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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

all 48 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 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) $7$
2.2021 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) $6$
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 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) $876$
5.2007 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) $2.5$
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 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) $4:1$
14.2021 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) $12\sqrt{3}$
15.2021 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) $6$
16.2018 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) $w^2h$
17.2018 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) $2$
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) $\frac{\sqrt{5}}{5}$
21.2010 AMC 10A #17
A solid cube has side length $3$ inches. A $2$ -inch by $2$ -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
1. A) $7$
2. B) $8$
3. C) $10$
4. D) $12$
5. E) $15$
22.2008 AMC 10B #19
A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?
1. A) $24\pi - 36 \sqrt {2}$
2. B) $24\pi - 24 \sqrt {3}$
3. C) $36\pi - 36 \sqrt {3}$
4. D) $36\pi - 24 \sqrt {2}$
5. E) $48\pi - 36 \sqrt {3}$
23.2007 AMC 10A #21
A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
1. A) $3$
2. B) $6$
3. C) $8$
4. D) $9$
5. E) $12$
24.2004 AMC 10A #19
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
1. A) $120$
2. B) $180$
3. C) $240$
4. D) $360$
5. E) $480$
25.2003 AMC 10B #17
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius? (Note: a cone with radius $r$ and height $h$ has volume $\pi r^2 h / 3$ and a sphere with radius $r$ has volume $4 \pi r^3 / 3$ .)
1. A) $2:1$
2. B) $3:1$
3. C) $4:1$
4. D) $16:3$
5. E) $6:1$
26.2003 AMC 10A #10
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
27.2002 AMC 10A #18
A $3$ x $3$ x $3$ cube is made of $27$ normal dice. Each die's opposite sides sum to $7$ . What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
1. A) $60$
2. B) $72$
3. C) $84$
4. D) $90$
5. E) $96$
28.2001 AMC 10 #17
Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides?
1. A) $\text{ A cone with slant height of } 10 \text{ and radius } 6$
2. B) $\text{ A cone with height of } 10 \text{ and radius } 6$
3. C) $\text{ A cone with slant height of } 10 \text{ and radius } 7$
4. D) $\text{ A cone with height of } 10 \text{ and radius } 7$
5. E) $\text{ A cone with slant height of } 10 \text{ and radius } 8$
29.2001 AMC 10 #21
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $10$ and altitude $12$ , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
1. A) $\frac{8}3$
2. B) $\frac{30}{11}$
3. C) $3$
4. D) $\frac{25}{8}$
5. E) $\frac{7}{2}$
30.2025 AMC 10B #19
A container has a $1\times 1$ square bottom, a $3\times 3$ open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes $35$ minutes to fill the container up to the midline of the trapezoids.

How many more minutes will it take to fill the remainder of the container?
1. A) $70$
2. B) $85$
3. C) $90$
4. D) $95$
5. E) $105$
31.2023 AMC 10B #20
Four congruent semicircles are drawn on the surface of a sphere with radius $2$ , as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$ . What is $n$ ?
1. A) $32$
2. B) $12$
3. C) $48$
4. D) $36$
5. E) $27$
32.2023 AMC 10A #18
A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
1. A) $5$
2. B) $6$
3. C) $7$
4. D) $8$
5. E) $9$
33.2022 AMC 10A #21
A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$ . The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
1. A) $6$
2. B) $7$
3. C) $5+2\sqrt{2}$
4. D) $8$
5. E) $9$
34.2021 AMC Fall 10A #22
Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$ . Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$ ?
1. A) $\frac{3}{2}$
2. B) $\frac{90-40\sqrt{3}}{11}$
3. C) $2$
4. D) $\frac{144-25\sqrt{3}}{44}$
5. E) $\frac{5}{2}$
35.2020 AMC 10B #20
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$ , together with its interior. For real $r\geq0$ , let $S(r)$ be the set of points in $3$ -dimensional space that lie within a distance $r$ of some point in $B$ . The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$ , where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
1. A) $6$
2. B) $19$
3. C) $24$
4. D) $26$
5. E) $38$
36.2019 AMC 10A #21
A sphere with center $O$ has radius 6. A triangle with sides of length $15$ , $15$ , and $24$ is situated in space so that each of its sides are tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
1. A) $2\sqrt{3}$
2. B) $4$
3. C) $3\sqrt{2}$
4. D) $2\sqrt{5}$
5. E) $5$
37.2015 AMC 10A #21
Tetrahedron $ABCD$ has $AB=5$ , $AC=3$ , $BC=4$ , $BD=4$ , $AD=3$ , and $CD=\tfrac{12}5\sqrt2$ . What is the volume of the tetrahedron?
1. A) $3\sqrt2$
2. B) $2\sqrt5$
3. C) $\dfrac{24}5$
4. D) $3\sqrt3$
5. E) $\dfrac{24}5\sqrt2$
38.2014 AMC 10A #19
Four cubes with edge lengths $1$ , $2$ , $3$ , and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$ ?
1. A) $\dfrac{3\sqrt{33}}5$
2. B) $2\sqrt3$
3. C) $\dfrac{2\sqrt{33}}3$
4. D) $4$
5. E) $3\sqrt2$
39.2013 AMC 10A #22
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$ . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
1. A) $\sqrt2$
2. B) $\frac{3}{2}$
3. C) $\frac{5}{3}$
4. D) $\sqrt3$
5. E) $2$
40.2012 AMC 10B #23
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
1. A) $\frac{\sqrt{3}}{3}$
2. B) $\frac{2\sqrt{2}}{3}$
3. C) $1$
4. D) $\frac{2\sqrt{3}}{3}$
5. E) $\sqrt{2}$
41.2011 AMC 10B #22
A pyramid has a square base with sides of length $1$ and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
1. A) $5\sqrt{2} - 7$
2. B) $7 - 4\sqrt{3}$
3. C) $\frac{2\sqrt{2}}{27}$
4. D) $\frac{\sqrt{2}}{9}$
5. E) $\frac{\sqrt{3}}{9}$
42.2011 AMC 10A #24
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
1. A) $\frac{1}{12}$
2. B) $\frac{\sqrt{2}}{12}$
3. C) $\frac{\sqrt{3}}{12}$
4. D) $\frac{1}{6}$
5. E) $\frac{\sqrt{2}}{6}$
43.2009 AMC 10B #22
A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$ ?
1. A) $\frac{24}{5}$
2. B) $\frac{32}{5}$
3. C) $8+\sqrt5$
4. D) $5+\frac{16\sqrt5}{5}$
5. E) $10+5\sqrt5$
44.2008 AMC 10A #21
A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$ , as shown. What is the area of quadrilateral $ABCD$ ?
1. A) $\frac{\sqrt{6}}{2}$
2. B) $\frac{5}{4}$
3. C) $\sqrt{2}$
4. D) $\frac{5}{8}$
5. E) $\frac{3}{4}$
45.2007 AMC 10B #23
A pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
1. A) $2$
2. B) $2+\sqrt{2}$
3. C) $1+2\sqrt{2}$
4. D) $4$
5. E) $4+2\sqrt{2}$
46.2006 AMC 10A #24
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
1. A) $\frac{1}{8}$
2. B) $\frac{1}{6}$
3. C) $\frac{1}{4}$
4. D) $\frac{1}{3}$
5. E) $\frac{1}{2}$
47.2004 AMC 10A #25
Three mutually tangent spheres of radius $1$ rest on a horizontal plane. A sphere of radius $2$ rests on them. What is the distance from the plane to the top of the larger sphere?
1. A) $3 + \frac {\sqrt {30}}{2}$
2. B) $3 + \frac {\sqrt {69}}{3}$
3. C) $3 + \frac {\sqrt {123}}{4}$
4. D) $\frac {52}{9}$
5. E) $3 + 2\sqrt2$
48.2014 AMC 10B #23
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
1. A) $\dfrac32$
2. B) $\dfrac{1+\sqrt5}2$
3. C) $\sqrt3$
4. D) $2$
5. E) $\dfrac{3+\sqrt5}2$

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