Question

Derive a formula for the area \(A_{5}\) of a regular pentagon, with side \(a\) (using plane geometry). Discuss the differences between Heron's formula \(A_{5}=\frac{5}{3} a^{2}\) and your formula.

Short answer

Question: Explain the similarities and differences between the derived formula for the area of a regular pentagon and Heron's formula. Answer: The derived formula for the area of a regular pentagon and Heron's formula both involve calculating area using side lengths of a triangle, specifically, incorporating the side length "a" squared. However, the main difference is the presence of the additional factor, sin(54°), in the derived formula for a pentagon. Additionally, Heron's formula does not yield the correct area for a regular pentagon, while our derived formula, which involves plane geometry and dividing the pentagon into congruent triangles, accurately represents the area of a regular pentagon.

Step-by-step solution

Divide the pentagon into triangles

We are given a regular pentagon with side length a. To find the area, we will divide the pentagon into five congruent isosceles triangles by drawing lines from each vertex to the center of the pentagon. Each triangle will have a base equal to the side length a and two congruent sides, also of length a.

Determine the central angle of each triangle

The sum of the interior angles of a polygon can be found using the formula \((n-2) \times 180^{\circ}\), where n is the number of sides. In our case, n = 5, so the sum of the interior angles of the pentagon is \((5-2) \times 180 = 3 \times 180 = 540^{\circ}\). Since there are five congruent triangles, the central angle of each triangle is \(540^{\circ}/5 = 108^{\circ}\).

Calculate the height of each triangle

Let's focus on one of the isosceles triangles. Let \(O\) be the center of the pentagon, \(A\) and \(B\) be two vertices of the triangle, and \(C\) be the midpoint of side \(AB\). We have \(\angle AOC= 1/2 *108^{\circ} = 54^{\circ}\), and \(\triangle AOC\) is a right-angled triangle (\(\angle ACB\) is a right angle). Using the sine function, we can find the height from the center to a side \(h_{A}=\frac{a}{2}\): $$ \sin(54^{\circ}) = \frac{h_{A}}{a} $$ Thus, the height \(h_{A} = a\sin(54^{\circ})\).

Calculate the area of a triangle

Now we can find the area of one triangle. The area of an isosceles triangle can be found using the formula \(A_{triangle}=\frac{1}{2}bh_{A},\) where b is the base, and \(h_{A}\) is the height. In our case, b = a, so: $$ A_{triangle}=\frac{1}{2}a(a\sin(54^{\circ})) = \frac{1}{2}a^{2}\sin(54^{\circ}) $$

Calculate the area of the pentagon

Since the pentagon is made up of five congruent triangles, its area can be found by multiplying the area of one triangle by 5: $$ A_{5}=5A_{triangle} = 5\frac{1}{2}a^{2}\sin(54^{\circ}) = \frac{5}{2}a^{2}\sin(54^{\circ}) $$ We derived the formula for the area of a regular pentagon: $$ A_{5} = \frac{5}{2}a^{2}\sin(54^{\circ}) $$

Comparison with Heron's formula

Comparing our derived formula to Heron's formula, we can see that the main difference is the presence of the \(\sin(54^{\circ})\) factor in our formula. Also, Heron's formula gives \(\frac{5}{3} a^{2}\), which is not the correct formula for the area of a regular pentagon. Our derived formula, using plane geometry, provides a more accurate representation of the area of a regular pentagon.

Key concepts

Regular Pentagon Area

Finding the area of a regular pentagon requires a bit of geometry magic! A regular pentagon has five equal sides and five equal angles. To make it easier to understand, think of slicing the pentagon like a pizza into five triangular pieces. These triangles are congruent, meaning they are identical, sharing the same base and height.
This method allows us to calculate the area of one triangle and then multiply by five to find the total area of the pentagon. The elegant formula for the area of a regular pentagon with side length \(a\) is \(A_5 = \frac{5}{2}a^2\sin(54°)\). This formula incorporates basic trigonometric functions and highlights the beauty of geometry.

Isosceles Triangle

An isosceles triangle is a triangle with at least two equal sides. In the case of a regular pentagon, each "pizza slice" is an isosceles triangle.

• All triangles have one angle at the pentagon's center, known as the central angle.
• The other two angles are equal because it's an isosceles triangle.

When you draw lines from the pentagon's vertices to the center, these lines become the equal sides of each isosceles triangle. Understanding these triangles helps unravel the geometry of the pentagon.

Central Angle

The central angle in a regular pentagon is crucial for dividing it into isosceles triangles. It can be found using the formula for interior angles of a polygon: \((n-2) \times 180°\), where \(n\) is the number of sides.
For a pentagon, this sums to \(540°\) because it has five sides. Dividing this sum by the five triangles gives a central angle of \(108°\). This angle helps in calculating other properties, like the height of the triangles, using trigonometric functions.

Trigonometric Functions

Trigonometric functions, particularly the sine function, play a pivotal role in finding the area of a regular pentagon. These functions relate the angles of a triangle to its side lengths.
Consider the isosceles triangle in the pentagon:

• The sine of \(54°\) is used to find the height of the triangle from its base to its apex at the pentagon's center.
• The formula \(h = a \sin(54°)\) helps determine this height, where \(h\) is the height, and \(a\) is the side length of the pentagon.

This calculated height is then used to find the area of each triangle, contributing to the pentagon's area.