Eine gerade quadratische Pyramide hat die Grundfläche ABCD.
A = [0, 0, 3], B = [4, 4, 5], C = [?, ?, ?], D = [4, -2, -1]
Die Spitze S liegt in der Ebene E: z = 8.
a) Ermitteln Sie die Koordinaten von C und S.
C = B + AD = B + D - A = [4, 4, 5] + [4, -2, -1] - [0, 0, 3] = [8, 2, 1]
M = 1/2·(B + D) = 1/2·([4, 4, 5] + [4, -2, -1]) = [4, 1, 2]
N = AB ⨯ AD = ([4, 4, 5] - [0, 0, 3]) ⨯ ([4, -2, -1] - [0, 0, 3]) = [-12, 24, -24] = -12·[1, -2, 2]
S = M + r·N = [4, 1, 2] + r·[1, -2, 2] = [x, y, 8]
x = 7 ∧ y = -5 ∧ r = 3
S = [7, -5, 8]
b) Berechnen sie das Volumen der Pyramide.
V = 1/3 * (AB ⨯ AD) ⋅ AS
V = 1/3·(([4, 4, 5] - [0, 0, 3]) ⨯ ([4, -2, -1] - [0, 0, 3])) ⋅ ([7, -5, 8] - [0, 0, 3]) = -108
Das Volumen beträgt 108 VE.
