Question

The surface of a table to be built will be in the shape shown below. The distance from the center of the shape to the center of each side is 7.8 inches and the length of each side is 9 inches.

A hexagon labeled ABCDEF is shown will all 6 sides equal in length. ED is labeled as 9 inches. A perpendicular is drawn from the center of the hexagon to the side ED. This perpendicular is labeled as 7.8 inches.

Part A: Describe how you can decompose this shape into triangles. (2 points)

Part B: What would be the area of each triangle? Show every step of your work. (5 points)

Part C: Using your answers above, determine the area of the table's surface. Show every step of your work. (3 points)

Answer 1

Step-by-step explanation:

Part A

Connect the center with each vertex of the hexagon. This will give us 6 equal triangles.

Part B

The area of each triangle is:

• A = 1/2bh

The base is the side of hexagon, the height is the perpendicular to the base from the center

Find the area by substituting the values

• A = 1/2(7.8)(9) = 35.1 in²

Part C

The area of table's surface is the sum of areas of 6 triangles

• A = 6*35.1 = 210.6 in²

Answer 2

Answer:

A)  see attached

B)  Area of each triangle = 35.1 in²

C)   210.6 in²

Step-by-step explanation:

Part A

The shape can be decomposed into triangles by drawing line segments from the center of the hexagon to each vertex (see attached image).

Part B

As the hexagon is a regular polygon, the 6 triangles are congruent (same shape and size) and are equilateral (all sides are the same length).

Area of triangle

 $\sf A=\dfrac{1}{2} \times base \times height$

The base of each triangle is the side length of the hexagon: 9 in.

The height of each triangle is the perpendicular:  7.8 in.

$\implies \sf Area\:of\:each\:triangle=\dfrac{1}{2} \times 9 \times 7.8$

$\implies \sf Area\:of\:each\:triangle=35.1\:in^2$

Part C

As the hexagon is made up of 6 congruent equilateral triangles, to determine the area of the table's surface, simply multiply the found area of one equilateral triangle by 6:

$\implies \sf Table's\:surface\:area=6 \times 35.1$

$\implies \sf Table's\:surface\:area=210.6\:\:in^2$

Therefore, the surface area of the table is 210.6 in²