Problem 218 - Calculus Volume 1

Question

An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from \(a\) unit to 2\(a\) units?

Step-by-Step Solution

Answer

The surface area increases by \(15\sqrt{3} a^2\).

Step 1: Understanding the Problem

An icosahedron consists of 20 equilateral triangles. To find out how the surface area changes as the side length doubles, we first need to calculate the initial surface area.

Step 2: Calculate the Initial Surface Area

The area of an equilateral triangle with side length \(a\) is given by the formula \(\frac{\sqrt{3}}{4}a^2\). Since there are 20 such triangles in an icosahedron, the total surface area is \(20 \times \frac{\sqrt{3}}{4} a^2 = 5\sqrt{3} a^2\).

Step 3: Calculate the New Surface Area

When the side length of each triangle is doubled to \(2a\), the area of one triangle becomes \(\frac{\sqrt{3}}{4}(2a)^2 = \frac{\sqrt{3}}{4} \times 4a^2 = \sqrt{3} a^2\). Therefore, the total surface area of the icosahedron with doubled side lengths is \(20 \times \sqrt{3} a^2 = 20\sqrt{3} a^2\).

Step 4: Determine the Increase in Surface Area

Subtract the initial surface area from the new surface area: \(20\sqrt{3} a^2 - 5\sqrt{3} a^2 = 15\sqrt{3} a^2\). Thus, the surface area increases by \(15\sqrt{3} a^2\).

Key Concepts

Platonic solid

Platonic solids are a fascinating and important topic in the world of geometry. They are a special class of three-dimensional shapes composed of faces that are identical in size and shape. Each face is a polygon, and more specifically, each face of a Platonic solid is an identical regular polygon. Furthermore, all edges are of equal length, and all angles are equal.
The icosahedron, for instance, is one of these Platonic solids. It consists of 20 equilateral triangle faces. Remarkably, there are only five types of Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Among these, the icosahedron is notable for having the most faces.

• Common features of Platonic solids include congruent faces, edges of equal length, and equal angles.
• These characteristics make them symmetrical and aesthetically pleasing.

Understanding Platonic solids gives insight into both mathematical beauty and symmetry found in nature and art.

Surface area

The concept of surface area applies to three-dimensional objects, like an icosahedron. The surface area is the total area that the surface of the object occupies.
Each face of the icosahedron is an equilateral triangle, making it easier to calculate the surface area. To find the surface area of an entire icosahedron, we calculate the area of one triangle and multiply it by the number of triangles. The formula for the area of one equilateral triangle with side length \(a\) is:\[\frac{\sqrt{3}}{4}a^2\]Thus, the overall surface area of the icosahedron with 20 triangles is:\[20 \times \frac{\sqrt{3}}{4} a^2 = 5\sqrt{3} a^2\]
When the sides of each triangle are doubled, the new surface area is calculated similarly, reflecting how even slight changes in dimensions can significantly impact the total surface area.

Equilateral triangle

An equilateral triangle is a triangle with all three sides of equal length, and all its angles are also equal, each being 60 degrees. This unique attribute not only makes the triangle stable in structure but also simpler to work with when calculating areas.
The formula for the area of an equilateral triangle is key in geometric calculations involving Platonic solids like the icosahedron:\[\frac{\sqrt{3}}{4}a^2\]
This equation is derived from the general formula for the area of a triangle (\( \frac{1}{2} \times \text{base} \times \text{height} \)) and utilizes the properties specific to the equilateral triangle to simplify calculations.

• All angles in an equilateral triangle are equal, making it a highly symmetrical and predictable shape.
• The equality of sides and angles ensures uniformity in geometric constructions.

These properties are valuable in constructing and understanding complex geometric shapes like the icosahedron.

Geometry

Geometry, the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs, plays a crucial role in many aspects of both theoretical and applied mathematics. The study of shapes like the icosahedron falls under this vast field.
Icosahedrons are a great example of how geometry helps us understand and manipulate space. By using geometric principles, we can precisely calculate properties such as the area and volume of various shapes. These calculations help in various fields, from architecture to digital modeling.

• Understanding geometric principles allows us to solve complex problems involving space and structure.
• Geometry is foundational in subjects such as calculus, trigonometry, and even physics.

The constant exploration and manipulation of geometric shapes push the boundaries of what we know, making it a continuous journey of discovery.

Mathematical problem solving

Mathematical problem solving is a skill that involves applying mathematical concepts and techniques to find solutions to problems. When faced with a problem such as calculating how the surface area of an icosahedron changes as its dimensions change, this is where this skill shines.
Problem-solving in mathematics often involves

• Understanding the problem: Comprehending precisely what is being asked.
• Planning: Deciding on the best approach or method to use.
• Executing: Carrying out the plan methodically and accurately.
• Reviewing: Checking the solution for errors or possible improvements.

In our specific problem with the icosahedron, solving involved recognizing the type of geometric shape, employing area formulas, and performing algebraic manipulations. Mastering problem-solving unravels complex challenges and deepens understanding of the interconnectedness of mathematical principles.