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Area of ​​a cube with a side length of a, when projected in the diagonal direction

작성자 Uploader : aproduite 작성일 Upload Date: 2019-10-22변경일 Update Date: 2020-03-14조회수 View : 305

As shown in the figure, when a cube with a side length of a is projected in the diagonal direction, the projected area can be obtained as follows.

The shape projected in the direction of the line segment AG becomes a regular hexagon.

In ΔADG, the height h of the triangle with the line segment AG as the base is the length of one side of the regular hexagon.

The length of the line segment DG is 2^(1/2) * a.

Since ΔADG is a right triangle,

h = 2^(1/2)*a^2/(a^2 + 2*a^2)^(1/2)

= 2^(1/2)*a/(3)^(1/2)

= (2/3)^(1/2)*a

The area A of a regular hexagon with a side length of h is as follows.

A = (3/2)*3^(1/2)*h^2
= (3/2)*3^(1/2)*(2/3)*a^2
= 3^(1/2)*a^2


*** 참고문헌[References] ***
A = 3^(1/2)*a^2
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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