29. 3D Geometry · Mastering AMC 8

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Cube

In this section you will be able to find a cube's volume, surface area and the long diagonal that cuts through its middle.

A cube-shaped box has sides 1 foot long. What is the longest straight stick you could hide inside it? Longer than 1 foot. Even longer than the diagonal of a face!

A cube has 6 square faces, 12 equal edges and 8 corners. With side $a$ , it holds $a^3$ unit cubes, and its 6 faces have area $6a^2$ .

The space diagonal goes from one corner to the opposite corner, straight through the middle. Use Pythagoras twice: first across a face, then up.

Exploreplay first, formulas later

Try this: Press Face diagonal. The bottom face is a square with side 1. Which right triangle gives the diagonal √2?
Try this: Now press Space diagonal. The new right triangle has legs √2 and 1. Why is its hypotenuse √3?
Try this: Make all three sides 2. Does the space diagonal become 2√3? Try side 3 too.

Name it

Theorem 29.1.1 · Volume and Surface Area of a cube

\text{Volume of a cube} = (\text{side length})^3 = a^3

\text{Surface area of a cube} = 6 \times (\text{side length})^2 = 6a^2

\text{Length of space diagonal of a cube} = \sqrt{3} \times \text{side length} = \sqrt{3}a

Along the way, a face diagonal has length $\sqrt{2}a$ .

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Example 29.1

AMC 12

What is the volume of a cube whose surface area is twice that of a cube with volume 1?

1
A cube has volume 1. How long is its side?

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Contest cornerreal contest problems

Problem 29.8.5

AMC 8

In the cube $ABCDEFGH$ with opposite vertices $C$ and $E$ , $J$ and $I$ are the midpoints of edges $\overline{FB}$ 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 29.8.6

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?

Checkpoint for section 29.1

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

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Prism

In this section you will be able to find the volume and surface area of any prism, and the space diagonal of a box.

A cereal box is 2 inches deep, 8 wide and 12 tall. How much cardboard does it take? Unfold it and you get six rectangles that come in three matching pairs.

A prism has two matching bases joined by rectangles. A box (a rectangular prism) is the prism whose base is a rectangle.

For every prism, stack copies of the base: the volume is base area × height. Unfold it to see the surface area: two bases plus one long strip as wide as the base’s perimeter.

Exploreplay first, formulas later

Try this: Change the length, width and height. Which readout grows fastest when you double one side?
Try this: Press Unfold. How many rectangles do you see? Which pairs have the same size?
Try this: Switch to the triangular prism. Is the volume still base area times height?

Name it

Theorem 29.2.1 · Volume and Surface Area of a rectangular prism

\text{Volume of a rectangular prism} = l \times b \times h = \text{product of all three dimensions}

\text{Surface area of a rectangular prism} = 2(lb + bh + lh)

\text{Length of space diagonal of a rectangular prism} = \sqrt{l^2 + b^2 + h^2}

Theorem 29.2.2 · Volume of a Prism

\text{Volume of a Prism} = \text{base area} \cdot \text{height}

Theorem 29.2.3 · Surface Area of a Prism

\text{Surface Area of a Prism} = 2 \cdot \text{base area} + \text{base perimeter} \times \text{height}

Unfold the prism: the two bases, plus one long strip whose width is the base perimeter.

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Example 29.2

AMC 8

The figure below shows a polygon $ABCDEFGH$ , consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$ . What is the volume of the prism?

1
Rectangle $ABJH$ folds into a side of the prism. Which of its sides is a leg of the right triangle $BJG$ , and how long is it?

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Contest cornerreal contest problems

Problem 29.8.2

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?

Checkpoint for section 29.2

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

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Pyramid

In this section you will be able to find the volume of a pyramid as one third of the prism with the same base and height.

Fill a hollow pyramid with water and pour it into a box with the same base and the same height. How many pours fill the box? Guess first, then try it.

A pyramid has a base and slanted triangle faces that meet at one point, the apex. Its height is measured straight down from the apex.

Any pyramid holds exactly one third of the matching prism. A tetrahedron is a pyramid with a triangle base.

Exploreplay first, formulas later

Try this: Press Pour one scoop. How high does the water rise in the prism?
Try this: How many scoops fill the prism exactly? What fraction is one pyramid?
Try this: Change the base side and the height. Does the number of scoops change?

Name it

Theorem 29.3.1 · Volume of a Pyramid

\text{Volume of any pyramid} = \frac{1}{3} \cdot \text{base area} \cdot \text{height}

The height $h$ is measured straight down from the apex to the base.

Theorem 29.3.2 · Volume of a Regular Square Pyramid (all sides equal)

When the pyramid has a square base, and all the sides are equal

\text{Volume of a regular pyramid} = \frac{\sqrt{2}}{6}s^3

Theorem 29.3.3 · Volume of a Regular Tetrahedron (all sides equal)

\text{Volume of a regular tetrahedron} = \frac{\sqrt{2}}{12}s^3

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Example 29.3

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 $OABC$ ?

1
Call $OA = a$ , $OB = b$ , $OC = c$ . The three axes are perpendicular. Which theorem links $a$ , $b$ and $AB = 5$ ?

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Checkpoint for section 29.3

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 by stacking circles and unrolling the side.

Peel the label off a soup can and lay it flat. What shape do you get? A rectangle! Its width is the distance around the can.

A cylinder is a prism with a circle for its base. So its volume is still base area × height: $\pi r^2 h$ .

Its surface is two circles plus the side. The side unrolls into a rectangle that is $2\pi r$ wide and $h$ tall.

Exploreplay first, formulas later

Try this: Change the radius from 2 to 4. The volume grows by how many times? Why not just 2?
Try this: Press Unfold. The side becomes a rectangle. How wide is it?
Try this: Turn on Cut. Every slice is the same circle. How does that explain πr²h?

Name it

Theorem 29.4.1 · Volume and Surface Area of a cylinder

\text{Volume of a cylinder} = \pi r^2 h

\text{Surface area of a cylinder} = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)

The two circles give $2\pi r^2$ . The side unrolls into a rectangle $2\pi r$ wide and $h$ tall.

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Problem 29.8.1

AMC 8

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume one of Felicia's cans?

Problem 29.8.10(additional)

BmMT

Shivani is planning a road trip in a car with special new tires made of solid rubber. Her tires are cylinders that have width 6 inches and have diameter 26 inches, but need to be replaced when the diameter is less than 22 inches. The tire manufacturer claims that $0.12\pi$ cubic inches of its tire will wear away with every single rotation. Assuming that the tire manufacturer is correct about the wear rate of their tires, and that the tire maintains its cylindrical shape and width (losing volume by reducing radius), how many revolutions can each tire make before she needs to replace it?

Contest cornerreal contest problems

Problem 29.8.3

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?

Problem 29.8.9(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?

Checkpoint for section 29.4

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

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Cone

In this section you will be able to find a cone's volume, slant height and surface area, and unroll its side into a sector.

Cut a party hat along a straight line and flatten it. You don’t get a triangle. You get a slice of a pizza: a sector of a circle!

A cone is a pyramid with a circle base. So it holds one third of the matching cylinder: $\frac13\pi r^2h$ .

The slant height $s$ runs from the tip down the side to the rim. Unroll the side: the sector’s radius is $s$ and its arc is the base’s circumference.

Exploreplay first, formulas later

Try this: Read the radius, the height and the slant height. Which right triangle connects them?
Try this: Press Unfold. The side becomes a sector. How long is its curved edge?
Try this: Make the radius equal to the slant height... you can't! Why must the sector always be less than a full circle?

Name it

Theorem 29.5.1 · Volume and Surface Area of a cone

\text{Volume of a cone} = \frac{1}{3}\pi r^2 h

which basically means

\text{Volume of a Cone} = \frac{1}{3}\pi \cdot \text{radius}^2 \cdot \text{height}

\text{Surface area of a cone} = \pi r^2 + \pi r s = \pi r(r + s)

where $s$ is the lateral or slant height, which can also be written as

\pi \cdot \text{radius}^2 + \pi \cdot \text{radius} \times \text{slant height}

Concept 29.5.3 · Cone Unfolding

The sides of a cone can be unfolded to form a sector a circle as well. The circumference of the base of the cone is equal to the length of the arc of the sector and the slant height of the cone is equal to the radius of the sector.

So the sector's angle is $\dfrac{2\pi r}{2\pi s} \times 360^\circ = \dfrac{r}{s} \times 360^\circ$ .

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Example 29.4

EMCC

An ant sits on the circumference of the circular base of a party hat (a cone without a circular base for the ant to walk on) of radius 2 and height $\sqrt{5}$ . If the ant wants to reach a point diametrically opposite of its current location on the hat, what is the minimum possible distance the ant needs to travel?

1
First, the slant height. What is $s = \sqrt{r^2 + h^2}$ ?

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Contest cornerreal contest problems

Problem 29.8.8(additional)

EMCC

The tiny island nation of Konistan is a cone with height 12 meters and base radius 9 meters, with the base of the cone at sea level. If the sea level rises 4 meters, what is the surface area of Konistan that is still above water, in square meters?

Checkpoint for section 29.5

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 find a sphere's volume and surface area, and use the centres of touching spheres.

Four marbles lie on a table, each touching two others. What is the smallest ball that could hold them all? The trick is to look at the centres.

A sphere is every point at distance $r$ from a centre. Two touching spheres have centres exactly $r_1 + r_2$ apart.

To find a big sphere that holds small ones, find the farthest small centre, then add the small radius.

Exploreplay first, formulas later

Try this: Double the radius. How many times bigger is the volume? And the surface area?
Try this: Turn on Cut and move the cut. Where is the cross-section circle biggest?
Try this: Set the radius to 3. Why are the volume and the surface area the same number of π?

Name it

Theorem 29.6.1 · Volume and Surface Area of a sphere

\text{Volume of a sphere} = \frac{4}{3}\pi r^3

\text{Surface area of a sphere} = 4\pi r^2

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Example 29.5

Each of 4 spheres of radius 1 are tangent to 2 of the other spheres such that all 4 of their centers lie on a plane. What is the radius of the smallest sphere that contains these 4 spheres?

1
Two tangent spheres of radius 1: how far apart are their centres?

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Example 29.6

AMC 12

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

1
A unit sphere in the first octant touches all three coordinate planes. Where is its centre?

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Checkpoint for section 29.6

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

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Similar Triangles in 3D

In this section you will be able to use similar solids: lengths scale by k, areas by k², volumes by k³.

Slice the tip off an ice cream cone, parallel to the top. The tip is a tiny cone with exactly the same shape. How much smaller is it, really?

A cut parallel to the base makes a smaller similar solid. Side view: two similar triangles.

If every length is multiplied by $k$ , every area is multiplied by $k^2$ and every volume by $k^3$ . A frustum is the big solid minus the small one.

Exploreplay first, formulas later

Try this: Move the cut. When the small cone is half as tall, what fraction of the volume is it?
Try this: Set 3 slices and cut after slice 2. Are the lengths, areas and volumes 2/3, 4/9 and 8/27?
Try this: Turn on Show every piece. Why do the pieces grow 1, 7, 19, …?

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Example 29.7

MATHCOUNTS

A cylinder whose height is 3 times its radius is inscribed in a cone whose height is 6 times its radius. What fraction of the cone's volume lies inside the cylinder? Express your answer as a common fraction.

1
Call the cone's radius $R$ (height $6R$ ) and the cylinder's radius $r$ (height $3r$ ). Look at the small cone above the cylinder. What are its height and radius?

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Example 29.8

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.

1
The top piece is a small cone. Call its volume $1$ . The top two pieces together make a cone twice as tall. What is its volume?

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Contest cornerreal contest problems

Problem 29.8.7(additional)

MATHCOUNTS

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

Problem 29.8.11(additional)

EMCC

A piece of paper is in the shape of an equilateral triangle $ABC$ with side length 12. Points $A_B$ and $B_A$ lie on segment $AB$ , such that $AA_B = 3$ , and $BB_A = 3$ . Define points $B_C$ and $C_B$ on segment $BC$ and points $C_A$ and $A_C$ on segment $CA$ similarly. Point $A_1$ is the intersection of $A_CB_C$ and $A_BC_B$ . Define $B_1$ and $C_1$ similarly. The three rhombi — $AA_BA_1A_C$ , $BB_CB_1B_A$ , $CC_AC_1C_B$ — are cut from triangle $ABC$ , and the paper is folded along segments $A_1B_1$ , $B_1C_1$ , $C_1A_1$ , to form a tray without a top. What is the volume of this tray?

Checkpoint for section 29.7

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

29.8

Practice Problems

In this section you will be able to mix every 3D tool, and a 2D warm-up, on contest problems.

One more from the AMC 8! It looks flat, but the same tools work: right triangles and areas. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 29.8.4

AMC 8

A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$ , and the other of length $b$ . What is the value of $ab$ ?