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Chapter 9: Problem 2

Find the length of the diagonal of a square with perimeter \(12 \mathrm{cm}\)

Short Answer

Expert verified
The length of the diagonal of the square is \(3\sqrt{2} \mathrm{cm}\).

Step by step solution

01

Find Length of Side

Given that the perimeter of a square is \(12 \mathrm{cm}\), we need to find the length of one side. Since the perimeter of a square is four times the length of one side (because all sides are equal in a square), the length of one side can be calculated as \( \frac{Perimeter}{4}\). Therefore, the length of the side is \( \frac{12}{4} \) which equals \(3 \mathrm{cm}\).
02

Calculate Length of Diagonal

The length of the diagonal of a square can be calculated using the formula \(d = \sqrt{2} \times\) length of side. Therefore, the diagonal is \(d = \sqrt{2} \times 3 \) which equals \(3\sqrt{2} \mathrm{cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter of a Square
Understanding the perimeter of a square is fundamental in geometric calculations. The perimeter is defined as the total distance around the boundary of the square. Since a square has four equal sides, calculating its perimeter is straightforward: simply multiply the length of one side by four. For instance, if a square has a perimeter of 12 cm, we can determine that each side measures 3 cm by dividing the total perimeter by 4.

This basic arithmetic ties into many other geometric concepts and is the starting point for finding other properties of the square, such as area and the length of its diagonals.
Geometric Formulas
Geometric formulas are essential tools to solve a wide array of problems in mathematics. While we focused on finding the perimeter and diagonal of a square, there are many other formulas that one might use in geometry. These formulas relate various dimensions of shapes鈥攕uch as area, volume, surface area, and more鈥攖o their sides, angles, and other characteristics. For a square, simple formulas based on its sides can unlock a plethora of other measurements, indicating the highly systematic nature of geometric properties.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry, particularly instrumental in the calculations involving right triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be put into the formula \( c^2 = a^2 + b^2 \) where \( c \) represents the hypotenuse, and \( a \) and \( b \) are the other two sides. In our square's case, we utilised a derivative of this theorem as the diagonal of a square forms two congruent right-angled triangles, thus allowing us to compute its length.
Square Properties
A square is a fascinating shape with properties that make it unique in geometry. Being a special case of a rectangle, each square has four right angles and, additionally, all sides of equal length make it a regular polygon. Not only does this make calculation of the perimeter quite simple, but the square's symmetry lends itself to easy calculation of the diagonal through the use of the Pythagorean theorem. Moreover, the diagonals of a square are not only equal in length but also bisect each other at right angles, serving as axes of symmetry. These properties help make finding the diagonal of a square an exercise in applying simple geometric principles.

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Most popular questions from this chapter

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