In the figure, the perimeter of hexagon abcdef is approximately (blank) units, and its area is (blank) square units.

The permimeter is 93 units.

The area is 500 square units

Step-by-step explanation:

Perimeter  (image 1, the one with yellow lines)

If we want to find out the perimeter of this hexagon we must sum every side of the shape. The image shows some measurements, but there are some others that are missing.

We already know the following measurements:

AB: 10 units

DE: 10 units

The measurements of BC, CD, EF and FA are missing and if we want to find out these sizes it’s necessary to use Pythagoras theorem.

For BC and CD, we assume that triangles formed with the sides are similar because they have two equivalent sides and an identical angle.

Let’s name these triangles as BCX and CDX,

BX = DX: 10 units

CX: 10 units

Using Pythagoras theorem;

BX² + CX² = BC²

10² + 10² = 200

BC = √200 = 14.14 units

Knowing that BX and DX are identical sides and both of triangles share the CX side, we assume the size of CD is also 14.14 units.

Then, we do the same with triangle formed by EFZ and FAZ

EZ = AZ: 10 units

FZ: 20 units

10² + 20² = 500

FA = EF = √500 = 22.36 units

Summing the sides:

AB + BC + CD + DE + EF + FA

10 units + 14.14 units + 14.14 units + 10 units + 22.36 units + 22.36 units = 93 units

The perimeter of hexagon ABCDEF is 93 units

Area (image 2, the one with red lines)

There isn’t a formula for finding out the area of a hexagon, so we can discover it by splitting this in simple shapes.

After we split the hexagon we can see three shapes, two triangles and one rectangle.

Shape 1: Triangle FAE: The area of a triangle is given by the following formula:

(Base ×Height)/2= (20 units × 20 units)/2= 200 units

Shape 2: Rectangle ABDE: The area of a triangle is given by the following formula:

Base * Height  

10 units * 20 units = 200 units

Shape 3: Triangle CBD: We already know the formula.

(20 units × 10 units)/2= 100 units

Then, we sum all of the results.

Area of triangle FAE + Area of rectangle ABDE + Area of triangle CBD

200 units + 200 units + 100 units = 500 units

The area of the hexagon is 500 units.