Equable Cylindrical Cone

EQUABLE CYLINDRICAL CONE

Balmoral Software

Solutions: 1

Consider the "cylindrical cone" consisting of a cone of radius R and height h surmounting a cylinder of the same radius and height. There is one such polytope that is equable. The total volume of the cone + cylinder is V, where

V/π = (1/3)R²h + R²h = (4/3)R²h

The lateral surface area of the cone is S_cone, where

The lateral surface area of the cylinder is S_cyl, where

S_cyl/π = 2Rh [1]

The surface area of the cylinder base is S_base, where

S_base/π = R² [2]

The above four quantities are positive integers by convention. We have

so R is rational. By [2], R² is an integer, so it follows that R is an integer.

The equability assumption is

Any equable solution will have a positive left side. Squaring both sides, we have

(16R² - 48R + 27)h = 24R² - 36R [3]

If R = 1, then h = 12/5 and S_cyl/π = 24/5 is not an integer. If R = 2, then h < 0. Therefore, R ≥ 3 and

so both the right side and the coefficient of h in [3] are strictly positive, and a rational solution for h is

[4]

From [1],

for some integer k. We previously established that the denominator is positive, and the numerator is also positive since R ≥ 3, so k is positive. The preceding equation can be written as a quadratic in R:

(16k - 72)R² - (48k - 81)R + 27k = 0 [5]

Since R is an integer, the discriminant of this quadratic must be a square:

576k² + 6561 = m² for some integer m
(m - 24k)(m + 24k) = 6561

The possibilities are summarized below:

m - 24k < m + 24k Sum = 2m m k = [(m + 24k) - m]/24

1 < 6561 6562 3281 (6561 - 3281)/24 is not an integer

3 < 2187 2190 1095 (2187 - 1095)/24 is not an integer

9 < 729 738 369 (729 - 369)/24 = 15

27 < 243 270 135 (243 - 135)/24 is not an integer

m - 24k	<	m + 24k	Sum = 2m	m	k = [(m + 24k) - m]/24
1	<	6561	6562	3281	(6561 - 3281)/24 is not an integer
3	<	2187	2190	1095	(2187 - 1095)/24 is not an integer
9	<	729	738	369	(729 - 369)/24 = 15
27	<	243	270	135	(243 - 135)/24 is not an integer

The only solution is k = 15, and thus [5] can be written

168R² - 639R + 405 = 0,

which has the sole integer solution R = 3. From [4], we have h = 4 and

V = 48 π
S_cone = 15 π
S_cyl = 24 π
S_base = 9 π

The equable cylindrical cone is a solid of revolution of a non-equable quadrilateral consisting of a 4 x 3 right triangle with its shortest side appended to a 4 x 3 rectangle, having perimeter 20 and area 18 and rotated along its longest side.

Non-equable case

It's interesting that without the equability restriction, the integer surface area sections have the property that

S_cone² = S_base² + (S_cyl/2)²

Therefore, any Pythagorean triple a² + b² = c², for which

[6]

generates the non-equable solution

For any Pythagorean triple, an alternate solution with a and b exchanged is possible as long as [6] is satisfied. Applying the limits on a and b, we have

h² > 9/16 + 81/624 ≈ 0.6923

Any element of a Pythagorean triple is at least 3, so R² = a ≥ 3 and

which reduces to

V > S_cone

Therefore, all solutions for which S_cone ≤ M for some maximum M will include as a subset all solutions with V ≤ M. The 23 solutions having volume less than 2000 π are listed below:

R h S_cone/π S_base/π S_cyl/π V/π

2 3/2 5 4 6 8

3 4 15 9 24 48 *

4 3 20 16 24 64

6 5/2 39 36 30 120

3 40/3 41 9 80 160

4 15/2 34 16 60 160

6 9/2 45 36 54 216

12 17/12 145 144 34 272

4 63/4 65 16 126 336

6 8 60 36 96 384

5 12 65 25 120 400

8 6 80 64 96 512

6 77/6 85 36 154 616

12 7/2 150 144 84 672

6 35/2 111 36 210 840

12 5 156 144 120 960

10 15/2 125 100 150 1000

6 80/3 164 36 320 1280

8 15 136 64 240 1280

9 12 135 81 216 1296

10 21/2 145 100 210 1400

7 24 175 49 336 1568

12 9 180 144 216 1728

*: Equable

R	h	S_cone/π	S_base/π	S_cyl/π	V/π
2	3/2	5	4	6	8
3	4	15	9	24	48 *
4	3	20	16	24	64
6	5/2	39	36	30	120
3	40/3	41	9	80	160
4	15/2	34	16	60	160
6	9/2	45	36	54	216
12	17/12	145	144	34	272
4	63/4	65	16	126	336
6	8	60	36	96	384
5	12	65	25	120	400
8	6	80	64	96	512
6	77/6	85	36	154	616
12	7/2	150	144	84	672
6	35/2	111	36	210	840
12	5	156	144	120	960
10	15/2	125	100	150	1000
6	80/3	164	36	320	1280
8	15	136	64	240	1280
9	12	135	81	216	1296
10	21/2	145	100	210	1400
7	24	175	49	336	1568
12	9	180	144	216	1728

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