求錐體ABCDE的體積

Question

設ABCDE為一個以菱形為底的錐體，其中E是錐體的頂點。已知菱形ABCD的邊長為11，AE=CE=12，且BE=DE=11。求錐體的體積。

Answer 1

假設菱形ABCD的兩對角線為 a,b
錐體高度 h 則
122 = h2 + (a/2)2 ... (1)
112 = h2 + (b/2)2 ... (2)
(1) - (2)
23 = (a/2)2 - (b/2)2 ... (3)
112 = (a/2)2 + (b/2)2 ... (4)
[(3)+(4)]/2 => 72 = (a/2)2
=> a/2 = 6√2
=> a = 12√2
[(4)-(3)]/2 => 49 = (b/2)2
=> b/2 = 7
=> b = 14
代入 (1) => h2 = 144 - 72 = 72, h = 6√2
菱形面積 = ab/2 = 12√2 * 14/2 = 84√2
錐體的體積 = (1/3) * 84√2 * 6√2 = 336
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