Problem 27 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Find the area of a square whose diagonal is \(24.0 \mathrm{cm}\).

Step-by-Step Solution

Answer

The area of the square is 288 cm².

Step 1: Understanding the Relation Between Diagonal and Side of a Square

A square's diagonal and sides are related by the formula \(d = s\sqrt{2}\), where \(d\) is the diagonal and \(s\) is the side of the square. This formula comes from the Pythagorean Theorem, as the diagonal forms a right triangle with two sides of the square.

Step 2: Solving for the Side of the Square

To find the side \(s\) of the square, we rearrange the diagonal-side relation formula to \(s = \frac{d}{\sqrt{2}}\). Substitute \(d = 24.0 \text{ cm}\) into the formula to find \[s = \frac{24.0}{\sqrt{2}} = \frac{24.0 \times \sqrt{2}}{2} = 12\sqrt{2} \text{ cm}.\]

Step 3: Calculating the Area of the Square

The area \(A\) of a square is calculated using the formula \(A = s^2\). Using the side calculated in the previous step, the area is given by \[A = (12\sqrt{2})^2 = 144 \times 2 = 288 \text{ cm}^2.\]

Key Concepts

Area Calculation

Calculating the area of geometrical shapes is a foundational skill in geometry. For a square, this task is straightforward since all sides are equal. The area of a square is determined by the formula:

• \(A = s^2\)

Where \(s\) represents the length of one side of the square. This formula makes it easy to find the area, knowing the side length.
When using this formula, ensure all measurements are in the same unit to maintain accuracy in your results. In problems involving diagonals, such as this one, first determine the side length using properties specific to squares, before calculating the area.

Pythagorean Theorem

The Pythagorean theorem is a powerful tool in geometry, particularly in dealing with right-angled triangles. It states that in a right triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. The formula for this theorem is:

• \(a^2 + b^2 = c^2\)

where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides of the triangle.
In the context of a square, the Pythagorean theorem helps establish the relationship between a square's diagonal and its sides. Since the diagonal divides the square into two right-angled triangles, each with two sides equal to \(s\), the theorem is applied as:

• \(s^2 + s^2 = d^2\)
• This simplifies to \(d = s\sqrt{2}\), where \(d\) is the diagonal.

This relationship aids in finding the side \(s\) when given the diagonal.

Square Properties

Understanding the properties of squares is crucial for accurately solving problems involving this shape. Key properties include:

• All four sides are equal in length.
• All interior angles are \(90^\circ\).
• Diagonals intersect at right angles and are of equal length.

Squares are a special case of rectangles and thus share some properties, such as having two pairs of parallel sides. However, the defining features are its equal side lengths and right-angled diagonals.
With the knowledge that diagonals are equal, and intersect at right angles, you can solve various related problems by forming two identical right-angled triangles within the square. This is critical for determining other dimensions like the side length from the diagonal measurement, as demonstrated in the exercise.