Question

The petals in the flower form a polygon that is approximately a regular pentagon with an apothem \(3.4\) centimeters long. a. Find the approximate area of the pentagon. b. There are five triangles in the pentagon that are not part of the flower. Assume that they have equal areas and have a height of \(1.5\) centimeters. Find the total area of the triangles. c. What is the approximate area of the flower?

Short answer

The pentagon's area is approximately 55.17 cm², the triangles have a total area of 18.75 cm², the flower's area is about 36.42 cm².

Step-by-step solution

Understanding Pentagon Area Formula

To find the area of a regular pentagon, use the formula: \( \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \). The perimeter can be determined knowing the side length.

Finding the Side Length

We know that the apothem \( a = 3.4 \) cm and the formula for the side length related to the apothem in a regular pentagon is \( s = 2a \tan(\frac{\pi}{5}) \). Calculate \( s \) using this relation.

Calculation of Perimeter

The perimeter of the pentagon is \( 5 \times s \), where \( s \) is the side length found in the previous step. Plug in the value of \( s \) to find the perimeter.

Calculating Pentagon Area

Use the area formula: \( \frac{1}{2} \times \text{Perimeter} \times 3.4 \). Substitute the perimeter from Step 3 to find the area of the pentagon.

Area of One Triangle

Each of the five triangles inside the pentagon has a height of 1.5 cm and forms part of an isosceles triangle with the pentagon's side as the base. Calculate the area of one triangle using \( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \).

Total Area of Five Triangles

Multiply the area of one triangle by 5 to obtain the total area of all the triangles.

Area of Flower

Subtract the total area of the five triangles (from Step 6) from the area of the pentagon (from Step 4) to find the approximate area of the flower.

Key concepts

Understanding a Regular Pentagon

A regular pentagon is a five-sided polygon where all sides are of equal length and all internal angles are the same, specifically 108 degrees each. This symmetry is what makes the pentagon 'regular.'
Regular pentagons have a center point from which all vertices are equidistant. This centrality aids in many calculations, such as finding the area or the relation between the side and the apothem.
The regular pentagon can be easily broken down into five symmetrical triangles for simpler calculations, which is useful in solving geometry problems.

• All sides and angles are equal.
• Internal angle = 108 degrees.
• The sum of internal angles = 540 degrees.

Calculating the Area of a Regular Pentagon

To find the area of a regular pentagon, we make use of its symmetrical properties, employing a specific formula. The area is calculated with: \[\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\] The perimeter is simply the total length around the pentagon and is calculated by multiplying the side length by five.
Once you have the perimeter, you multiply it by the apothem and divide by two to find the area. This formula arises because the pentagon can be split into five triangles, and we can sum the area of these triangles.

• Perimeter = 5 x side length.
• Area is calculated using the perimeter and apothem.

Role of the Apothem in Area Calculation

The apothem is a crucial part of a regular pentagon, acting as the backbone of the area calculation. It is the distance from the center of the pentagon to the midpoint of one of its sides.
The apothem helps in linking the side length to the overall shape of the pentagon using trigonometric relations.
Since the apothem connects to the formula for the side of the pentagon through the tangent function, understanding its role is essential. You calculate the side using: \[s = 2a \tan\left(\frac{\pi}{5}\right)\] where \(a\) is the apothem. This relation helps find the perimeter of the pentagon.

• Apothem is perpendicular from center to side.
• Key in calculating side length and perimeter.

Calculating Triangle Areas Inside a Pentagon

In a regular pentagon, particularly when solving for a flower or other similar problems, understanding the triangles within is useful.
Each triangle shares a side with the pentagon, using the pentagon's sides as their base. The height of these triangles is often known or given, enabling straightforward area calculation using \[\text{Area of one triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}\] By simply multiplying the area of one triangle by five, you can find the total area of all triangles within the pentagon.

• Each triangle has an equal base (side of pentagon).
• Area determined by base and given or computed height.