Praxis Core Academic Skills for Educators: Mathematics (5733)GeometryHard
A machinist is cutting a metal plate in the shape of a regular hexagon. If one side of the hexagon measures 6 cm, what is the area of the hexagon?
- A108√3 cm²
- B72√3 cm²
- C54√3 cm²
- D36√3 cm²
Show answer & explanationAnswer & explanation
Correct answer: C. 54√3 cm²
A regular hexagon can be divided into 6 equilateral triangles. The side length of each equilateral triangle is equal to the side length of the hexagon (s=6 cm). The area of an equilateral triangle with side 's' is (√3/4)s². So, the area of one triangle is (√3/4) * 6² = (√3/4) * 36 = 9√3 cm². Since there are 6 such triangles, the total area of the hexagon is 6 * 9√3 = 54√3 cm².
Why the other options are wrong
- A. This is the area of an equilateral triangle with side 6, multiplied by 12, or just 3s^2 * sqrt(3).
- B. This is the area of one equilateral triangle multiplied by 8 instead of 6, or an incorrect formula.
- D. This is 6 times 6, times sqrt(3) / 2, an incorrect formula for a hexagon.
Area of a Regular Hexagon
The total two-dimensional space enclosed within the boundaries of a regular six-sided polygon.
- A regular hexagon can be decomposed into 6 congruent equilateral triangles.
- If 's' is the side length, the radius of the circumcircle is also 's'.
- Formula: Area = (3√3/2)s², or 6 * (√3/4)s².
Memory trick: Divide and conquer, into triangles you'll see, for symmetric shapes, area is key!