Problem 3 - Precalculus Mathematics for Calculus

Question

Give a formula for the area of the geometric figure. (a) A square of side \(x: A=\) _______. (b) A rectangle of length \(l\) and width \(w: A=\) _______. (c) A circle of radius \(r: A=\) _______.

Step-by-Step Solution

Answer

(a) \(A = x^2\), (b) \(A = l \times w\), (c) \(A = \pi r^2\).

Step 1: Understanding the Square Area Formula

The area of a square is determined by multiplying the side of the square by itself, as all sides of a square have the same length. The formula is therefore derived as follows: If the side of the square is denoted by \(x\), then the area \(A\) of the square is \(A = x^2\).

Step 2: Calculating the Rectangle Area

A rectangle's area is calculated by multiplying its length by its width. This is because a rectangle consists of two sets of equal-length sides, and the area measures how much space is inside the boundary of the rectangle. With length \(l\) and width \(w\), the area \(A\) is \(A = l \times w\).

Step 3: Using the Circle Area Formula

The formula for the area of a circle is based on its radius and the constant \(\pi\) (approximately 3.14159). The area is defined as \(\pi\) times the square of the radius \(r\). Therefore, the formula for the area \(A\) is \(A = \pi r^2\). These formulas give a measure of the space contained within their respective shapes.

Key Concepts

Area of Square

A square is a special quadrilateral where all four sides are equal in length. It's quite easy to determine the area of a square because of this property. The formula for the area of a square is found by multiplying the length of one side by itself. This is often expressed mathematically as:

\( A = x^2 \)
where \( x \) is the length of a side of the square.

It’s important to remember that squaring a number involves multiplying the number by itself. So, if you have a square with sides that are 5 units long, the area is:

\( 5 \times 5 = 25 \) square units.

Understanding this helps to visualize how the space inside the square is calculated. It’s literally the square of the side’s length!

Rectangle Area

Rectangles are geometric shapes with opposite sides that are equal in length and have four right angles. The method for calculating the area of a rectangle is straightforward.

To find the area, simply multiply the length \( l \) by the width \( w \) of the rectangle. By doing this, we can determine how much space is enclosed within the rectangle. The formula can be expressed as:

\( A = l \times w \)

For example, if a rectangle has a length of 6 units and a width of 4 units, the area will be:

• \( 6 \times 4 = 24 \) square units.

So, whenever you need to find the space within a rectangle, remember that it’s simply the product of the length and width.

Circle Area

Circles are unique because they don’t have sides like other shapes, and instead, their space is determined by the radius. The radius \( r \) is a line from the center of the circle to any point on the circle itself.

The area of a circle is calculated using the mathematical constant \( \pi \), which is approximately equal to 3.14159. The formula for the circle's area is:

\( A = \pi r^2 \)

This means you square the radius (multiply it by itself) and then multiply that result by \( \pi \). For example, if a circle’s radius is 3 units, its area would be:

• \( \pi \times 3^2 = \pi \times 9 \approx 28.27 \) square units.

Understanding this formula helps us appreciate how circles are measured, despite their round shape!