36. 3-D Geometry · Mastering AMC 10/12

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Cube

In this section you will be able to find a cube's volume, surface area and space diagonal from its side.

A cube’s space diagonal is a right triangle on top of a right triangle: first $\sqrt2,a$ across a face, then $\sqrt3,a$ through the middle.

Six equal square faces give $S = 6a^2$ , and stacking $a$ layers of $a \times a$ unit cubes gives $V = a^3$ .

Look:

The face diagonal $\sqrt2,a$ and one edge $a$ are the legs of a right triangle whose hypotenuse is the space diagonal: $\sqrt{2a^2 + a^2} = \sqrt3,a$ .

Exploreplay first, formulas later

Try this: Press Unfold. How many squares make the net? Check that the surface area is 6 times one face.
Try this: Raise side a to 3. Does the readout show volume 27, surface area 54 and space diagonal 3√3?
Try this: Look at the face diagonal and the space diagonal. Why is one √2 times the side and the other √3 times?

Name it

Theorem 36.1.1 · Volume and Surface Area of a Cube

For a cube with side length $a$ :

\text{Volume} = a^3, \qquad \text{Surface area} = 6a^2,

and the space diagonal (from a corner to the opposite corner, through the inside) has length

\sqrt{3}\,a.

Contest cornerreal contest problems

Problem 36.10.1

AMC 10A

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?

Problem 36.10.35(additional)

AMC 10

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?

Checkpoint for section 36.1

Get 2 of 3 right to master this section. One try each, no hints.

36.20%

Rectangular Prism

In this section you will be able to find a box's volume, surface area and space diagonal from its three dimensions.

Volume multiplies the three dimensions; surface area adds six faces that come in three matching pairs.

A box with length $l$ , width $b$ and height $h$ has $V = lbh$ and $S = 2(lb + bh + lh)$ .

Look:

The space diagonal uses the Pythagorean Theorem twice: $\sqrt{l^2 + b^2}$ across the floor, then up by $h$ .

Exploreplay first, formulas later

Try this: The box is 3 × 4 × 5. Read V and S. Which three face areas, each counted twice, make S?
Try this: Press Unfold. Find the three pairs of matching rectangles in the net.
Try this: The space diagonal is √(3² + 4² + 5²) = 5√2. Change one dimension and predict the new diagonal first.

Name it

Theorem 36.2.1 · Volume and Surface Area of a Rectangular Prism

For a box with length $l$ , width $b$ and height $h$ :

V = lbh, \qquad S = 2(lb + bh + lh),

and its space diagonal has length

\sqrt{l^2 + b^2 + h^2}.

Try it

Problem 36.10.4

AMC 8

Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

Contest cornerreal contest problems

Problem 36.10.16(additional)

AMC 10

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?

Problem 36.10.33(additional)

AMC 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 $M$ ?

Problem 36.10.25(additional)

AMC 10

This problem also appears in the book as 36.10.16.

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?

Problem 36.10.40(additional)

AIME

A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m + n$ .

Checkpoint for section 36.2

Get 2 of 3 right to master this section. One try each, no hints.

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Cylinder

In this section you will be able to find a cylinder's volume and surface area, and unroll its side into a rectangle.

Peel the label off a soup can: it’s a rectangle $2\pi r$ long and $h$ tall.

A cylinder is a stack of circles, so $V = \pi r^2 h$ . Its surface is two discs plus that rectangle: $S = 2\pi r^2 + 2\pi r h$ .

Look:

On the unrolled rectangle, paths and stripes become straight lines and slanted bands. That’s how you measure anything drawn on a curved side.

Exploreplay first, formulas later

Try this: Press Unfold. The side becomes a rectangle. How long is it, and why is it 2πr?
Try this: Read the surface area. Where do the two extra πr² terms come from?
Try this: Double the radius. Does the volume double or quadruple?

Name it

Theorem 36.3.1 · Volume and Surface Area of a Cylinder

For a cylinder with radius $r$ and height $h$ :

V = \pi r^2 h, \qquad S = 2\pi r^2 + 2\pi r h = 2\pi r(r + h).

The two discs give $2\pi r^2$ ; the side unrolls into a rectangle $2\pi r$ by $h$ .

Try it

Problem 36.10.2

AMC 8

Jerry cuts a wedge from a $6$ -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?

Contest cornerreal contest problems

Problem 36.10.19(additional)

AMC 10A

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?

Problem 36.10.32(additional)

AIME

A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$ , where $n$ is a positive integer. Find $n$ .

Problem 36.10.43(additional)

AIME

A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$ .

Checkpoint for section 36.3

Get 2 of 3 right to master this section. One try each, no hints.

1

A cylinder has radius $3$ and height $5$ . What is its total surface area? (Answer in terms of $\pi$ .)

2

A cylinder has volume $72\pi$ and radius $3$ . What is its height?

3

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?

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Cone

In this section you will be able to use a cone's volume, surface area and slant height, and work with frustums.

A cone is exactly one third of the cylinder around it.

$V = \frac13 \pi r^2 h$ . The slant height $s = \sqrt{r^2 + h^2}$ is the hypotenuse of the right triangle you see from the side, and the side area is $\pi r s$ .

Look:

Cut a cone parallel to its base and the small cone on top is similar to the whole one. If its height is $k$ times as big, its volume is $k^3$ times as big.

Exploreplay first, formulas later

Try this: Compare Cone with Cylinder (same r and h). How many cones of water fill the cylinder?
Try this: The cone has r = 3, h = 4. Find the slant height in the side view before you read it.
Try this: Press Unfold. The side becomes a sector. Why is its radius the slant height?
Try this: Try Frustum: a cone with its tip cut off. How would you get its volume from two cones?

Name it

Theorem 36.4.1 · Volume and Surface Area of a Cone

For a cone with radius $r$ , height $h$ and slant height $s$ :

V = \frac13 \pi r^2 h, \qquad S = \pi r^2 + \pi r s = \pi r(r + s).

In words: one third of $\pi \cdot \text{radius}^2 \cdot \text{height}$ , and the base $\pi r^2$ plus the side $\pi \cdot \text{radius} \cdot \text{slant height}$ .

Remark 36.4.2 · Slant Height

The slant height $s$ is the distance from the tip of the cone to the edge of its base. The height, the radius and the slant height make a right triangle, so

s = \sqrt{r^2 + h^2} = \sqrt{\text{radius}^2 + \text{height}^2}.

Try it

Problem 36.10.7

AMC 10/12

An inverted cone with base radius $12$ cm and height $18$ cm is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24$ cm. What is the height in centimeters of the water in the cylinder?

Contest cornerreal contest problems

Problem 36.10.20(additional)

AMC 10

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.

Problem 36.10.26(additional)

AMC 12

An ice-cream novelty item consists of a cup in the shape of a $4$ -inch-tall frustum of a right circular cone, with a $2$ -inch-diameter base at the bottom and a $4$ -inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?

Problem 36.10.5

MATHCOUNTS

The two cones shown have parallel bases and common apex $T$ . $TW = 32$ m, $WV = 8$ m, and $ZY = 5$ m. What is the volume of the frustum with circle $W$ and circle $Z$ as its bases? Express your answer in terms of $\pi$ .

Problem 36.10.15(additional)

MATHCOUNTS

A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fraction.

Problem 36.10.8

AMC 10/12

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 $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ 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?

Problem 36.10.29(additional)

AMC 10

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?

Problem 36.10.37(additional)

AIME

When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi$ cm $^3$ . When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi$ cm $^3$ . What is the length (in cm) of the hypotenuse of the triangle?

Problem 36.10.39(additional)

AIME

A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$ -inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is $m - n\sqrt[3]{p}$ , from the base where $m$ , $n$ , and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m + n + p$ .

Problem 36.10.13

AMC 12

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt3$ inches, its top diameter is $6$ inches, and its bottom diameter is $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

Checkpoint for section 36.4

Get 2 of 3 right to master this section. One try each, no hints.

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Sphere

In this section you will be able to use a sphere's volume and surface area, and see how they grow with the radius.

A sphere’s surface is exactly four of its great circles: $4\pi r^2$ .

$V = \frac43 \pi r^3$ and $S = 4\pi r^2$ . Volume grows like $r^3$ and area like $r^2$ , the same rule as for any similar solids.

Exploreplay first, formulas later

Try this: Change r from 3 to 6. The volume grows 8 times, the surface area 4 times. Why?
Try this: Turn on Cut and move it to the middle. The section is a great circle. How many great circles make the surface area?
Try this: Switch to Cone and set h = 6, so it is as tall as the sphere. How does its volume compare with the sphere's?

Name it

Theorem 36.5.1 · Volume and Surface Area of a Sphere

For a sphere with radius $r$ :

V = \frac43 \pi r^3, \qquad S = 4\pi r^2.

Try it

Problem 36.10.22(additional)

AMC 10

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 $4\pi r^3/3$ .)

Problem 36.10.36(additional)

AMC 10

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?

Contest cornerreal contest problems

Problem 36.10.6

AMC 10

Four congruent semicircles are drawn on the surface of a sphere with radius $2$ , as shown, creating a closed curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$ . What is $n$ ?

Checkpoint for section 36.5

Get 2 of 3 right to master this section. One try each, no hints.

36.60%

Tetrahedron

In this section you will be able to find the volume of any tetrahedron, and of a regular one from its edge.

Any tetrahedron is $\frac13 \cdot \text{base} \cdot \text{height}$ ; a regular one is $\frac{\sqrt2}{12}s^3$ .

Pick the most convenient face as the base. When three edges meet at right angles (like a box corner), use the right triangle as the base and the third edge as the height: $V = \frac16 abc$ .

Exploreplay first, formulas later

Try this: The edge is 6. The apex sits above the center of the base, 6/√3 from each corner. Check the height 2√6 ≈ 4.9.
Try this: Read the volume 18√2 and compare it with √2/12 · 6³.
Try this: Press Unfold: four equilateral triangles. What is the surface area for edge 6?

Name it

Theorem 36.6.1 · Volume of a Tetrahedron

For any tetrahedron:

V = \frac13 \cdot \text{base area} \cdot \text{height},

where the height is the distance from the fourth vertex to the plane of the base.

Theorem 36.6.2 · Volume of a Regular Tetrahedron

A regular tetrahedron (all six edges equal to $s$ ) has volume

V = \frac{\sqrt2}{12} s^3.

Contest cornerreal contest problems

Problem 36.10.10

AMC 10

What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$ , $AC = 3$ , $AD = 4$ , $BC = \sqrt{13}$ , $BD = 2\sqrt5$ , and $CD = 5$ ?

Problem 36.10.23(additional)

AMC 12

Triangle $ABC$ , with sides of length $5$ , $6$ , and $7$ , has one vertex on the positive $x$ -axis, one on the positive $y$ -axis, and one on the positive $z$ -axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$ ?

Problem 36.10.42(additional)

AIME

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 36.10.14

AIMEChallenge

Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41}$ , $AC = BD = \sqrt{80}$ , and $BC = AD = \sqrt{89}$ . There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m\sqrt n}{p}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m + n + p$ .

Checkpoint for section 36.6

Get 2 of 3 right to master this section. One try each, no hints.

36.70%

Pyramid

In this section you will be able to find the volume of a pyramid, and use similar pyramids cut parallel to the base.

Pyramids are one third of prisms. With a square base and all edges equal, $V = \frac{\sqrt2}{6}s^3$ .

A plane parallel to the base cuts off a small pyramid similar to the whole one. With scale factor $k$ , lengths scale by $k$ , areas by $k^2$ and volumes by $k^3$ .

Exploreplay first, formulas later

base a6

height h4.24

turn30°

tilt30°

unfold0%

Square pyramid: base side 6, height 4.24. Drag the picture or use turn and tilt.

Volume 36√2

V = \tfrac{\sqrt2}{6}s^3 = 36\sqrt{2}

Surface area 98.35

S = s^2 + 4 \cdot \tfrac{\sqrt3}{4}s^2 = 98.35

face height: $3\sqrt{3}$
height: $3\sqrt{2}$

Try this: All eight edges are 6. Read the volume 36√2 = √2/6 · 6³. What is the height?
Try this: Press Unfold, then fold it back up. Which right triangle gives the face height?
Try this: Turn on Cut and set the cut halfway up. The top is a small pyramid: what fraction of the volume is it?

Name it

Theorem 36.7.1 · Volume of a Pyramid

For any pyramid:

V = \frac13 \cdot \text{base area} \cdot \text{height}.

A pyramid fills exactly one third of the prism with the same base and height.

Theorem 36.7.2 · Volume of a Regular Square Pyramid

When the pyramid has a square base and all eight edges are equal to $s$ :

V = \frac{\sqrt2}{6} s^3.

Contest cornerreal contest problems

Problem 36.10.18(additional)

AMC 10

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?

Problem 36.10.27(additional)

AMC 12

A pyramid has a square base with side 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?

Problem 36.10.28(additional)

AMC 10

This problem also appears in the book as 36.10.27.

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?

Problem 36.10.17(additional)

AMC 12B

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?

Problem 36.10.11

AMC 12

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$ . Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA}$ , $\overline{MC}$ , and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD$ ?

Checkpoint for section 36.7

Get 2 of 3 right to master this section. One try each, no hints.

36.80%

Volume of Complex Polyhedrons

In this section you will be able to find the volume of a solid built from pieces you know: pyramids, prisms and cut-off corners.

Fold six triangles up around a hexagon. Where does the apex land?

Most “strange” solids are a pyramid, a prism, or a box with pieces added or cut away. Find the height with a right triangle, then use $\frac13 \cdot \text{base} \cdot \text{height}$ on each piece.

Exploreplay first, formulas later

Try this: Raise the unfold slider back to 0%. When do the triangle tips meet?
Try this: The tips meet above the center. Which right triangle gives the height?
Try this: Make the slanted edge b smaller, close to a. What happens to the height and the volume?

Watch itpredict, then reveal

Example 36.1

AMC 10

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$ . The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

1
Fold the six triangles up until their tips meet. Where is the apex, compared with the hexagon?

open on YouTube

Contest cornerreal contest problems

Problem 36.10.21(additional)

AMC 12

This problem also appears in the book as 36.1.

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$ . The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

Problem 36.10.9

AMC 10A

A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$ . The edges of 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?

Problem 36.10.12

AMC 12A

The base of the pentahedron shown below is a $13 \times 8$ rectangle, and its lateral faces are two isosceles triangles with base of length $8$ and congruent sides of length $13$ , and two isosceles trapezoids with bases of length $7$ and $13$ and nonparallel sides of length $13$ . What is the volume of the pentahedron?

Problem 36.10.38(additional)

AIME

The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All other edges have length $s$ . Given that $s = 6\sqrt2$ , what is the volume of the solid?

Checkpoint for section 36.8

Get 2 of 3 right to master this section. One try each, no hints.

36.90%

Cross Sections

In this section you will be able to slice a 3-D figure through its centers and solve the 2-D picture you get.

3-D problems become 2-D when you slice through the centers.

Spheres touching each other or a flat face: slice through their centers, and each sphere becomes a circle of the same radius. Touching spheres have centers $r_1 + r_2$ apart.

Look:

A sphere cut by a plane at distance $d$ from its center gives a circle of radius $\sqrt{R^2 - d^2}$ . That one right triangle solves most “sphere and plane” problems.

Exploreplay first, formulas later

Try this: The plane goes through three edge midpoints and cuts a regular hexagon. Change point 1 and watch the shape change.
Try this: Set the points to E, mid BF and C. What shape is the section? Check its diagonals.
Try this: Try three corners next to one vertex, like B, D and E. What triangle do you get?

Watch itpredict, then reveal

Example 36.2

AIME

Eight spheres of radius $100$ are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a + b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c$ .

1
Look down from above. Where are the eight centers, and how far apart are neighbors?

Predict first:
Your prediction

open on YouTube

Example 36.3

AIME

A right prism with height $h$ has bases that are regular hexagons with sides of length $12$ . A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$ .

1
$A$ 's three neighbors are $B$ and $F$ in the base and $A'$ straight above $A$ . Which face of the pyramid lies in the base, and which one doesn't contain $A$ ?

open on YouTube

Example 36.4

AMC 10

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 is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

1
Each side touches the sphere at one point. What does the triangle's plane cut out of the sphere, and how is it related to the triangle?

open on YouTube

Example 36.5

AMC 10

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
Look from above. The three centers sit at height $r$ . What shape do they form, and what is its side?

Predict first:
Click a choice or type its letter (A–E).

open on YouTube

Example 36.6

AMC 12

A $4 \times 4 \times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$ . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$ ?

1
Put a corner of the box at the origin. Where is the center of the small sphere in that corner?

open on YouTube

Contest cornerreal contest problems

Problem 36.10.3

AMC 8

In the cube $ABCDEFGH$ with opposite vertices $C$ and $E$ , $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD}$ , respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2$ ?

Problem 36.10.24(additional)

AMC 10

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 $ABCD$ ?

Problem 36.10.41(additional)

AMC 10

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?

Problem 36.10.30(additional)

AMC 10/12

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?

Problem 36.10.31(additional)

AMC 10

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?

Problem 36.10.44(additional)

AIMEChallenge

Cube $ABCDEFGH$ , labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .

Checkpoint for section 36.9

Get 2 of 3 right to master this section. One try each, no hints.

36.10

Practice Problems

In this section you will be able to mix the chapter's tools on one more contest problem.

Mixed practice: pick the formula, find the hidden right triangle, or slice the solid.

Most practice problems for this chapter appear in the lesson they use. This last one mixes two ideas: face centers and the $\frac13$ rule.

Contest cornerreal contest problems

Problem 36.10.34(additional)

AMC 10

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