Math Core

Solid geometry problems look intimidating because the pictures are hard to draw. But almost every AMC 3D problem is solved by reducing it to 2D: slice the solid along a well-chosen plane, unfold its surface flat, or use the Pythagorean Theorem in three dimensions. Add a few volume formulas and the scaling rule for similar solids, and you have the whole toolkit.

Formulas to know

Solid	Volume	Surface area
box $a \times b \times c$	$abc$	$2(ab + bc + ca)$
prism or cylinder	(base area) $\times$ height	cylinder: $2\pi r^2 + 2\pi r h$
pyramid or cone	$\tfrac13$ (base area) $\times$ height	cone: $\pi r^2 + \pi r \ell$ ( $\ell$ = slant height)
sphere	$\tfrac43 \pi r^3$	$4\pi r^2$
regular tetrahedron, edge $s$	$\dfrac{s^3 \sqrt2}{12}$	$\sqrt3\, s^2$

The space diagonal of an $a \times b \times c$ box is $\sqrt{a^2 + b^2 + c^2}$ : use the Pythagorean Theorem once on the base, then again up the height.

Three ways to go from 3D to 2D

Slice through the solid along a plane of symmetry (through the axis of a cone or cylinder, or through a diagonal of a cube). Spheres become circles and cones become triangles.
Unfold the surface into a net when the question is about a path along the surface.
Right triangles in space: a length in 3D is often the hypotenuse of a right triangle whose legs you can find in 2D.

Slicing a cone (radius 6, height 8) with an inscribed sphere through its axis: the sphere becomes the incircle of a 12-10-10 triangle.

Volumes and similar solids

Scaling

If two solids are similar with length ratio $k$ , their surface areas are in ratio $k^2$ and their volumes are in ratio $k^3$ .

Worked example: Water in a cone

A cone-shaped cup (vertex down) is filled with water to half its height. What fraction of the cup is full?

The water forms a smaller cone similar to the cup with ratio $\tfrac12$ . Its volume is $\left(\tfrac12\right)^3 = \tfrac18$ of the cup's.

Worked example: Rolling a sector into a cone

A sector of a circle with radius $12$ and central angle $120^\circ$ is rolled into a cone. Find its volume.

The radius $12$ becomes the slant height. The arc length, $\tfrac13 \cdot 2\pi \cdot 12 = 8\pi$ , becomes the base circumference, so the base radius is $4$ . The height is $\sqrt{12^2 - 4^2} = 8\sqrt2$ , and the volume is $\tfrac13 \pi \cdot 16 \cdot 8\sqrt2 = \dfrac{128\sqrt2\,\pi}{3}$ .

Paths on surfaces

Worked example: The ant on a box

An ant walks on the surface of a $1 \times 1 \times 2$ box from one corner to the opposite corner. How long is the shortest path?

Unfold two adjacent faces into a flat rectangle, so the path becomes a straight line. There are essentially two ways to unfold:

the two unit edges side by side: a $2 \times 2$ rectangle, diagonal $\sqrt{2^2 + 2^2} = 2\sqrt2$ ;
a unit edge next to the long edge: a $3 \times 1$ rectangle, diagonal $\sqrt{10}$ .

The shortest path is $2\sqrt2$ . In general, for an $a \times b \times c$ box with $c$ the longest edge, the answer is $\sqrt{(a + b)^2 + c^2}$ .

Corners and cross-sections

Cutting a corner off a box with a plane through three edges gives a tetrahedron with three right angles at one vertex. If its edges from that corner are $a$ , $b$ , $c$ , then its volume is $\dfrac{abc}{6}$ (a pyramid with a right-triangle base $\tfrac12 ab$ and height $c$ ). Use this volume, computed another way with the slanted face as the base, to find the distance from the corner to the plane.

Worked example: A triangle on a sphere

The three vertices of a $6$ - $8$ - $10$ triangle lie on a sphere of radius $13$ . How far is the center of the sphere from the plane of the triangle?

The plane cuts the sphere in a circle through all three vertices, which is the circumcircle of the triangle. A right triangle's circumcircle has the hypotenuse as a diameter, so its radius is $5$ . The center of the sphere lies directly above the center of that circle, so by the Pythagorean Theorem the distance is $\sqrt{13^2 - 5^2} = 12$ .

Common mistake

Don't forget the $\tfrac13$ for pyramids and cones, and don't confuse the height (perpendicular to the base) with the slant height (along the side). They are the legs and hypotenuse of a right triangle: $\ell^2 = r^2 + h^2$ .

Practice

Practice 1

A cube has surface area $150$ . What is the length of its space diagonal (the segment joining two opposite vertices)?

Practice 2

A sphere is inscribed in a cube with volume $216$ . What is the volume of the sphere?

Practice 3

The three faces of a rectangular box that meet at one corner have areas $12$ , $15$ and $20$ . What is the volume of the box?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A closed conical tank stands with its vertex pointing up. Water fills it to half the tank's height. What fraction of the tank's volume is water?

Practice 5

A cylinder of height $6$ is inscribed in a sphere of radius $5$ (both circular edges of the cylinder lie on the sphere). What is the volume of the cylinder?

Practice 6

What is the volume of a regular tetrahedron with edge length $6$ ?

Practice 7

A cylinder has circumference $4$ and height $9$ . A string is wound around it exactly $3$ times, going from a point on the bottom edge to the point directly above it on the top edge, rising steadily. How long is the string?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A box has edges $2$ , $3$ and $6$ . A plane passes through the three vertices adjacent to one corner $V$ , cutting off a tetrahedron. What is the distance from $V$ to the plane?

Practice 9

A cone has base radius $6$ and height $8$ . What is the radius of the largest sphere that fits inside the cone?

Real contest practice

2019 AMC 10A, Problem 21: a sphere tangent to the sides of a triangle; the plane cuts the sphere in the incircle.
2015 AMC 10A, Problem 21: the volume of a tetrahedron with given edge lengths.
2012 AMC 10B, Problem 23: a tetrahedron sliced off the corner of a cube.