Problem 21 - Intermediate Algebra

Question

Find the area of each figure. See Example 2 . A square with sides \(17.2 \mathrm{mi}\) long

Step-by-Step Solution

Answer

The area of the square is \( 295.84 \text{ mi}^2 \).

Step 1: Identify the Geometry Involved

In this exercise, we are dealing with a square. A square is a four-sided polygon (quadrilateral) with equal side lengths and all interior angles being right angles (90 degrees).

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

The area of a square can be calculated using the formula \( A = s^2 \), where \( s \) is the length of one side of the square.

Step 3: Apply the Given Measurements

We are given the side length of the square as \( 17.2 \) miles. Plug this value into the formula: \[ A = (17.2)^2 \].

Step 4: Calculate the Square of the Side Length

Perform the calculation \( 17.2 \times 17.2 = 295.84 \). Thus, \( A = 295.84 \).

Step 5: Write the Final Answer with Units

The area of the square is \( 295.84 \) square miles, often written as \( 295.84 \text{ mi}^2 \).

Key Concepts

Understanding Geometry

Geometry is a vast and fascinating branch of mathematics that deals with shapes, sizes, and the properties of space. Its central focus is on shapes like circles, triangles, polygons, and in our case, squares. A key part of geometry is understanding the properties that define each shape.

• It explores the properties and relations of points, lines, surfaces, and solids.
• Geometry can be divided into several types such as plane geometry, solid geometry, and spherical geometry.
• In plane geometry, which is the most basic form, we deal with two-dimensional shapes like squares.

Understanding geometry helps us to artistically and practically engage with the world, as it applies to fields from art to architecture.

Properties of a Quadrilateral

A quadrilateral is a polygon with four sides and four vertices. In simpler words, it is a four-sided shape. Squares are a special type of quadrilateral with unique properties.

• Sides: All four sides of a square are equal in length.
• Angles: Every angle in a square is a right angle, measuring \(90^{\circ}\).
• Diagonals: The diagonals of a square bisect each other at right angles and are equal in length.

These properties make squares a fascinating study in quadrilateral geometry, as they are exceptionally symmetrical and balanced.

The Magic of Right Angles

Right angles are crucial in the study of geometry and give a square its name and shape. A right angle is an angle of exactly \(90^{\circ}\). Squares have four right angles, which define their shape precisely.

• It looks like the corner of a typical piece of paper.
• Each angle in a square forms a right angle, providing a visually appealing and symmetrical look.
• The presence of these angles contributes to the square's rigidity and uniformity. Without them, a square would not maintain its shape.

Understanding right angles helps in comprehending not just squares, but all rectangular shapes and patterns as well.

Applying the Square Formula

The square formula is a fundamental concept in geometry used to calculate the area of a square. The simplicity of the formula, \(A = s^2\), where \(s\) is the side length, makes calculations straightforward.

• The formula revolves around the idea of squaring one side of the square, reflecting how each side is used twice in building the area.
• For example, with side length \(17.2\) miles, substituting into the formula gives \((17.2)^2 = 295.84\)
• Thus, the area is \(295.84\) square miles, calculated simply by multiplying the side by itself.

This formula is not only simple but also emphasizes the elegant structure and mathematical beauty of squares.