This whole thing was stolen from another board... but it is certainly applicable here.
I have emerged from my winter cave with the results of much research, and possibly this site's longest post. For anyone interested...
I thought it would be interesting to measure some of my favorite mountains in WA to see which ones were biggest/steepest from top to bottom. Instead of looking at the elevation of a peak, which does not tell you how tall a mountain is, only how high above sea level it is, I measured summit to immediate valley -- vertical relief. A tall mountain may not be impressive if it isn't steep, so I measured steepness, too.
I borrowed a formula devised by Edward Earl from the Yahoo Prominence group to come up with values that factor both height and steepness.
He calls it "Cliff Measure" or CM = (rise^2)/(rise+run).
To make the numbers more intuitive:
A 1,000 ft flagpole gets a CM of 1,000. A 1,000 ft hill with a 45-degree slope gets a score of 500. Basically take the vertical relief of the peak and subtract from that number based on the steepness (or lack of steepness) of the slope.
It's difficult to figure out the maximum CM for a peak without writing a program to do it, so I made an educated guess. Usually that meant measuring from the summit down the mountain until the terrain starts to flatten out in the valley below.
However, David Metzler, a mathematician, mountain afficionado and all-around good guy, did figure out some maximum CMs for comparison to some of the planet's best:
Peak Max CM
---------------------------------------
Rainier 3320 feet
Davis Pk (N Cascades) 2717
Half Dome 2714
El Capitan 2431
Whitney 1936
Logan 6043
Denali 6027
Annapurna Fang 7792
Gurja Himal 7769
--------------------------------------------------------------
Here are some of my calculations, best read with a fixed-width font
---Cascades---
Peak Vertical Relief (ft) Distance (mi) Aspect CM
Davis Pk * 5,623 1.19 ne 2655.6
Castle Rock 7,100 2.30 east 2619.5
Rainier ** 11,400 7.50 nw 2548.2
McGregor 6,700 2.10 south 2523.6
Tricouni 6,082 1.64 east 2509.3
Goode 6,000 1.60 ne 2491.7
Despair 5,763 1.47 east 2455.7
Agnes 5,771 1.48 north 2451.5
Elephant Butte 5,220 1.14 north 2424.4
Rainier 12,000 9.00 sw 2419.4
Eldorado 6,386 2.00 west 2406.5
Jack 7,450 3.00 west 2383.1
HiddenLakeLO 5,830 1.60 sw 2380.5
Blum 6,750 2.35 north 2378.2
Hozomeen 5,040 1.09 north 2353.0
Big Devil 6,635 2.30 north 2344.3
Colonial 6,500 2.20 ne 2332.2
Luna 5,100 1.15 north 2328.1
Johannesburg 5,450 1.40 north 2312.9
Flora 7,300 3.00 east 2302.9
Primus 6,500 2.25 north 2298.7
Ruby 6,200 2.00 west 2293.6
Snowking ne ridge 4,514 0.85 ne 2263.5
Snowfield subpk 6,137 2.00 south 2255.7
Davis Pk 6,100 2.00 south 2233.5
Whitehorse 6,100 2.00 north 2233.5
Primus 6,535 2.40 east 2223.5
Pt 6872 (l beaver crk) 4,522 0.89 north 2217.6
Sloan Pk 5,540 1.60 se 2194.1
Forbidden 6,100 2.10 south 2164.9
Big Four 4,200 0.75 north 2161.8
Crater 6,250 2.24 south 2160.9
Triumph 5,000 1.25 north 2155.2
Pugh 6,194 2.20 west 2154.2
Eldorado 7,000 3.00 west 2145.4
Terror 7,000 3.00 south 2145.4
Baring 3,675 0.50 north 2138.7
Pyramid Mtn *** 7,145 3.20 east 2123.5
Booker 5,660 1.80 south 2112.6
White Chuck 5,801 1.92 south 2111.5
Three Fingers 4,000 0.70 east 2079.0
Jack 6,665 2.80 north 2071.1
Luna 5,815 2.00 south 2065.0
Johannesburg 6,800 3.00 west 2042.4
Pioneer RidgeW 4,245 0.88 west 2026.7
Shuksan 4,590 1.10 north 2026.2
1660m 4,350 0.95 south 2020.3
Formidable 5,969 2.29 west 2020.0
Dome 6,000 2.25 south 2013.4
Garfield 4,000 0.75 south 2010.1
Index 3,231 0.38 west 1993.2
Baring 3,138 0.35 north 1974.9
Shuksan 8,260 5.00 se 1968.5
Booker 4,019 0.83 ne 1922.6
Hozomeen 6,450 2.90 west 1911.7
Baring 5,265 1.75 sw 1911.1
Glacier Pk 7,034 3.60 west 1900.0
Church 5,150 1.70 south 1877.6
Baker 10,150 8.50 se 1872.1
Goode 5,300 1.85 sw 1864.2
Snowfield 7,100 3.80 east 1855.8
Blum 7,180 4.00 west 1821.6
Garfield 4,460 1.25 south 1798.5
Logan 7,000 3.85 nw 1793.0
Shuksan 4,770 1.50 ne 1793.0
Triumph 6,300 3.00 east 1792.7
Pioneer Ridge 4,100 1.00 north 1792.1
Challenger 5,000 1.70 east 1788.8
Buckner 6,000 2.70 south 1777.3
Shuksan **** 8,431 6.00 south 1772.1
Index 3,220 0.50 north 1769.4
* Davis Peak near Newhalem, believed to be the only place in the lower 48 that drops over one vertical mile within one horizontal mile. Max CM is 2,717 according to David Metzler. You can see the upper NE face from the Diablo Lake overlook.
** Metzler calculated Rainier's max CM to be 3,320, by far the best in the lower 48 - Point Success to Kautz Creek: 9,022 feet in 2.89 miles.
*** This measures to the edge of Lake Chelan. Interestingly, the lake's deepest point is near this peak, which makes 8,631 ft of vertical, the deepest gorge in North America. To the middle of the lake it is 3.75 miles, for a CM roughly of 2620, just guessing.
**** Probably the greatest vertical relief for a non-volcanic peak in WA. The CM isn't very high because this measures over the relatively flat and expansive Sulphide Glacier. This measures to the surface of dam-made Baker Lake.
Here are a few other major peaks in the lower 48. Again, not necessarily max CM:
---Olympics---
Vertical Distance Aspect CM
Constance 6,823 3.04 south 2035.19
Carrie 5,645 2.45 sw 1714.98
Tom 5,624 2.50 sw 1680.78
Anderson 4,315 1.36 east 1619.65
Brothers 3,726 1.00 se 1541.54
Olympus 7,000 5.43 nw 1373.69
Olympus 5,503 3.70 se 1209.43
---Sierras---
I used the book "The High Sierra: Peaks, Passes and Trails" to identify the major peaks in the range. The biggest peaks in the Sierras are the fault block escarpments like San Jacinto and the eastern wall of the high sierra. These peaks get significant extra vertical from abutting large valleys that drop slowly. I measured both to the edge of the valley and to the deepest points in the valleys. The huge steep walls of Yosemite also make high CMs, with Half Dome the highest in CA at CM=2714 according to Metzler.
CM Peak Vertical Distance Aspect
2649.9 San Jacinto 8,867 3.94 north
2286.5 El Cap 3,134 0.22 south (the nose)
2246.8 San Jacinto 10,175 6.80 east
2241.3 Lone Pine 6,985 2.80 east (to edge of valley)
2234.0 Williamson 8,383 4.37 east (to edge of valley)
2035.9 El Cap 3,550 0.50 south
2003.5 Telescope 10,024 7.60 west (to edge of valley)
1799.6 Whitney * 7,919 5.10 east (to edge of valley)
1732.0 Telescope 9,049 7.24 east (to edge of valley)
1698.0 Telescope 11,256 12.00 east
1642.1 Morrison 4,635 1.60 north
1491.4 Whitney 10,695 12.50 east
1295.1 Humphreys 7,000 5.84 east
1218.2 Mt Sill 4,585 2.40 east
1204.8 Whitney 2,000 0.25 east face
1150.0 Ritter 3,885 1.75 east
1117.0 Matterhorn Pk 5,200 3.60 north
1102.2 Black Kaweah 4,405 2.50 south
1060.4 N Palisade 4,000 2.10 north
990.6 Winchell 5,056 3.93 west
638.4 Darwin 4,005 4.00 ne
472.2 Darwin 2,981 3.00 west
* Whitney Max CM is 1936
---Rockies---
Obviously not comprehensive, I don't have software for this region, maybe someone with topo software and a lot of time on their hands can fill out this list.
Peak Vertical Distance Aspect CM
MT
Kinnerly* 5,550 1.13 north 2680.8
WY
Grand Teton 6,910 3.00 east 2098.8
Squaretop 3,595 0.72 north 1747.2
CO
Sopris W 6,307 2.62 west 1975.0
N Maroon 3,814 0.87 east 1730.2
Maroon 3,756 0.89 east 1668.5
Sneffels 4,550 1.43 nw 1710.8
Longs 2,455 0.48 ne 1207.9 to chasm lake
Longs 5,950 5.08 east 1080.2
Blanca 5,745 6.84 se 788.4
Blanca 1,945 0.30 north 1071.9
Pikes Pk 3,010 1.21 north 963.9
Pikes Pk ** 5,110 3.83 east 1030.7 to Hell Gate
* Glacier NP, one of the top contenders in the lower 48 for big, steep faces.
** I deemed this the maximum vertical for Pikes, though it is subjective.