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Solid Geometry

Volume, surface area, cubes, cones, spheres, cylinders, cross-sections, nets

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Solid GeometryAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Volume and surface. Prism or cylinder: $V = Bh$ , where $B$ is the area of the base and $h$ the height. Pyramid or cone: $V = \frac13 Bh$ . Sphere of radius $r$ : $V = \frac43\pi r^3$ , surface area $4\pi r^2$ . Example: with radius $3$ and height $4$ , a cylinder holds $36\pi$ and a cone $12\pi$ .
Scaling. Multiplying every length by $k$ multiplies areas by $k^2$ and volumes by $k^3$ . A cut parallel to the base of a cone or pyramid leaves a smaller cone or pyramid of the same shape. Example: doubling a cube's edges makes its surface area $4$ times and its volume $8$ times as big.
Diagonals. A cube with edge $s$ has face diagonal $s\sqrt2$ and space diagonal (corner to opposite corner, through the inside) $s\sqrt3$ . An $a\times b\times c$ box has space diagonal $\sqrt{a^2+b^2+c^2}$ . Example: a $2\times3\times6$ box has diagonal $\sqrt{4+9+36} = 7$ .
Slicing a cube. A flat cut through a cube can give a triangle, a four-sided shape (square, rectangle, rhombus, parallelogram, trapezoid), a pentagon, or a hexagon. It never has more than six sides, since a cube has six faces. Cutting through the midpoints of six edges gives a regular hexagon.
Paths on the surface. The shortest path along the outside becomes a straight line when you unfold the solid flat (into a net).

How AMC 10 tests it

Early: the volume or surface area of a box, or how the surface area changes when unit cubes are removed from a corner (no change: $3$ faces are lost and $3$ are uncovered) or from an edge (it goes up by $2$ ).
Water poured between containers, or a partly filled cone standing on its tip. Set the volumes equal and use $k^3$ scaling.
Problems 15–22: a sphere or cylinder inside a cone (similar triangles in the cross-section through the axis), or an ant's shortest path on a box.

Standard approaches

Draw the flat cross-section through the axis (the center line). Most solid problems become a triangle problem there.
Use scaling before computing: similar solids with length ratio $k$ have volume ratio $k^3$ .
For odd shapes, start from a full prism or box and subtract pyramids or corner tetrahedra (triangular pyramids).
For surface paths, unfold the net and use the Pythagorean Theorem. Try several unfoldings and keep the shortest.

Worked example

An inverted cone (vertex down) has height $12$ and base radius $4$ . It contains water to a depth of $9$ . The water is poured into an empty cylinder of radius $3$ . What is the depth of the water in the cylinder?

(A) $2$ (B) $\dfrac52$ (C) $3$ (D) $\dfrac{27}{8}$ (E) $4$

Solution. Step 1. The water forms a smaller cone of the same shape. Its height is $9$ out of $12$ , so every length is scaled by $\frac{9}{12} = \frac34$ , and its radius is $4\cdot\frac34 = 3$ .

Step 2. The water's volume is
$\frac13\pi\cdot 3^2\cdot 9 = 27\pi.$
(Check by scaling: the full cone holds $\frac13\pi\cdot 4^2\cdot 12 = 64\pi$ , and $64\pi\cdot\left(\frac34\right)^3 = 27\pi$ .)

Step 3. In the cylinder, water of depth $h$ has volume $\pi\cdot 3^2\cdot h = 9\pi h$ . Setting $9\pi h = 27\pi$ gives $h = 3$ .

The answer is $\boxed{\textbf{(C)}\ 3}$ .

Pitfalls

Dropping the $\frac13$ for a cone or pyramid, or using it for a cylinder.
Scaling volume by $k$ or $k^2$ instead of $k^3$ when a cone is partly filled. Example: a cone standing on its tip, filled to half its height, holds $\frac18$ of its volume, not $\frac12$ . (Standing on its base, it would hold $\frac78$ .)
Mixing up a cone's slant height (along the slope) with its height (straight up). Example: radius $3$ and height $4$ give slant height $5$ .

Traps that recur

Using the sphere's diameter as the slant height, or forgetting the middle term Rr in the frustum volume formula. (2014 AMC 10B #23)
Assuming that filling to half the height fills half the container, so the rest takes about the same 35 minutes again and a guess is pushed toward the 70 in (A). (2025 AMC 10B #19)
Counting edges as 12 times 4 = 48 without halving for the two faces each edge borders. (2023 AMC 10A #18)
Assuming each semicircle is half a great circle of radius 2, which gives 8pi = pisqrt(64), not among the choices. (2023 AMC 10B #20)

1.2021 AMC 10B #10
An inverted cone with base radius $12 \text{cm}$ and height $18\text{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24\text{cm}$ . What is the height in centimeters of the water in the cylinder?
1. A) $1.5$
2. B) $3$
3. C) $4$
4. D) $4.5$
5. E) $6$
2.2021 AMC Fall 10A #3
What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
3.2020 AMC 10B #2
Carl has $5$ cubes each having side length $1$ , and Kate has $5$ cubes each having side length $2$ . What is the total volume of these $10$ cubes?
1. A) $24$
2. B) $25$
3. C) $28$
4. D) $40$
5. E) $45$
4.2007 AMC 10A #3
An aquarium has a rectangular base that measures $100$ cm by $40$ cm and has a height of $50$ cm. It is filled with water to a height of $40$ cm. A brick with a rectangular base that measures $40$ cm by $20$ cm and a height of $10$ cm is placed in the aquarium. By how many centimeters does the water rise?
1. A) $0.5$
2. B) $1$
3. C) $1.5$
4. D) $2$
5. E) $2.5$
5.2007 AMC 10B #1
Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted?
1. A) $678$
2. B) $768$
3. C) $786$
4. D) $867$
5. E) $876$
6.2003 AMC 10A #3
A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?
1. A) $4.5\%$
2. B) $9\%$
3. C) $12\%$
4. D) $18\%$
5. E) $24\%$
7.2020 AMC 10A #10
Seven cubes, whose volumes are $1$ , $8$ , $27$ , $64$ , $125$ , $216$ , and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
1. A) $644$
2. B) $658$
3. C) $664$
4. D) $720$
5. E) $749$
8.2020 AMC 10B #10
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
1. A) $3\pi \sqrt5$
2. B) $4\pi \sqrt3$
3. C) $3 \pi \sqrt7$
4. D) $6\pi \sqrt3$
5. E) $6\pi \sqrt7$
9.2017 AMC 10A #11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$ . What is the length $\textit{AB}$ ?
1. A) $6$
2. B) $12$
3. C) $18$
4. D) $20$
5. E) $24$
10.2017 AMC 10B #6
What is the largest number of solid $2\text{ in.}$ by $2\text{ in.}$ by $1\text{ in.}$ blocks that can fit in a $3\text{ in.}$ by $2\text{ in.}$ by $3\text{ in.}$ box?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
11.2010 AMC 10A #12
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
1. A) $0.04$
2. B) $\frac{0.4}{\pi}$
3. C) $0.4$
4. D) $\frac{4}{\pi}$
5. E) $4$
12.2005 AMC 10A #11
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$ ?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
13.2021 AMC Fall 10A #17
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$ , which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$ , $B$ , and $C$ are $12$ , $9$ , and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$ ?
1. A) $9$
2. B) $6\sqrt{3}$
3. C) $8\sqrt{3}$
4. D) $17$
5. E) $12\sqrt{3}$
14.2021 AMC 10A #13
What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$ , $AC = 3$ , $AD = 4$ , $BC = \sqrt{13}$ , $BD = 2\sqrt{5}$ , and $CD = 5$ ?
1. A) $3$
2. B) $2\sqrt{3}$
3. C) $4$
4. D) $3\sqrt{3}$
5. E) $6$
15.2021 AMC 10A #12
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$ . Into each cone is dropped a spherical marble of radius $1 \text{ cm}$ , which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
1. A) $1:1$
2. B) $47:43$
3. C) $2:1$
4. D) $40:13$
5. E) $4:1$
16.2018 AMC 10B #10
In the rectangular parallelepiped shown, $AB=3$ , $BC=1$ , and $CG=2$ . Point $M$ is the midpoint of $\overline{FG}$ . What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$ ?
1. A) $1$
2. B) $\frac{4}{3}$
3. C) $\frac{3}{2}$
4. D) $\frac{5}{3}$
5. E) $2$
17.2018 AMC 10B #15
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$ . What is the area of the sheet of wrapping paper?
1. A) $2(w+h)^2$
2. B) $\frac{(w+h)^2}2$
3. C) $2w^2+4wh$
4. D) $2w^2$
5. E) $w^2h$
18.2015 AMC 10B #17
When the centers of the faces of the right rectangular prism shown below are joined to create an octahedron, what is the volume of the octahedron?
1. A) $\dfrac{75}{12}$
2. B) $10$
3. C) $12$
4. D) $10\sqrt2$
5. E) $15$
19.2013 AMC 10A #14
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$ . How many edges does the remaining solid have?
1. A) $36$
2. B) $60$
3. C) $72$
4. D) $84$
5. E) $108$
20.2012 AMC 10B #17
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
1. A) $\frac{1}{8}$
2. B) $\frac{1}{4}$
3. C) $\frac{\sqrt{10}}{10}$
4. D) $\frac{\sqrt{5}}{6}$
5. E) $\frac{\sqrt{5}}{5}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.