What is the approximate area of the hexagon?

Question

A regular hexagon has an apothem measuring 14 cm and an approximate perimeter of 96 cm.

Answer 1

Draw one of the six equilateral triangles into which the hexagon can be divided;

Bisect its central angle of 60 degrees to form a pair of congruent 30-60-90 triangles.

The longer leg of either of these triangles is 14 cm; the shorter leg is r/2 and the hypotenuse is r units.

Comparing it to the model 30-60-90 triangle that has corresponding sides √3, 1 and 2.

You may therefore write the ratio of corresponding parts 

r/14= 2/√3 so r = 28/√3 which is about 16.1658 in decimal although let's keep it exact.

Now use the formula for area of one equilateral triangle whose side is r, and finally multiply it by 6 to get the area of the hexagon.  

 

This is the best method.  The approximate perimeter value that you were given should be 6(16.1658) and so that is not a reliable value.

Answer 2

In a regular hexagon, if you draw a line segment from the center to each vertex, it divides the hexagon into 6 congruent triangles.  So if you can find the area of one of the triangles, you just multiply by 6 to get the area of the hexagon.   The area of a triangle is (1/2*base*height.  The height is the apothen = 14 cm.  Find the base using the perimeter:

 

Perimeter = 6*base

Perimeter/6 - base

96/6 = base

16 = base

 

So the area of one triangle is

 

Area = (1/2)*base*height

Area = (1/2)*(16 cm)*(14 cm)

Area = 112 cm²

 

Multiply that by 6 to get the area of the hexagon.

Answer 3

A = (0.5)(a)(p)

A = (0.5)(14)(96)