The Perfect Slice

Geometry Level 1

The perfect slice through a cube will reveal a regular hexagonal cross section. If this is a $2\text{ in} \times 2\text{ in} \times 2\text{ in}$ cube, what is the surface area of the hexagonal cross section?

\frac{\sqrt{3}}{2}

2\sqrt{3}

\frac{\sqrt{2}}{3}

3\sqrt{3}

1 solution

Zandra Vinegar Staff
Oct 16, 2015

The hexagonal cross section is made out of 6 equilateral triangles, each of which has side-length $\sqrt{2}$ . The area of an equilateral triangle of side length $s$ is given by: $\frac{\sqrt{3}}{4}s^2$ With $s = \sqrt{2}$ , this makes the area of each equilateral triangle $\frac{\sqrt{3}}{4}(\sqrt{2})^2 = \frac{\sqrt{3}}{2}$ .

Six such triangles, therefore have area $3\sqrt{3}$ altogether.

How do you know the side of the triangle?

Mandeep Singh - 5 years, 3 months ago

The perfect slice bisects every edge of the cube (otherwise you would not get a regular hexagon). That creates a right triangle with legs of 1 and 1 at the corner. By Pythagorean theorem, the hypotenuse of the triangle is the square root of two.

Edward Aubry - 5 years, 3 months ago

We’re we told that it is cut through the midpoints of the edges?

Lauren Day - 1 year, 8 months ago

How did you come to know that it's an equilateral triangle?

Omkar Chavan - 5 years, 3 months ago

A regular hexagon is composed of six equilateral triangles. The proof is tied to the formula for the measures of the interior angles of a regular polygon, 180(n-2)/n. For a hexagon, n=6, so the interior angles are all 120 degrees. If you connect opposite vertices, that bisects those angles to 60 degrees, as well as creating 6 central angles, which all have to be 360/6 = 60 degrees as well. 60-60-60 makes an equilateral triangle.

Edward Aubry - 5 years, 3 months ago

The Perfect Slice

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