Topics / Geometry

Solid Geometry

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

Solid GeometryAMC 10 Step by Step

Name: Date: September 13, 2026

Geometry · 48 problems have this as their main idea

primary-topic problems (3.7% of all)

What you need to know

Prism and cylinder: $V = Bh$ . Pyramid and cone: $V = \frac13 Bh$ . Sphere: $V = \frac43\pi r^3$ , surface area $4\pi r^2$ .
Scaling every length by $k$ multiplies areas by $k^2$ and volumes by $k^3$ . A plane parallel to the base of a cone or pyramid cuts off a similar solid.
Cube of side $s$ : face diagonal $s\sqrt2$ , space diagonal $s\sqrt3$ ; a box $a\times b\times c$ has diagonal $\sqrt{a^2+b^2+c^2}$ .
Cross-sections of a cube can be triangles, rectangles, pentagons, or regular hexagons; shortest surface paths become straight lines on an unfolded net.

How AMC 10 tests it

Early: volume or surface area of a box, or how the surface area changes when unit cubes are removed from a corner (unchanged) or an edge.
Water poured between containers, or a partly filled inverted cone; equate volumes and use $k^3$ scaling.
Problems 15–22: a sphere or cylinder inscribed in a cone (similar triangles in the axial cross-section), or an ant's shortest path on a box.

Standard approaches

Draw the two-dimensional cross-section through the axis; most solid problems become a triangle problem there.
Apply the scaling law before computing: similar solids with ratio $k$ have volume ratio $k^3$ .
For odd shapes, subtract pyramids or corner tetrahedra from a full prism.
For surface paths, unfold the net and use the Pythagorean theorem; compare several unfoldings.

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. The water forms a cone similar to the whole cone with ratio $\frac{9}{12} = \frac34$ , so its radius is $4\cdot\frac34 = 3$ and its volume is
$\frac13\pi\cdot 3^2\cdot 9 = 27\pi.$
In the cylinder, $\pi\cdot 3^2\cdot h = 27\pi$ gives $h = 3$ . The answer is $\boxed{\textbf{(C)}\ 3}$ .

Pitfalls

Dropping the $\frac13$ in cone and pyramid volumes, or using it for cylinders.
Scaling volume by $k$ or $k^2$ instead of $k^3$ when a cone is partly filled.
Confusing the slant height with the height of a cone.

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)

Problems, easiest first

2021 AMC Fall 10A · #3Solid Geometry

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?

2021 AMC 10B · #10Solid Geometry

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?

2020 AMC 10B · #2Solid Geometry

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?

2007 AMC 10B · #1Solid Geometry

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?

2007 AMC 10A · #3Solid Geometry

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?

2003 AMC 10A · #3Solid Geometry

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?

2020 AMC 10B · #10Solid Geometry

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?

2020 AMC 10A · #10Solid Geometry

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 …

2017 AMC 10B · #6Solid Geometry

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?

2017 AMC 10A · #11Solid Geometry

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 $\textit{AB}$ ?

2010 AMC 10A · #12Solid Geometry

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?

2005 AMC 10A · #11Solid Geometry

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 $n$ ?

2021 AMC 10A · #12Solid Geometry

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 …

2021 AMC Fall 10A · #17Solid Geometry

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 $10$ …

2021 AMC 10A · #13Solid Geometry

What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$ , $AC = 3$ , $AD = 4$ , $BC = \sqrt{13}$ , $BD = 2\sqrt{5}$ , and $CD = 5$ ?

2018 AMC 10B · #15Solid Geometry

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 …

2018 AMC 10B · #10Solid Geometry

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 $M$ ?

2015 AMC 10B · #17Solid Geometry

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?

2013 AMC 10A · #14Solid Geometry

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?

2012 AMC 10B · #17Solid Geometry

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?

2010 AMC 10A · #17Solid Geometry

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?

2008 AMC 10B · #19Solid Geometry

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?

2007 AMC 10A · #21Solid Geometry

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?

2004 AMC 10A · #19Solid Geometry

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?

2003 AMC 10B · #17Solid Geometry

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 …

2003 AMC 10A · #10Solid Geometry

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?

2002 AMC 10A · #18Solid Geometry

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?

2001 AMC 10 · #17Solid Geometry

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?

2001 AMC 10 · #21Solid Geometry

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.

2025 AMC 10B · #19Solid Geometry

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 …

2023 AMC 10B · #20Solid Geometry

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 $n$ ?

2023 AMC 10A · #18Solid Geometry

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?

2022 AMC 10A · #21Solid Geometry

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?

2021 AMC Fall 10A · #22Solid Geometry

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 $r$ ?

2020 AMC 10B · #20Solid Geometry

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 …

2019 AMC 10A · #21Solid Geometry

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?

2015 AMC 10A · #21Solid Geometry

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?

2014 AMC 10A · #19Solid Geometry

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 $3$ ?

2013 AMC 10A · #22Solid Geometry

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 …

2012 AMC 10B · #23Solid Geometry

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 …

2011 AMC 10A · #24Solid Geometry

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?

2011 AMC 10B · #22Solid Geometry

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?

2009 AMC 10B · #22Solid Geometry

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 $c+s$ ?

2008 AMC 10A · #21Solid Geometry

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 $ABCD$ ?

2007 AMC 10B · #23Solid Geometry

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?

2006 AMC 10A · #24Solid Geometry

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

2004 AMC 10A · #25Solid Geometry

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?

2014 AMC 10B · #23Solid Geometry

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?