)	$2\pi r^2 + 2\pi r h$

A closed cylinder is two circular ends ( $2\pi r^2$ ) plus the curved wall, which unrolls into a rectangle of height $h$ and width equal to the circumference $2\pi r$ , giving $2\pi r h$ . Reading a cylinder as "two circles plus an unrolled rectangle" makes its surface area easy to reconstruct on the spot.

The Space Diagonal of a Box

The longest straight segment inside a rectangular box runs corner to opposite corner — the space diagonal. It comes from applying the Pythagorean relationship twice: first to the base to get a face diagonal, then to the right triangle formed by that face diagonal and the vertical edge. The result is a clean one-step formula:

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

For a box with edges $3$ , $4$ , and $12$ , the diagonal is $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$ . Recognizing that a "3D" question is really two stacked right triangles is the single most useful move in solid geometry.

Combining 2D Facts Inside 3D

The hardest GRE solid questions never test a solid in isolation — they hide a right triangle, a circle, or a proportion inside the figure. A sphere inscribed in a cube shares the cube's edge as its diameter. A cylinder inscribed in a box has diameter equal to the box's width. The longest rod that fits in a box is its space diagonal. The strategy is always the same: identify the 2D relationship, extract it, solve it, then feed the result back into the 3D formula.

The Scaling Rule for Solids

When every linear dimension is multiplied by a factor $k$ , surface area scales by $k^2$ and volume by $k^3$ . Doubling the edge of a cube does not double its volume — it multiplies the volume by $2^3 = 8$ . The GRE loves this because intuition wrongly expects volume to grow in step with length.

$\text{length} \times k \;\Rightarrow\; \text{area} \times k^2 \;\Rightarrow\; \text{volume} \times k^3.$

A Reliable Game Plan

Decide first which quantity the question wants — volume or surface area — because a figure usually gives enough to compute either. Square only the radius in a cylinder. Reach for the $\tfrac{1}{3}$ on cones and the $\tfrac{4}{3}$ on spheres. For "longest rod" or "corner to corner," use the space-diagonal formula. When a dimension changes by a factor, apply $k^2$ or $k^3$ rather than scaling linearly. And when a solid seems to hide something, look for the 2D right triangle or circle inside it.

Worked Examples

— follow the steps, hardest last

Volume of a rectangular box

easy

A rectangular box has edge lengths $5$ , $6$ , and $10$ . What is its volume?

Rectangular box

Note: Figure not drawn to scale.

Volume of a cylinder

medium

A cylinder has radius $4$ and height $9$ . What is its volume, in terms of $\pi$ ?

Surface area of a closed cylinder

medium

A closed cylinder has radius $3$ and height $10$ . What is its total surface area, in terms of $\pi$ ?

Space diagonal of a box

hard

What is the length of the longest straight rod that fits inside a rectangular box with edge lengths $2$ , $3$ , and $6$ ?

From volume back to a dimension

hard

A cylinder has volume $50\pi$ and height $2$ . What is the radius of its base?

A sphere inscribed in a cube

hard

A sphere is inscribed in a cube so that it touches all six faces. If the cube has edge length $6$ , what is the volume of the sphere, in terms of $\pi$ ?

Watch Out For

Squaring the height in a cylinder, writing $\pi r h^2$ or $\pi(rh)^2$ instead of $\pi r^2 h$ .
Only the radius is squared: $V = \pi r^2 h$ . The height multiplies the base area once.
Dropping the $\tfrac{1}{3}$ on a cone or the $\tfrac{4}{3}$ on a sphere, reporting a cylinder-style value.
Write the fraction first: $V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h$ and $V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$ . A cone is one-third of its matching cylinder.
Finding only a face diagonal and calling it the space diagonal, using two edges instead of all three.
The space diagonal uses all three edges: $d = \sqrt{\ell^2 + w^2 + h^2}$ . Leaving one edge out gives just a face diagonal.
Assuming that doubling an edge doubles the volume of a solid.
Volume scales by the cube of the linear factor. Doubling an edge multiplies volume by $2^3 = 8$ ; surface area by $2^2 = 4$ .

Practice Problems

A cube has a surface area of $96$ . What is its volume?

A cylinder has a volume of $45\pi$ and a height of $5$ . What is the radius of its base?

What is the length of the longest straight rod that fits inside a rectangular box with edge lengths $4$ , $4$ , and $7$ ?

Quantity A: the volume of a cylinder with radius $4$ and height $3$

Quantity B: the volume of a cube with edge length $5$

A closed cylinder has radius $2$ and height $5$ . What is its total surface area, in terms of $\pi$ ?

A rectangular box has a volume of $48$ and integer edge lengths.

Quantity A: the length of the box's longest edge

Quantity B: $6$

Frequently Asked Questions

How do I find the space diagonal of a rectangular box?: Use $d = \sqrt{\ell^2 + w^2 + h^2}$, the three-dimensional Pythagorean formula. It comes from applying the ordinary Pythagorean relationship twice — first to the base to get a face diagonal, then to the right triangle formed by that face diagonal and the vertical edge.
What is the surface area of a closed cylinder?: It is $2\pi r^2 + 2\pi r h$: the two circular ends contribute $2\pi r^2$, and the curved wall unrolls into a rectangle of area $2\pi r h$ (height $h$ times circumference $2\pi r$). If the cylinder is open at one or both ends, drop the corresponding $\pi r^2$ term.
Does the GRE give me the cone and sphere volume formulas?: No — the GRE does not supply a formula sheet, so memorize $V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h$ and $V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$. A cone holds exactly one-third of the cylinder that shares its base and height, which makes the cone formula easy to reconstruct.
Why does doubling an edge multiply a solid's volume by 8?: Volume is three-dimensional, so it scales with the cube of any change in a linear dimension. Doubling every edge multiplies by $2$ in all three directions, and $2 \times 2 \times 2 = 8$. Surface area, being two-dimensional, would scale by only $2^2 = 4$.

Keep Going

Circles

Master GRE circle geometry — radius, diameter, circumference, area, arcs and sectors, inscribed and central angles, tangents, and circles in the coordinate plane.

Triangles and Polygons

Command GRE triangle and polygon rules — the angle sum, exterior-angle theorem, triangle inequality, area, the Pythagorean special right triangles, similarity, and the interior-angle sum of any n-gon.

Lines and Angles

Master GRE angle relationships — complementary, supplementary, and vertical angles, the straight-line and full-turn sums, and parallel lines cut by a transversal — then chase unknown angles with confidence.

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.

Three-Dimensional Figures

Master GRE solid geometry — volume and surface area of boxes, cubes, and cylinders, cone and sphere formulas, the space diagonal of a box, and combining 2D facts inside 3D figures.

GREquantitative reasoninghard21 min read6 worked examples6 practice

Key Formulas

Box and cube volume

$V = \ell w h; \;\; V_{\text{cube}} = s^3$

A cube is a box with all edges equal.

Cylinder volume

$V = \pi r^2 h$

Base-circle area times height. Only the radius is squared.

Cone and sphere volume

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h; \;\; V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$

A cone is one-third of its matching cylinder.

Surface areas

$6s^2$ (cube); $2\pi r^2 + 2\pi r h$ (cylinder)

A closed cylinder is two circles plus an unrolled rectangle.

Space diagonal of a box

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

Two stacked right triangles: base diagonal, then the vertical edge.

Right circular cylinder

Note: Figure not drawn to scale.

Video Lesson

Explanation

From Flat Facts to Solid Figures

Volume: How Much Fits Inside

Volume is measured in cubic units. The three solids you must know cold:

Solid	Volume	Key idea
Rectangular box	$V = \ell w h$	multiply the three edge lengths
Cube (edge $s$ )	$V = s^3$	a box with all edges equal
Cylinder	$V = \pi r^2 h$	base-circle area times height

Two more formulas appear less often but are worth memorizing so you are never stuck:

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3.$

A cone holds exactly one-third of the cylinder that shares its base and height — dropping that $\tfrac{1}{3}$ is one of the most common errors in the whole topic.

Surface Area: Wrapping the Outside

Surface area is the total area of every face, measured in square units — think of the paper needed to wrap the solid.

Solid	Surface area
Rectangular box	$2(\ell w + \ell h + w h)$
Cube (edge $s$ )	$6s^2$
Cylinder (closed)	$2\pi r^2 + 2\pi r h$

The Space Diagonal of a Box

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

Combining 2D Facts Inside 3D

The Scaling Rule for Solids

$\text{length} \times k \;\Rightarrow\; \text{area} \times k^2 \;\Rightarrow\; \text{volume} \times k^3.$

A Reliable Game Plan

Worked Examples

— follow the steps, hardest last

Volume of a rectangular box

easy

A rectangular box has edge lengths $5$ , $6$ , and $10$ . What is its volume?

Rectangular box

Note: Figure not drawn to scale.

Volume of a cylinder

medium

A cylinder has radius $4$ and height $9$ . What is its volume, in terms of $\pi$ ?

Surface area of a closed cylinder

medium

A closed cylinder has radius $3$ and height $10$ . What is its total surface area, in terms of $\pi$ ?

Space diagonal of a box

hard

What is the length of the longest straight rod that fits inside a rectangular box with edge lengths $2$ , $3$ , and $6$ ?

From volume back to a dimension

hard

A cylinder has volume $50\pi$ and height $2$ . What is the radius of its base?

A sphere inscribed in a cube

hard

A sphere is inscribed in a cube so that it touches all six faces. If the cube has edge length $6$ , what is the volume of the sphere, in terms of $\pi$ ?

Watch Out For

Squaring the height in a cylinder, writing $\pi r h^2$ or $\pi(rh)^2$ instead of $\pi r^2 h$ .
Only the radius is squared: $V = \pi r^2 h$ . The height multiplies the base area once.
Dropping the $\tfrac{1}{3}$ on a cone or the $\tfrac{4}{3}$ on a sphere, reporting a cylinder-style value.
Write the fraction first: $V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h$ and $V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$ . A cone is one-third of its matching cylinder.
Finding only a face diagonal and calling it the space diagonal, using two edges instead of all three.
The space diagonal uses all three edges: $d = \sqrt{\ell^2 + w^2 + h^2}$ . Leaving one edge out gives just a face diagonal.
Assuming that doubling an edge doubles the volume of a solid.
Volume scales by the cube of the linear factor. Doubling an edge multiplies volume by $2^3 = 8$ ; surface area by $2^2 = 4$ .

Practice Problems

A cube has a surface area of $96$ . What is its volume?

A cylinder has a volume of $45\pi$ and a height of $5$ . What is the radius of its base?

What is the length of the longest straight rod that fits inside a rectangular box with edge lengths $4$ , $4$ , and $7$ ?

Quantity A: the volume of a cylinder with radius $4$ and height $3$

Quantity B: the volume of a cube with edge length $5$

A closed cylinder has radius $2$ and height $5$ . What is its total surface area, in terms of $\pi$ ?

A rectangular box has a volume of $48$ and integer edge lengths.

Quantity A: the length of the box's longest edge

Quantity B: $6$

Frequently Asked Questions

How do I find the space diagonal of a rectangular box?: Use $d = \sqrt{\ell^2 + w^2 + h^2}$, the three-dimensional Pythagorean formula. It comes from applying the ordinary Pythagorean relationship twice — first to the base to get a face diagonal, then to the right triangle formed by that face diagonal and the vertical edge.
What is the surface area of a closed cylinder?: It is $2\pi r^2 + 2\pi r h$: the two circular ends contribute $2\pi r^2$, and the curved wall unrolls into a rectangle of area $2\pi r h$ (height $h$ times circumference $2\pi r$). If the cylinder is open at one or both ends, drop the corresponding $\pi r^2$ term.
Does the GRE give me the cone and sphere volume formulas?: No — the GRE does not supply a formula sheet, so memorize $V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h$ and $V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$. A cone holds exactly one-third of the cylinder that shares its base and height, which makes the cone formula easy to reconstruct.
Why does doubling an edge multiply a solid's volume by 8?: Volume is three-dimensional, so it scales with the cube of any change in a linear dimension. Doubling every edge multiplies by $2$ in all three directions, and $2 \times 2 \times 2 = 8$. Surface area, being two-dimensional, would scale by only $2^2 = 4$.

Keep Going

Circles

Master GRE circle geometry — radius, diameter, circumference, area, arcs and sectors, inscribed and central angles, tangents, and circles in the coordinate plane.

Triangles and Polygons

Lines and Angles

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.