Problem 3 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Area of a Square

A square is a four-sided figure with all sides equal. To find the area of a square with side length \(x\), you multiply the side by itself. This is because area is calculated as the length of a side multiplied by the length of another side (which is the same in a square).

Step 2: Calculating the Area of a Square

The formula for the area \(A\) of a square with side \(x\) is given by:\[ A = x \times x = x^2 \]

Step 3: Understanding the Area of a Rectangle

A rectangle is a four-sided figure with opposite sides equal in length. The area of a rectangle is calculated by multiplying the length \(l\) by the width \(w\). This is because you take how many units long and wide the rectangle is to cover the entire space.

Step 4: Calculating the Area of a Rectangle

The formula for the area \(A\) of a rectangle with length \(l\) and width \(w\) is:\[ A = l \times w \]

Step 5: Understanding the Area of a Circle

A circle is a round shape with all points equidistant from the center. The area of a circle is calculated using the constant \(\pi\), which is approximately 3.14159, and the square of its radius \(r\). This covers the full space inside the boundary of the circle.

Step 6: Calculating the Area of a Circle

The formula for the area \(A\) of a circle with radius \(r\) is:\[ A = \pi \times r^2 \]

Key Concepts

Square Area

A square is a unique type of quadrilateral where all four sides are of equal length. When calculating the area of a square, we utilize the property that each side length is identical. The area is essentially how much space is contained within the boundary of the square. To find this, we take one side of the square and multiply it by itself. In other terms, we square the side length. This results in the formula for the area of a square:

• Area formula: \[ A = x^2 \]

Where \(x\) is the length of a side of the square. Squaring \(x\) gives us the total area covered within its boundaries. This formula is straightforward because of the square's symmetry and equal side lengths.
Keep in mind:- Every side of a square is equal.- The area grows rapidly as the side length increases, illustrating geometric scale principles.

Rectangle Area

A rectangle is similar to a square but has two pairs of equal sides rather than all four sides being the same length. The area of a rectangle is determined by its length and width, representing its dimensions. Length is usually the longer side, and width is the shorter one, although the orientation doesn't change the overall area calculation. To find the area, you multiply the length by the width.

• Area formula: \[ A = l \times w \]

Where \(l\) is the length and \(w\) is the width of the rectangle. This calculation is simple but crucial for understanding the space within the rectangle's parameters.
When working with rectangles, remember:- Opposite sides are equal.- The order of multiplication does not matter; \(l \times w\) is the same as \(w \times l\).- Rectangles can encompass squares where the length equals the width.

Circle Area

A circle is a completely round figure with every point on its edge being equidistant from its center. The concept of pi (\(\pi\)) plays a pivotal role in calculating the area of a circle. Pi is an irrational number often approximated as 3.14159. To find the area of a circle, we use its radius, which is the distance from the center to any point on the circle. By multiplying pi by the square of the radius, we find the area enclosed within the circular boundary.

• Area formula: \[ A = \pi \times r^2 \]

Here, \(r\) is the radius. Squaring the radius \(r\) and then multiplying by pi gives the area.
Points to consider:- The radius is half of the circle's diameter, so \(r = \frac{d}{2}\).- Circles have rotational symmetry, which means the area is evenly spread around the center.- Pi is crucial and unique to circles, differentiating them from polygonal shapes.