Question

You are buying plywood to board up your windows in preparation for Hurricane Euclid. In the master bedroom you have a Norman window (in the shape of a rectangle with a semicircular top). You need to calculate the area of the window. If the rectangular part of the window is 4 feet wide and 5 feet tall, what is the area of the entire window? Explain.

Short answer

The area of the window is \( 20 + 2\pi \) square feet.

Step-by-step solution

Identify Parts of the Window

The window consists of a rectangle and a semicircle. The rectangle is 4 feet wide and 5 feet tall. The semicircle is on top of the rectangle and has the same width as the rectangle.

Calculate Area of the Rectangular Part

The area of a rectangle is given by the formula: \( \text{Area} = \text{width} \times \text{height} \). Here, the width is 4 feet and the height is 5 feet. Thus, the area of the rectangular part is \( 4 \times 5 = 20 \) square feet.

Calculate the Radius of the Semicircle

The width of the semicircle is the same as the rectangle, which is 4 feet. This is the diameter. The radius is half of the diameter, so the radius is \( \frac{4}{2} = 2 \) feet.

Calculate Area of the Semicircular Part

The area of a circle is \( \pi r^2 \). The area of a semicircle is half that of a full circle, \( \frac{1}{2} \pi r^2 \). For our semicircle with radius 2, the area is \( \frac{1}{2} \pi (2)^2 = 2\pi \) square feet.

Calculate Total Area of the Window

Add the area of the rectangular part and the semicircular part together: \( 20 + 2\pi \) square feet. Thus, the total area of the window is \( 20 + 2\pi \) square feet.

Key concepts

area calculation

Calculating the area of a window is essential for various tasks, like buying material to cover it or estimating the amount of light it will let through. The area determines how much surface space the window covers. Using the right formulas and understanding the shape involved are important steps in solving these problems. For a window combining shapes, like our Norman window, you have to consider each part separately.

To calculate the area, follow the basic steps:

• Identify and separate distinct shapes in the window.
• Use appropriate geometric formulas for each shape.
• Add the areas of these individual shapes to find the total area.

Following these steps ensures you accurately determine the total area of a complex window structure.

window shape

Understanding the shape of your window is crucial for calculating its area. "Norman" windows are common in architecture and made up of simple geometric forms to create a visually appealing structure. These windows typically have a rectangular base with a semicircular top. This combination gives a classic architectural look while also providing structural elegance.

To determine the area, break down the window into its basic shapes:

• Rectangular base
• Semicircular top

Each part has its calculation method, and by splitting them, you simplify the problem-solving process. Knowing the window's shape lets you apply the correct formulas for an accurate calculation.

semicircle

A semicircle is half of a circle and a common geometric shape in architectural designs like windows. Calculating its area might seem a bit tricky, but it's just a step away from calculating the area of a full circle. Since the semicircle is half a circle, the formula involves dividing the area of a circle by two.

The steps are:

• Determine the diameter, which in this case, would be the width of the rectangle.
• Find the radius by halving the diameter.
• Use the formula for the area of a semicircle: \( \frac{1}{2} \pi r^2 \).

By understanding these steps, you can efficiently find the area of any semicircular structure. It's a straightforward computation once you're familiar with the basic process.

rectangle

The rectangle is one of the simplest and the most used shapes in geometry. Its area is calculated by multiplying its width by its height—simple yet powerful. This calculation helps in various practical applications, such as finding the amount of material required to cover or construct the rectangular part of a window.

For any rectangle:

• The width and height are all that's needed to compute the area.
• The formula is straightforward: \( \text{Area} = \text{width} \times \text{height} \).
• In our case, with a width of 4 feet and a height of 5 feet, the area amounts to 20 square feet.

Understanding rectangles and their properties is fundamental in geometry and essential for solving many real-world problems.

mathematics problem solving

Problem-solving in mathematics is more than just handling numbers; it's about understanding concepts, applying methods, and critically thinking to arrive at a solution. For a geometry problem like calculating the area of a Norman window, break down your process as follows:

• Visualize the structure: Identify individual shapes and their dimensions.
• Select the correct formulas for each part of the problem.
• Perform calculations step-by-step, confirming each as you go.
• Combine results for the final solution.

This approach ensures accuracy and a clear understanding of how different shapes and formulas contribute to the whole. Practicing these steps will also improve efficiency and boost confidence in handling similar problems in mathematics.