Question

What is the area of a pentagon with a perimeter of 30 centimeters and an apothem of 4.13 centimeters? A. \(34.13 \mathrm{~cm}^2\) B. \(61.95 \mathrm{~cm}^2\) C. \(68.26 \mathrm{~cm}^2\) D. \(123.90 \mathrm{~cm}^2\)

Short answer

The area of the pentagon is 61.95 cm\textsuperscript{2}.

Step-by-step solution

Understand the Problem

To find the area of a regular pentagon, you need to know the formula and the given values. The formula for the area is: \[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \].

Identify Given Values

From the problem, the given values are: - Perimeter (P) = 30 cm - Apothem (a) = 4.13 cm

Substitute the Values into the Formula

Substitute the given values into the formula to find the area: \[ \text{Area} = \frac{1}{2} \times 30 \text{ cm} \times 4.13 \text{ cm} \].

Calculate the Area

Perform the calculation: \[ \text{Area} = \frac{1}{2} \times 30 \times 4.13 \] \[ \text{Area} = \frac{1}{2} \times 123.9 \] \[ \text{Area} = 61.95 \text{ cm}^2 \].

Select the Correct Answer

Among the answer choices given, the correct answer is: B. 61.95 \( \text{cm}^2 \).

Key concepts

pentagon area formula

The area of a regular pentagon can be calculated using a simple formula. This formula relies on two crucial measurements: the perimeter and the apothem.
To find the area, use:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \].

• Perimeter: This is the total length around the pentagon, summing up all five sides.
• Apothem: This is a line from the center of the pentagon to the midpoint of one of its sides, perpendicular to that side.

In our problem, we are given a pentagon with a perimeter (P) of 30 cm and an apothem (a) of 4.13 cm. By plugging these values into our formula:
\[ \text{Area} = \frac{1}{2} \times 30 \text{ cm} \times 4.13 \text{ cm} \].
The calculation is carried out step-by-step until we get the result:
\[ 61.95 \text{ cm}^2 \].

geometry

Understanding the basic concepts of geometry is pivotal when diving into the calculations of shapes like the pentagon. Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids.
Pentagons are five-sided polygons. There are regular and irregular pentagons:

• Regular Pentagon: All sides and interior angles are equal.
• Irregular Pentagon: Sides and angles may vary.

In our exercise, we deal with a regular pentagon. Characteristics of regular pentagons include:

• Equal side lengths.
• Equal interior angles of 108 degrees each.
• Symmetrical shape.

The concepts of angles, symmetry, and equal side lengths help simplify our calculations and understanding of other properties, such as apothem and perimeter.

perimeter and apothem

Two key measurements, the perimeter and the apothem, are essential in the formula for finding the area of a regular pentagon.

• Perimeter (P): The perimeter is the total distance around the pentagon. For a regular pentagon, it's calculated by multiplying the length of one side by 5.
• For example, if one side is 6 cm, the perimeter is \(5 \times 6 \text{ cm} = 30 \text{ cm}\).
• Apothem (a): The apothem is the line segment from the center of the pentagon to the midpoint of one of its sides.
• It forms a right angle with the side. It can be thought of as the 'radius' of an inscribed circle within the pentagon.

These values are integral to our formula:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \].
By substituting known values into the formula and simplifying, we find the area.
In our example, with a perimeter of 30 cm and an apothem of 4.13 cm:
\[ \text{Area} = \frac{1}{2} \times 30 \text{ cm} \times 4.13 \text{ cm} \ = 61.95 \text{ cm}^2 \].