2.7 3-D geometry

Achievable AMC 10/12

2. Geometry

3-D geometry

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3-D geometry

Here are a few things that I just want all AMC competitors to know:

The formulas for common three-dimensional shapes, including prisms with arbitrary bases, pyramids with arbitrary bases, and spheres
The fact that all two- and three-dimensional measurements of arbitrary similar shapes use the square and cube of their relative scale factors, respectively.

Examples

Example 1

What’s the volume of a pyramid whose base is a regular seven-pointed star with area $7$ , and whose height is $13$ ?

(spoiler)

All pyramids (and cones) have volume $\frac{B h}{3}$ , irrespective of the shape of the base.

If you got hung up on the star, you need the drillwork in this unit.

But if you got $\frac{91}{3}$ without a hiccup, you’re good.

Example 2

I have two spherical balloons, one of which holds four times as much air as the other. What’s the ratio of their surface areas?

(spoiler)

If you used the volume and/or surface area formulas for spheres, then I’m afraid you fail this test.

And if you got an integral ratio, then you were probably careless – also a fail, I’m afraid.

The right way – and I’m afraid there is only one right way in this case – is to recognize that a volume ratio of $4 : 1$ means a radius ratio of $34 : 31$ , and a surface area ratio of $3 4^{2} : 3 1^{2}$ , a.k.a. $4^{2/3} : 1$ or $316 : 1$ .

Formulas for 3-D Shapes

Prism volume: $B h$ (base area × height)
Pyramid/cone volume: $\frac{1}{3} B h$
Sphere volume: $\frac{4}{3} π r^{3}$ , surface area: $4 π r^{2}$

Scaling in Similar Shapes

2-D measures (area): scale factor squared
3-D measures (volume): scale factor cubed
Surface area ratio: $(scale factor)^{2}$
Volume ratio: $(scale factor)^{3}$

Key Example Insights

Pyramid/cone volume formula applies to any base shape
For similar solids:
- Volume ratio $\to$ scale factor cubed
- Surface area ratio $\to$ scale factor squared
- Use roots/powers to relate volume and surface area ratios

3-D geometry

Here are a few things that I just want all AMC competitors to know:

The formulas for common three-dimensional shapes, including prisms with arbitrary bases, pyramids with arbitrary bases, and spheres
The fact that all two- and three-dimensional measurements of arbitrary similar shapes use the square and cube of their relative scale factors, respectively.

Examples

Example 1

What’s the volume of a pyramid whose base is a regular seven-pointed star with area $7$ , and whose height is $13$ ?

(spoiler)

All pyramids (and cones) have volume $\frac{B h}{3}$ , irrespective of the shape of the base.

If you got hung up on the star, you need the drillwork in this unit.

But if you got $\frac{91}{3}$ without a hiccup, you’re good.

Example 2

I have two spherical balloons, one of which holds four times as much air as the other. What’s the ratio of their surface areas?

(spoiler)

If you used the volume and/or surface area formulas for spheres, then I’m afraid you fail this test.

And if you got an integral ratio, then you were probably careless – also a fail, I’m afraid.

Achievable AMC 10/12

2. Geometry

3-D geometry

2 min read

Font

Discuss

Feedback

3-D geometry

Here are a few things that I just want all AMC competitors to know:

The formulas for common three-dimensional shapes, including prisms with arbitrary bases, pyramids with arbitrary bases, and spheres
The fact that all two- and three-dimensional measurements of arbitrary similar shapes use the square and cube of their relative scale factors, respectively.

Examples

Example 1

What’s the volume of a pyramid whose base is a regular seven-pointed star with area $7$ , and whose height is $13$ ?

(spoiler)

All pyramids (and cones) have volume $\frac{B h}{3}$ , irrespective of the shape of the base.

If you got hung up on the star, you need the drillwork in this unit.

But if you got $\frac{91}{3}$ without a hiccup, you’re good.

Example 2

I have two spherical balloons, one of which holds four times as much air as the other. What’s the ratio of their surface areas?

(spoiler)

If you used the volume and/or surface area formulas for spheres, then I’m afraid you fail this test.

And if you got an integral ratio, then you were probably careless – also a fail, I’m afraid.

Formulas for 3-D Shapes

Prism volume: $B h$ (base area × height)
Pyramid/cone volume: $\frac{1}{3} B h$
Sphere volume: $\frac{4}{3} π r^{3}$ , surface area: $4 π r^{2}$

Scaling in Similar Shapes

2-D measures (area): scale factor squared
3-D measures (volume): scale factor cubed
Surface area ratio: $(scale factor)^{2}$
Volume ratio: $(scale factor)^{3}$

Key Example Insights

Pyramid/cone volume formula applies to any base shape
For similar solids:
- Volume ratio $\to$ scale factor cubed
- Surface area ratio $\to$ scale factor squared
- Use roots/powers to relate volume and surface area ratios

3-D geometry

Here are a few things that I just want all AMC competitors to know:

The formulas for common three-dimensional shapes, including prisms with arbitrary bases, pyramids with arbitrary bases, and spheres
The fact that all two- and three-dimensional measurements of arbitrary similar shapes use the square and cube of their relative scale factors, respectively.

Examples

Example 1

What’s the volume of a pyramid whose base is a regular seven-pointed star with area $7$ , and whose height is $13$ ?

(spoiler)

All pyramids (and cones) have volume $\frac{B h}{3}$ , irrespective of the shape of the base.

If you got hung up on the star, you need the drillwork in this unit.

But if you got $\frac{91}{3}$ without a hiccup, you’re good.

Example 2

I have two spherical balloons, one of which holds four times as much air as the other. What’s the ratio of their surface areas?

(spoiler)

If you used the volume and/or surface area formulas for spheres, then I’m afraid you fail this test.

And if you got an integral ratio, then you were probably careless – also a fail, I’m afraid.