Question

The Castel del Monte in Apulia, Italy, was built in the 13th century. The outer shape and the inner courtyard are both regular octagons. a. Find the total area of the castle, including the courtyard. b. Find the area of the courtyard if the octagon has an apothem of \(14.5\) feet and sides of 12 feet. c. What is the area of the inside of the castle, not including the courtyard?

Short answer

The total area is 696 sq ft; the courtyard is 696 sq ft; no additional interior area exists.

Step-by-step solution

Calculate the Area of One Octagon

To find the total area of one octagon, use the formula for the area of a regular octagon: \[A = \frac{1}{2} \times Perimeter \times Apothem\]First, find the perimeter: \[Perimeter = 8 \times side = 8 \times 12 = 96 \text{ feet}\]Now substitute the values into the area formula:\[A = \frac{1}{2} \times 96 \times 14.5 = 696 \text{ square feet}\]

Find the Total Area with Courtyard

The total area of the castle, including the courtyard, is simply the area calculated for one octagon, since both the outer and inner shapes of the castle are regular octagons sharing the same size properties. Hence, the total area is:\[696 \text{ square feet}\]

Calculate the Area of the Courtyard

Since the courtyard itself is a regular octagon with the given apothem and side, it already shares the same dimensions as the previously calculated area. Thus, the area of the courtyard is:\[696 \text{ square feet}\]

Find the Interior Area of the Castle

The interior area of the castle, not including the courtyard, is zero. This is because the question specifies both the outer structure and the courtyard are the same size and shape octagons, indicating that the courtyard occupies the entire interior footprint defined by the outer octagon.

Key concepts

Regular Octagon

A regular octagon is a polygon with eight equal sides and eight equal angles. This geometric shape is quite symmetrical and has various properties that make calculations easier. In a regular octagon, all interior angles are equal, each measuring 135 degrees.

• Regular octagons are often used in architecture and design due to their symmetry.
• They can be divided into triangles for easier calculations, with symmetry making many formulas applicable.

When examining structures like the Castel del Monte in Italy, understanding the regular octagon helps in visualizing both the aesthetic and functional aspects of such historical buildings. This pattern is frequently seen in courtyards and other architectural designs to create balance and harmony.

Area Calculation

Calculating the area of a regular octagon involves a specific formula: \[A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\] This formula is derived from the fact that an octagon can be divided into eight identical isosceles triangles, each with a base equal to the side of the octagon and two equal sides extending to the apothem.

• "Perimeter" refers to the sum of all sides of the octagon.
• "Apothem" is the line drawn from the center to the midpoint of one side.

By plugging in the values into the formula, you can find the total surface area occupied by the octagon in any planar design.

Apothem

The apothem of a regular octagon is a crucial part in calculating its area. It is the line segment from the center of the octagon perpendicular to one of its sides and becomes the height of the triangles formed when dividing the octagon.

• The apothem provides a direct link between the perimeter and the area of an octagon.
• In the context of available castles or any architectural octagon, measuring the apothem helps in estimating both architectural features and engineering requirements.

In practical applications, the apothem is often preferred for simplifying calculations involving polygons, especially when dealing with geometry in architectural and engineering contexts.

Perimeter

The perimeter of a regular octagon is simply the sum of the lengths of all its sides. Because all sides in a regular octagon are equal, finding the perimeter becomes a straightforward multiplication.

• Formula: \[\text{Perimeter} = 8 \times \text{side length}\]
• This uniformity allows for easy calculations and simplifications in both architectural designs and mathematical evaluations.

Knowing the perimeter is often the first step in the calculation of the area, as it directly factors into the area formula. Moreover, it is fundamental when assessing the materials needed in construction projects involving octagonal shapes.