Problem 13 - Basic Technical Mathematics with Calculus

Question

Find the area of each figure. Square: \(s=6.4 \mathrm{mm}\)

Step-by-Step Solution

Answer

The area of the square is 40.96 mm².

Step 1: Understanding the Problem

We need to find the area of a square. We are given the side length of the square, which is denoted as \(s\). In this problem, \(s = 6.4\) mm.

Step 2: Recall the Formula for the Area of a Square

The formula to calculate the area of a square is given by \( ext{Area} = s^2 \), where \(s\) is the length of a side of the square.

Step 3: Substitute the Side Length into the Formula

Insert the given side length \(s = 6.4\, \text{mm}\) into the area formula. The expression becomes \( ext{Area} = 6.4^2 \).

Step 4: Calculate the Area

Compute the square of \(6.4\): \[ 6.4^2 = 6.4 imes 6.4 = 40.96 \]So, the area of the square is \(40.96\, \text{mm}^2\).

Key Concepts

Square Geometry

Squares are fascinating shapes in geometry! A square is a special type of quadrilateral with some unique properties. The most distinctive characteristic of a square is that it has four equal sides. This means that every side of a square is the same length. Also, each angle in a square is a right angle, which is exactly 90 degrees.

Squares are symmetric along both diagonals, which means if you fold the square along a diagonal, both halves will match perfectly. This symmetry is one of the reasons why squares are considered very stable shapes.

A square is also a type of rectangle and rhombus due to its equal sides and right angles. Because of these properties, squares are often used in practical applications like tiling floors or designing game boards. Understanding square geometry helps us in solving various mathematical and real-world problems.

Area Calculation Formula

Calculating the area of a square is straightforward with the area calculation formula. The area of any shape is defined as the amount of space it covers, measured in square units. For a square, the area is calculated by squaring the length of one of its sides.

The formula for the area of a square is:

• Area = \( s^2 \)

Where \( s \) is the length of a side. Therefore, the area formula is simply saying "multiply the side by itself."

For example, if a square has a side length of 6.4 mm, the area is:

• \( 6.4 \mathrm{mm} \times 6.4 \mathrm{mm} = 40.96 \mathrm{mm}^2 \)

This means the square covers an area of 40.96 square millimeters. This straightforward approach makes it easy for students to calculate areas accurately.

Basic Mathematics Concepts

Understanding basic mathematics concepts is essential for comprehending more complex topics later on. In geometry, key concepts include understanding shapes, dimensions, and how to calculate space.

Some basic math ideas used in calculating the area include:

• **Squaring a number**: This is multiplying a number by itself, like \( 6.4 \times 6.4 \).
• **Units of measurement**: It's crucial to pay attention to the units, like millimeters (mm) in this case, and how they change when calculating area (mm becomes \( \text{mm}^2 \)).
• **Arithmetic operations**: Skills like multiplication and understanding powers are essential.

These foundational concepts provide the building blocks for tackling more advanced mathematical problems. By learning how to calculate the area of a square, students also reinforce their arithmetic skills and prepare for more sophisticated geometry challenges.