From the image, we see a square inscribed in a circle. The diagonal of the square is labeled as $2\sqrt{2}$ units. We need to calculate the area and perimeter of the square.

Step 1: Relationship between the diagonal and side of the square

The diagonal ($d$) of a square is related to its side length ($s$) by the formula: $d = s\sqrt{2}.$ Here, $d = 2\sqrt{2}$. Substituting this into the formula: $2\sqrt{2} = s\sqrt{2}.$ Divide both sides by $\sqrt{2}$: $s = 2.$

Step 2: Area of the square

The area ($A$) of a square is given by: $A = s^2.$ Substitute $s = 2$: $A = 2^2 = 4 \, \text{square units}.$

Step 3: Perimeter of the square

The perimeter ($P$) of a square is given by: $P = 4s.$ Substitute $s = 2$: $P = 4 \times 2 = 8 \, \text{units}.$

Final Answer:

• Area: $4 \, \text{square units}$
• Perimeter: $8 \, \text{units}$

Would you like a detailed derivation of any step or further assistance?

Related Questions:

1. How is the diagonal formula derived for a square?
2. What is the area of the circle enclosing the square?
3. Can the square's diagonal be used to calculate its perimeter directly?
4. How would the solution change if the diagonal were $4\sqrt{2}$?
5. What is the ratio of the square's area to the circle's area?

Tip:

Always check whether the diagonal or side of a square is given when solving inscribed or circumscribed problems!