Pentagonal Prism Volume Calculation A Step-by-Step Guide

In the realm of geometry, prisms stand as fundamental three-dimensional shapes, characterized by their consistent cross-sectional area throughout their length. This article embarks on a journey to unravel the intricacies of a specific prism: one crafted with two regular pentagons as its bases. Our mission is to decipher the expression that precisely represents the volume of this prism, measured in cubic centimeters. To navigate this mathematical landscape, we'll dissect the properties of regular pentagons, explore the concept of apothems, and ultimately, piece together the formula that unlocks the prism's volume.

At the heart of our prism lies the regular pentagon, a five-sided polygon distinguished by its equal side lengths and equal interior angles. Each interior angle of a regular pentagon measures 108 degrees, a testament to its inherent symmetry. To calculate the area of a regular pentagon, we often turn to the apothem, a line segment drawn from the center of the pentagon perpendicular to one of its sides. This apothem serves as a crucial tool in our quest to determine the prism's volume.

The apothem, in essence, acts as the radius of the pentagon's inscribed circle, a circle that nestles perfectly within the pentagon, touching each side at its midpoint. Knowing the apothem, which in our case is 2.8 centimeters, is a key stepping stone in calculating the pentagon's area. The formula that connects the area (A) of a regular polygon to its apothem (a) and perimeter (P) is elegantly simple: A = (1/2) * a * P. This formula underscores the apothem's significance in area calculations.

To further unravel the pentagon's area, we need to determine its perimeter. Since a regular pentagon boasts five equal sides, knowing the length of one side is sufficient to calculate the perimeter. Let's denote the side length as 's'. The perimeter (P) then becomes 5s. Now, the challenge lies in relating the side length 's' to the given apothem of 2.8 centimeters. This is where trigonometry steps in, offering its powerful tools to bridge this gap.

To relate the apothem to the side length, we can dissect the regular pentagon into five congruent isosceles triangles, each radiating from the center of the pentagon. The apothem, by virtue of being perpendicular to the side, bisects both the side and the central angle of the pentagon. This creates a right-angled triangle within the isosceles triangle, with the apothem as one leg, half the side length (s/2) as another leg, and the radius of the circumscribed circle (the circle that passes through all vertices of the pentagon) as the hypotenuse.

The central angle of the pentagon, 360 degrees, is divided into five equal parts, making each central angle 72 degrees. The right-angled triangle we've carved out has an angle of 36 degrees (half of 72 degrees) opposite the side s/2. Now, we can invoke the trigonometric tangent function, which relates the opposite side to the adjacent side in a right-angled triangle. In our case, tan(36°) = (s/2) / 2.8. Solving for 's', the side length, is our next crucial step.

Rearranging the equation, we get s = 2 * 2.8 * tan(36°). Using a calculator, we find that tan(36°) is approximately 0.7265. Therefore, s ≈ 2 * 2.8 * 0.7265 ≈ 4.0684 centimeters. With the side length in hand, we can now calculate the perimeter of the pentagon: P = 5s ≈ 5 * 4.0684 ≈ 20.342 centimeters. This perimeter value is the final piece we need to unlock the pentagon's area.

Now that we have both the apothem (2.8 cm) and the perimeter (approximately 20.342 cm), we can confidently calculate the area of the regular pentagon using the formula A = (1/2) * a * P. Plugging in the values, we get A ≈ (1/2) * 2.8 * 20.342 ≈ 28.479 square centimeters. This area represents the foundation upon which our pentagonal prism is built.

The volume of any prism is simply the product of its base area and its height. In our case, the base is the regular pentagon we've been meticulously analyzing, and the height is the perpendicular distance between the two pentagonal bases. Let's denote the height of the prism as 'x' centimeters. Therefore, the volume (V) of the prism is given by V = A * x, where A is the area of the pentagon.

Substituting the area we calculated, A ≈ 28.479 square centimeters, into the volume formula, we get V ≈ 28.479 * x cubic centimeters. This expression, while numerically accurate, doesn't directly match the algebraic options presented. The key is to recognize that the side length 's' is implicitly present in the area calculation. We need to express the area in terms of 'x' to match the given answer choices.

Going back to the side length calculation, s ≈ 2 * 2.8 * tan(36°), we can see that 's' is a constant value. The perimeter, P = 5s, is also a constant. The area, A = (1/2) * 2.8 * P, is therefore a constant as well. The volume, V = A * x, is a linear function of 'x'. This understanding is crucial in selecting the correct algebraic expression.

The provided options are algebraic expressions involving 'x', presumably representing the height of the prism. We need to find the expression that, when evaluated, yields the volume we calculated, V ≈ 28.479 * x cubic centimeters. This means we're looking for an expression that has a coefficient of approximately 28.479 when 'x' is considered.

Let's examine the options:

A. 9x² + 7x B. 14x² + 7x C. 16x² + 14x

Notice that options A, B, and C are all quadratic expressions, meaning they involve a term with x². However, we know that the volume is a linear function of x, not a quadratic function. This immediately eliminates options A, B, and C.

To find the correct answer, we must go back to the basics of volume calculation for a prism and meticulously re-evaluate each step to ensure no approximations have led us astray. Let's revisit the area calculation of the pentagon.

Recall that the area of a regular pentagon can be expressed as A = (5/4) * s² * cot(π/5), where 's' is the side length. We found that s = 2 * 2.8 * tan(36°). Let's express 36° in radians: 36° = 36 * (π/180) = π/5 radians. Therefore, s = 5.6 * tan(π/5).

Substituting this into the area formula, we get:

A = (5/4) * (5.6 * tan(π/5))² * cot(π/5)

A = (5/4) * (5.6)² * tan²(π/5) * cot(π/5)

Since cot(π/5) = 1 / tan(π/5), we can simplify this to:

A = (5/4) * (5.6)² * tan(π/5)

Now, the volume V = A * x becomes:

V = (5/4) * (5.6)² * tan(π/5) * x

V = (5/4) * 31.36 * tan(π/5) * x

Using a calculator, we find that tan(π/5) ≈ 0.7265. Therefore:

V ≈ (5/4) * 31.36 * 0.7265 * x

V ≈ 28.479x

This confirms our previous calculation. However, the absence of a matching linear expression in the options suggests there might be an error in the provided choices or in the problem statement itself. We have meticulously followed the correct geometric principles and calculations, arriving at a linear expression for the volume.

Through a comprehensive analysis of the pentagonal prism, we have determined that its volume can be accurately represented by a linear expression of the form V ≈ 28.479x cubic centimeters, where 'x' is the height of the prism. The provided options, being quadratic expressions, do not align with this result. This discrepancy highlights the importance of verifying the accuracy of problem statements and answer choices in mathematical problem-solving. The journey through this geometric puzzle underscores the power of fundamental principles, trigonometric relationships, and meticulous calculations in unraveling the complexities of three-dimensional shapes.

Understanding the Problem The prompt "A prism is created using 2 regular pentagons as bases. The apothem of each pentagon is 2.8 centimeters. Which expression represents the volume of the prism, in cubic centimeters?" presents a classical geometry problem that requires us to apply our knowledge of prisms, regular pentagons, and their respective formulas. We're given a prism with regular pentagonal bases and the apothem of each pentagon. The goal is to determine the algebraic expression that accurately represents the prism's volume.

Key Concepts and Formulas

Before diving into the calculations, let's refresh the key concepts and formulas that will guide us:

• Prism: A prism is a three-dimensional geometric shape with two parallel and congruent bases connected by lateral faces that are parallelograms.
• Volume of a Prism: The volume (V) of a prism is given by the formula V = A * h, where A is the area of the base and h is the height of the prism (the perpendicular distance between the bases).
• Regular Pentagon: A regular pentagon is a five-sided polygon with all sides and all angles equal. Each interior angle of a regular pentagon measures 108 degrees.
• Apothem: The apothem of a regular polygon is the perpendicular distance from the center of the polygon to the midpoint of one of its sides.
• Area of a Regular Polygon: The area (A) of a regular polygon can be calculated using the formula A = (1/2) * a * P, where 'a' is the apothem and P is the perimeter of the polygon. Another formula, more directly related to the side length, is A = (5/4) * s² * cot(π/5), where 's' is the side length.
• Trigonometry: Trigonometric functions, particularly the tangent function, play a crucial role in relating the apothem and side length of the regular pentagon.

Step-by-Step Solution

Let's break down the problem into manageable steps:

1. Relate the Apothem to the Side Length

As discussed earlier, we can divide the regular pentagon into five congruent isosceles triangles. The apothem bisects both the side and the central angle of the pentagon, creating a right-angled triangle. If 's' is the side length of the pentagon, the apothem (2.8 cm) is one leg of the right-angled triangle, and half the side length (s/2) is the other leg. The angle opposite the side s/2 is 36 degrees (half of the central angle 72 degrees).

Using the tangent function, we have:

tan(36°) = (s/2) / 2.8

Solving for 's':

s = 2 * 2.8 * tan(36°)

s = 5.6 * tan(36°)

2. Calculate the Area of the Pentagon

Now that we have the side length in terms of tan(36°), we can use the formula A = (5/4) * s² * cot(π/5) to calculate the area. Remember that π/5 radians is equivalent to 36 degrees, and cot(36°) = 1 / tan(36°).

Substituting the value of 's', we get:

A = (5/4) * (5.6 * tan(36°))² * cot(36°)

A = (5/4) * (5.6)² * tan²(36°) * (1 / tan(36°))

A = (5/4) * (5.6)² * tan(36°)

3. Determine the Volume of the Prism

The volume (V) of the prism is the product of the base area (A) and the height (h), which we'll represent as 'x' centimeters.

V = A * x

Substituting the area expression, we get:

V = (5/4) * (5.6)² * tan(36°) * x

4. Simplify the Expression

Let's simplify the expression to match the algebraic form of the answer choices:

V = (5/4) * 31.36 * tan(36°) * x

This expression represents the volume of the prism in cubic centimeters.

Conclusion

By meticulously applying the principles of geometry and trigonometry, we've successfully derived an expression for the volume of the pentagonal prism. The expression V = (5/4) * 31.36 * tan(36°) * x accurately represents the volume in cubic centimeters, where 'x' is the height of the prism. This exercise underscores the importance of a systematic approach, formula mastery, and trigonometric insight in tackling geometric problems.

Original Keyword: Which expression represents the volume of the prism, in cubic centimeters?

Improved Keyword: What expression represents the volume, in cubic centimeters, of a prism with regular pentagonal bases and an apothem of 2.8 cm?

Pentagonal Prism Volume Calculation A Step-by-Step Guide