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Обозначим проекцию апофемы на основание за х.
Тогда проекция боковой стороны на основание будет 2х.
По Пифагору имеем - боковая сторона L равна:
L = √((2x)² + (√3)²) = √(4x² + 3).
Апофема А равна √(x² + (√3)²) = √(x² + 3).
Высота треугольника основания равна 3х.
Тогда сторона основания а = 3x/cos 30° = 3x/(√3/2) = 6x/√3 = 2√3x.
Но, так как сторона основания - это гипотенуза при двух катетах L, то можно выразить: a = √(2L²) = L√2 = √(4x² + 3)*√2 = √(8x² + 6).
Приравняем: √(8x² + 6) = 2√3x. Возведём в квадрат:
8x² + 6 = 12x или 4x² = 6 или 2x² = 3.
Отсюда находим х = √(3/2).
Теперь можно определить длину стороны основания, подставив значение х: а = 2√3*(√(3/2)) = 3√2.
Площадь основания So = a²√3/4 = 18√3/4 = 9√3/2 кв.ед.
ответ: V = (1/3)SoH = (1/3)*(9√3/2)*√3 = (9/2) куб.ед.