To calculate the perimeter of the given shape and find the unknown slanted length, we need to break down the problem.

The shape appears to be a trapezoid. It has the following dimensions:

• A vertical height of 6 units (on the right side).
• A smaller vertical segment of 3 units (on the left).
• A horizontal base of 4 units.

Step 1: Identify the Unknown Length

The unknown length is the slanted side of the trapezoid, which forms a right triangle with the vertical side of 3 units and the horizontal base difference of 4 units minus 0 units (since the horizontal starts from 0).

Thus, we have a right triangle where:

• One leg is 3 units (the vertical height of the smaller side),
• The other leg is 4 units (the horizontal distance).

Using the Pythagorean theorem to find the hypotenuse (which is the unknown slanted length):

$\text{Hypotenuse}^2 = 3^2 + 4^2$ $\text{Hypotenuse} = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the slanted side is 5 units.

Step 2: Calculate the Perimeter

The perimeter is the sum of all the sides of the shape:

• Bottom side: 4 units
• Left vertical side: 3 units
• Right vertical side: 6 units
• Slanted side: 5 units

$\text{Perimeter} = 4 + 3 + 6 + 5 = 18 \text{ units}$

The perimeter of the shape is 18 units.

Would you like further details on any of the steps?

Here are 5 related questions for further exploration:

1. How do you calculate the area of this trapezoid?
2. What is the relationship between the Pythagorean theorem and other triangle types?
3. Can the perimeter be calculated if the shape was irregular?
4. How would you calculate the diagonal of a rectangle with the same base and height?
5. What formulas are used to calculate the perimeter of other polygons like hexagons?

Tip: In any geometric figure, the Pythagorean theorem is a powerful tool for finding unknown sides when right angles are involved!