Question

Hexagon ABCDEF has has vertices A(-2,4), B(0,4), C(2,1), D(5,1), E(5,-2), F(-2,-2). Sketch the figure on the coordinate plane. What is the area of the hexagon?

Answer 1

The area of the hexagon is 30 square units.

Sketch of figure on the coordinate plane below.

Explanation:

Separate the hexagon into simpler shapes: a triangle, and two rectangles.

Find the area of the simpler figures. Count units to find the dimensions.

The triangle with vertices at (0, 4), (2, 1), and (0, 1):

A = 1 /2

BH

A = 1 /2

(2)(3)

A = 3

The area of the triangle is 3 square units.

The rectangle with vertices at (0, 1), (5, 1), (5, −2), and (0, −2):

A = LW

A = 5 × 3

A = 15

The area of the rectangle is 15 square units.

The rectangle with vertices at (−2, 4), (0, 4), (0, −2), and (−2, −2):

A = LW

A = 6 × 2

A = 12

The area of the rectangle is 12 square units.

Find the area of the hexagon.

A = 3 + 15 + 12

A = 30

The area of the hexagon is 30 square units.

Answer 2

A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle, which equals {\displaystyle {\tfrac {2}{\sqrt {3}}}} {\displaystyle {\tfrac {2}{\sqrt {3}}}} times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 reflection symmetries (six lines of symmetry