Given: ABCD is a rectangle m∠B1 DB = 60º, m∠ABD=45º DB1 = 37 Find: V

Question

Given: ABCD is a rectangle m∠B₁ DB = 60º, m∠ABD=45º DB₁ = 37 Find: V

Answer 1

m∠B1BD = 90˚ because it is the angle from the base of the rectangle to the side of the rectangle. It is like m∠B1BC but rotated. Since a triangle must have 180˚, it is possible to find the m∠BB1D because you know of the angles in triangle BB1D. m∠BB1D = 180 - (90 + 60) = 30. Remember that a 30˚-60˚-90˚ triangle is half of an equilateral triangle. In an equilateral triangle, all the angles are 60˚ and all the sides are equivalent. By cutting it in half, one side is half of the longest side. Using 1 and 1/2 for side lengths, it is possible to find the third side of a 30˚-60˚-90˚ triangle by the Pythagorean theorem.

a² + b² = c²

(1/2)² + b² = 1²

(1/4) + b² = 1

b² = 1 - 1/4

b² = 3/4

b = √(3) / 2

A 30˚-60˚-90˚ always has these angles so the sides will always be proportional to 1, 1/2, and √(3) / 2. DB1 is the longest side in triangle DB1B so it is proportional to the side length of 1. So DB1 = 37, BB1 = 37/2 and BD = 37 * √(3) / 2.

Looking at triangle ABD, m∠BAD is 90˚ because it is the corner of a rectangle. Doing the same thing as above, m∠DAB = 180 - (45 + 90) = 45˚. In a 45˚-45˚ triangle, the sides are congruent to 2, 1, and 1. BD is the longest side so sides AB and side AD are half of BD.

AB = (37 * √(3) / 2) / 2 = 37 * √(3) / 4

AD = (37 * √(3) / 2) / 2 = 37 * √(3) / 4

AB and AD are the width and length of the shape and BB1 is the height. So the volume is

V = AB * AD * BB1 = (37 * √(3) / 4) * (37 * √(3) / 4) * (37/2) = 4,748.71875

Ugly answer but don't be afraid of decimals!