Question

If the volume of a sphere, a cube, a tetrahedron and octahedron be same then which of the following has maximum surface area? (a) Sphere (b) Cube (c) Octahedron (d) Tetrahedron

Short answer

Answer: (c) Octahedron

Step-by-step solution

Find formulas for the surface area and volume of each shape

We will find formulas for the volume and surface area of each of the given shapes. Recall the formulas: - Sphere: Volume: \(V = \frac{4}{3}\pi r^3\), Surface area: \(A = 4\pi r^2\) - Cube: Volume: \(V = a^3\), Surface area: \(A = 6a^2\) - Tetrahedron: Volume: \(V = \frac{a^3}{6\sqrt{2}}\), Surface area: \(A = a^2\sqrt{3}\) - Octahedron: Volume: \(V = \frac{a^3\sqrt{2}}{3}\), Surface area: \(A = 2a^2\sqrt{2}\)

Express the surface area in terms of the volume

In order to compare the surface area for each shape while having the same volume, we'll express each surface area in terms of the volume using their respective formulas. - Sphere: Solving the volume formula for r: \(r = \sqrt[3]{\frac{3V}{4\pi}}\) Substituting r into the surface area formula: \(A_{sphere} = 4\pi\left(\sqrt[3]{\frac{3V}{4\pi}}\right)^2\) - Cube: Solving the volume formula for a: \(a = \sqrt[3]{V}\) Substituting a into the surface area formula: \(A_{cube} = 6\left(\sqrt[3]{V}\right)^2\) - Tetrahedron: Solving the volume formula for a: \(a = \sqrt[3]{6\sqrt{2}V}\) Substituting a into the surface area formula: \(A_{tetrahedron} = \left(\sqrt[3]{6\sqrt{2}V}\right)^2\sqrt{3}\) - Octahedron: Solving the volume formula for a: \(a = \sqrt[3]{\frac{3V}{\sqrt{2}}}\) Substituting a into the surface area formula: \(A_{octahedron} = 2\left(\sqrt[3]{\frac{3V}{\sqrt{2}}}\right)^2\sqrt{2}\)

Compare surface areas

Now that we've expressed the surface areas in terms of volume, we want to compare them to find which one has the maximum surface area for the same volume. For any positive volume V: - A sphere has a smaller surface area to volume ratio than a cube due to the presence of \(\pi\) in its surface area formula. - A tetrahedron has a smaller surface area to volume ratio than a cube because it has a \(\sqrt{3}\) factor in its surface area formula. - An octahedron has a larger surface area to volume ratio than a cube because it has a \(\sqrt{2}\) factor in its surface area formula. So out of all four shapes, the octahedron has the highest surface area to volume ratio and thus the maximum surface area. So the correct answer is: (c) Octahedron

Key concepts

Solid Geometry

Solid geometry is the study of three-dimensional figures like spheres, cubes, and pyramids. Unlike two-dimensional shapes that have only length and width, three-dimensional shapes also have depth or height, which enables them to enclose space and hence have volume. Solid figures are everywhere in our daily lives, from the basketball you might play with, to the ice cube cooling your drink, or even the pyramid-shaped tents you might camp in.

When learning solid geometry, it is essential to become familiar with terms like vertices (corners), edges (the line segments where two faces meet), and faces (flat or curved surfaces). For example, a cube has 6 faces, all of which are squares, 12 edges, and 8 vertices. Understanding these basics helps students visualize and solve problems involving three-dimensional objects.

Volume and Surface Area Formulas

In solid geometry, volume and surface area are two fundamental attributes of three-dimensional figures. Volume measures the capacity of the figure, representing the amount of space it occupies, usually quantified in cubic units. Surface area, on the other hand, refers to the total area that the surface of the object covers.

To calculate these values, we rely on mathematical formulas specific to each geometric shape. For instance:

• The volume of a cube is given by the formula: \(V = a^3\), where 'a' is the length of its sides.
• The surface area of a sphere is computed as: \(A = 4\rightarrow r^2\), with 'r' being the radius of the sphere.

The correct formula will allow you to compare figures based on these measurements effectively.

Comparing Geometric Shapes

Comparing geometric shapes like spheres, cubes, and tetrahedrons involves analyzing the relationship between their volume and surface area, particularly when figuring out which shape has the maximum or minimum surface area for a given volume. This process involves setting the volume for different shapes equal to a constant value and then determining the corresponding surface area for each.

By understanding the formulas and manipulating them algebraically, you can deduce the surface areas in terms of volume. This way, a direct comparison can be made. For example, while a cube's surface area formula involves only the side length squared, more complex shapes like octahedrons involve additional square roots in their calculations. These differences in formulas affect how the surface area scales with volume and can lead to interesting findings, such as in our textbook problem where the octahedron has the largest surface area for the same volume amongst the mentioned shapes.