Problem 16 - Prealgebra

Question

Find the perimeter and area of each rectangle. a square that is 5 meters on each side

Step-by-Step Solution

Answer

Perimeter: 20 meters, Area: 25 square meters.

Step 1: Identify Rectangle Dimensions

A square is a special type of rectangle where all sides are equal. In this problem, each side of the square is 5 meters.

Step 2: Calculate the Perimeter

The formula for the perimeter of a rectangle (or square) is sum of all its sides. For a square, this becomes \(P = 4 \times ext{side length}\). Thus, the perimeter of our square is \(P = 4 \times 5 = 20\) meters.

Step 3: Calculate the Area

The formula to calculate the area of a rectangle (or square) is length multiplied by width. Since all sides of a square are equal, the formula simplifies to \(A = ext{side length} \times ext{side length}\) or \(A = ext{side length}^2\). So, the area of our square is \(A = 5^2 = 25\) square meters.

Key Concepts

Geometry

Geometry is a fascinating branch of mathematics that deals with shapes, sizes, and the properties of space. At its core, geometry is about understanding how different shapes interact and relate to one another.
When it comes to rectangles and squares, two-dimensional geometry focuses on properties such as perimeter and area.
Rectangles and squares are special polygonal shapes. A square is actually a specific type of rectangle. Both have four sides and four right angles, but a square has the added attribute of all sides being equal.
With these fundamental properties, you can explore more about the calculations involving perimeter and area.
By mastering geometric concepts, you can solve practical problems and gain insights into the beauty of mathematical shapes around us.

Square Properties

Understanding the properties of squares is crucial when calculating perimeter and area. A square, by definition, is a four-sided polygon where all sides are equal in length and all interior angles measure 90 degrees.
This symmetry makes calculations straightforward.
Some important properties of squares are:

• All sides have equal length.
• Each internal angle is 90 degrees, giving a total of 360 degrees.
• The diagonals bisect each other at right angles and are equal in length.
• A square is both a rectangle (with equal opposite sides and equal angles) and a rhombus (with equal all sides).

These properties not only make calculations easier but also provide insights into how squares fit into the larger category of shapes known as quadrilaterals.

Mathematical Formulas

Mathematical formulas are essential tools for calculating perimeter and area. Specific shapes have specific formulas that make calculations easier and ensure accuracy.
For squares, these formulas are:

• Perimeter: The perimeter of a square is calculated as four times the side length. The formula is given by \( P = 4 imes \text{side length} \). Therefore, if a square's side is 5 meters, its perimeter is \( P = 4 imes 5 = 20 \) meters.
• Area: The area of a square is found by multiplying the side length by itself, simplifying to the square of the side length. The formula is \( A = \text{side length} \times \text{side length} \) or \( A = \text{side length}^2 \). For a square with 5-meter sides, the area computes to \( A = 5^2 = 25 \) square meters.

These formulas allow for quick calculations and provide a strong foundation in understanding more complex geometric properties and solving real-world geometry problems.