The perimeter of a square is 56 cm. What is the approximate length of its diagonal?
A. 10.6 cm
B. 14.0 cm
C. 15.0 cm
D. 19.8 cm
Answer (1)
Find the side length of the square using the perimeter: s = 4 56 = 14 cm.
Calculate the diagonal using the Pythagorean theorem: d = s 2 .
Substitute the side length: d = 14 2 cm.
Approximate the diagonal length: d ≈ 19.8 cm.
19.8 cm
Explanation
Problem Analysis The perimeter of a square is given as 56 cm. We need to find the approximate length of its diagonal.
Find the side length Let P be the perimeter of the square and s be the side length. The formula for the perimeter of a square is P = 4 s . We are given that P = 56 cm. To find the side length s , we can use the formula:
56 = 4 s
Divide both sides by 4:
s = 4 56 = 14 cm
Find the diagonal length Now that we have the side length, we can find the length of the diagonal d . The diagonal of a square divides it into two right-angled triangles. Using the Pythagorean theorem, we have:
d 2 = s 2 + s 2 = 2 s 2
So, d = 2 s 2 = s 2
Substitute s = 14 cm into the equation:
d = 14 2 cm
Approximate the diagonal length We know that 2 ≈ 1.414 . Therefore,
d ≈ 14 × 1.414 = 19.796 cm
Rounding to one decimal place, we get d ≈ 19.8 cm.
Final Answer The approximate length of the diagonal of the square is 19.8 cm.
Examples
Understanding the diagonal of a square is useful in many real-world scenarios. For example, if you're designing a square garden and want to build a diagonal pathway across it, knowing how to calculate the diagonal helps you determine the length of materials needed. If your garden has a side length of 10 meters, the diagonal pathway would be approximately 10 2 ≈ 14.14 meters long. This ensures you purchase the correct amount of paving stones or other materials.
