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Chapter 11: Problem 1

The perimeter of a regular polygon is 24 and the apothem is \(3 .\) Find the polygon's area.

Short Answer

Expert verified
The area of the regular polygon is 36 square units

Step by step solution

01

Understanding the formula

The formula for the area of a regular polygon is given by \[Area = \frac{1}{2} * Perimeter * Apothem\]. It's important to understand that in this case, the perimeter is the sum of the lengths of all sides, and the apothem is the perpendicular distance from the center to any side.
02

Plug in the values

Understood the formula, substitute the given values into the formula. The given values are Perimeter = 24 and Apothem = 3. So, our formula becomes \[Area = \frac{1}{2} * 24 * 3\].
03

Solve the result

To find a result solve the operations in the formula. First, calculate the product 24 and 3, which is 72. Then divide 72 by 2 to get the final result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter of a Regular Polygon
The perimeter of a regular polygon is the total length around the polygon. It's calculated by summing up the lengths of all its sides. For regular polygons, all sides have equal lengths. Therefore, you can simply multiply the length of one side by the number of sides.

For example, if a regular hexagon (a polygon with six equal sides) has each side measuring 4 units, then its perimeter would be calculated as:
  • 6 sides
  • Each side = 4 units
Perimeter = 6 x 4 = 24 units.
In the exercise, the perimeter is already given as 24, which is the length around the polygon. This essential value is required to calculate the polygon's area.
Apothem of a Regular Polygon
The apothem of a regular polygon is a significant element in geometry. It refers to the perpendicular distance from the center of the polygon to the midpoint of one of its sides. The apothem is especially notable in regular polygons because it helps in calculating the area.

To visualize the apothem, imagine dividing the polygon into equal triangles by drawing lines from the center to each vertex. The apothem is the height of these triangles, offering an efficient way to calculate the polygon's area.
In our scenario, the apothem is given as 3 units, simplifying further calculations related to the area of the polygon.
Area of a Regular Polygon
The area of a regular polygon can be derived using the perimeter and the apothem. The formula for calculating the area is:
  • \(Area = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\)
This formula captures the combined effort of both the perimeter and the apothem to determine how much space is enclosed within the polygon.

When you plug the given values of the perimeter (24) and apothem (3) into this formula, follow these steps:
  • Multiply the perimeter and apothem: 24 x 3 = 72
  • Divide the product by 2: \(\frac{72}{2} = 36\)
Thus, the area of the polygon is 36 square units. This calculated area represents the space the polygon covers.

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Most popular questions from this chapter

The hypotenuse of a right triangle is \(50,\) and one leg is 14 a. Find the area of the triangle. b. Find the altitude to the hypotenuse.

If the ratio of the areas of two similar polygons is \(9: 16,\) find the ratio of a pair of corresponding altitudes.

Find the length of one side and of the apothem of a. A square whose area is 121 b. An equilateral triangle whose area is \(36 \sqrt{3}\) sq \(\mathrm{m}\) c. A regular hexagon whose perimeter is \(24 \mathrm{cm}\)

If the midpoints of the sides of a quadrilateral are joined in order, another quadrilateral is formed. Find the ratio of the area of the larger quadrilateral to that of the smaller quadrilateral.

Given a trapezoid with bases 6 and 15 and height \(7,\) find the median and the area.
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