Problem 98 - Beginning and Intermediate Algebra

Question

The area of a plane figure is a measure of the amount of surface of the figure. Find the area of each figure. (The area of a triangle is \(\frac{1}{2}\) the product of its base and height. The area of a rectangle is the product of its length and width. Recall that area is measured in square units IMAGE CANNOT COPY!

Step-by-Step Solution

Answer

Calculate using given dimensions for each figure: triangle and rectangle.

Step 1: Identify the Figures

Without the image, we can't specify the exact figures, but let's assume we have a triangle and a rectangle, which are common figures for this type of question.

Step 2: Identify the Triangle Parameters

The area of a triangle is calculated using the formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Locate the base and the height of the triangle from the figure measurements provided.

Step 3: Calculate the Area of the Triangle

Substitute the values of the base and the height into the formula. For example, if the base is 4 units and the height is 3 units, the area is \( A = \frac{1}{2} \times 4 \times 3 = 6 \) square units.

Step 4: Identify the Rectangle Parameters

For the rectangle, use the formula \( A = \text{length} \times \text{width} \). Locate the length and width from the figure measurements provided.

Step 5: Calculate the Area of the Rectangle

Substitute the length and width values into the formula. For example, if the length is 5 units and the width is 2 units, the area is \( A = 5 \times 2 = 10 \) square units.

Key Concepts

Understanding the Area of a Triangle

The area of a triangle is a way to measure the amount of space inside the triangle's boundaries. To find the area, you need two important measurements: the base and the height. The base of a triangle is one of its sides, usually considered the bottom side. The height is a perpendicular line drawn from the base to the opposite vertex. The formula for the area of a triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

• The base and the height must be perpendicular to each other.
• If the base is not the horizontal side, it's still fine as long as you know its length and the height.

This formula works by calculating half of the area of a rectangle that would be formed if you continued the triangle twice with the base and height. It breaks down the figure into manageable parts, making it easy to use and remember.

Grasping the Area of a Rectangle

A rectangle is a four-sided figure with opposite sides being equal and angles being right angles. Finding the area of a rectangle is quite straightforward. You need to identify two key measurements: the length and the width. The length is the longer side of the rectangle, and the width is the shorter side, perpendicular to the length. Using these measurements, the area is calculated by the formula:\[ A = \text{length} \times \text{width} \]

• Both the length and width need to be in the same units.
• The result will give you the number of square units of space inside the rectangle.

With this formula, you are essentially multiplying the number of units along the length and the width, visualizing how the figure fills up in checkered square units. It's as easy as stretching a grid over the rectangle to see how much space it covers.

Mathematics in Measuring Area

Mathematics helps us quantify the world around us. One of the fundamental aspects is measuring areas, or the surfaces of figures like triangles and rectangles. Understanding how to calculate area allows us to solve real-world problems, from finding out how much paint is needed for a wall to determining the size of a garden. The area is always expressed in square units, such as square meters or square feet, because it represents a two-dimensional space. When calculating area:

• Ensure you are using the correct formula for the shape you are measuring.
• Always double-check that measurements are in consistent units (e.g., all in meters).
• Remember that the result gives you an idea of the extents within the perimeter of the shape.

By understanding these mathematical principles, you can confidently tackle problems involving surfaces and learn to appreciate the precision and logic that math brings into everyday tasks.