3D SolidsNets and Surface Area

Meaning

Let’s create a model of the car to estimate the number of sticky notes we will need.

We can use the unit cubes to create the model of the car:

DISCUSS: Do we need this blank? Perhaps we should do something else with it.

This means the surface area of the actual car is approximately 28 square meters centimeters inches feet.

DISCUSS: use equation editor instead?

If each sticky note is 0.006m2, then we will need to have =4467 sticky notes to cover the car!

Here, we have calculated the surface area of the car model to find the total number of sticky notes needed.

The surface area of a 3D solid is the number of square units that cover all the faces of the polyhedron, without any gaps or overlaps.

Surface Area = square units. The units used to measure the surface area are square meter (m2), square centimeter (cm2), square inches (in2), square feet (ft2), and so forth.

Surface Area vs Volume

Surface area and volume are different attributes of 3D figures. The key difference between them is that the surface area volume is a 2D measurement and the volume surface area is a 3D measurement of a solid.

You may build different solids by using the same number of cubes.

Use 6 cubes to create a solid with the greatest possible surface area.

A cuboid with the surface area of unit squares is the maximum surface area we can create by using the six cubes.

Now, create a solid with the smallest possible surface area.

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The minimum surface area is unit squares.

Both of the solids you have created have a volume of cubic units but have different surface areas. Notice the cubes that form solids with the more surface area are more spread out compact. More of their faces are exposed.

In nature, having a large or small surface area with respect to the volume determines the vital characteristics of living things.

The relation between the surface area and the volume of an object is so important that it affects where an animal can live when a cell has to divide, or the size of the wings of an airbus plane.

For instance, do you know why elephants have giant ears?

African elephants are the largest land animals on Earth. They can be identified by their larger ears with respect to other elephant species. They grow up to 4 meters and weigh 6 tones on average.

Elephants and all animals generate heat internally in proportion to their volume. Larger animals produce more heat with respect to small ones like mice.

Most of the reactions like heat transfer occur at the surface of the objects and living things. It means if a large animal lives in a hot environment, then it needs to lose heat by maximizing minimizing its surface area.

Alternatively, if you were living in the Antarctic you would want a small surface area to volume ratio. This would reduce heat loss and conserve it in the body. That’s why polar versions of animals are usually bigger smaller with respect to the other species in the same family.

The relationship between the volume and the surface area as a shape's dimensions changes is a mathematical principle called the square-cube law. (SA:V).

Animals can be thought of as simple cubes:

As you continue to increase the edge size of the cube, Volume Surface Area will grow faster.

Galileo might have been the first to formally recognize this when he stated in his 1638 book, Two New Science. In addition to biology, it has many applications in different scientific fields like mechanics, ecology, engineering.

Surface Area vs Volume

Let’s drag the examples of quantities related to volume and surface area to complete the diagram:

Nets

Artists and mathematicians like Leonardo Da Vinci devoted much effort to represent 3D objects on 2D paper. Da Vinci’s greatest accomplishment in this area is the illustrations for Luca Pacioli's 1509 book The Divine Proportion. Da Vinci drew roughly 60 different illustrations of polyhedra in the book.

Since we don't all have Da Vinci's artistic skills, we can use grid paper to help us draw polyhedra. Let’s start by drawing a cube on the dotted grid.

When the solids get complicated, drawing them on a 2D paper gets harder too. But there is another way to represent the 3D solids on a 2D plane.

We can use the nets of the solids which are composed of polygons that form the faces of a polyhedron.

They are the 2D coats that cover up the entire surface of 3D solids.

For instance, the net of a cube consists of squares. With the correct arrangement of the squares, they can fold up a cube.

Let’s try to draw the net of a cube:

There are many different ways to arrange six squares to fold up as a cube.

Let’s have a look at which one of the below can be folded to a cube.

INTERACTIVE-2.08: Net animations

The row of squares in Shape A can be folded into a ring and then the nearest square to the ring will close off a face, but the other square will have to overlap an existing face.

The row of squares in Shape B can be folded into a ring, and the other two squares can close off the other two faces. You can visualize those two faces as the bases of the cube.

When you fold the row of squares in Shape C into a ring, the fifth square will overlap the first one.

The row of squares in Shape D can be folded into a ring, and the other two squares can close off the other two faces. You can visualize those two faces as the bases of the cube.

One of the most common cubes we come across every day are dice.

Let’s try to build our own fair die by inserting the faces correctly.

Opposite faces of a fair die always add up to 7.

Great!

Lets try one more:

Now that we can identify opposite faces of a die using nets, let's try another puzzle:

The opposite face of the yellow is painted bluepurplepinkgreenred.

The opposite face of the green is painted pinkbluepurplegreenred.

The opposite face of the red is painted purplebluepinkgreenred.

Continue

INTERACTIVE-2.10: Net construction

Using nets to calculate the surface area

A net allows us to see all the faces of a 3D solid at once. We can use the nets to find the surface areas of the cuboids instead of counting the numbers of squares in each face one by one.

Let’s find the surface area of this cuboid by using its net

Use the slider to open the cuboid to its net. Then drag the side length measures to corresponding sides to find the surface area.

INTERACTIVE-2.12: solid <-> net

The pink rectangles have a total area of m2

The yellow rectangles have a total area of m2

The orange rectangles have a total area of m2

So the cuboid has a total area of m2.

Can we come up with shortcuts to find the surface area of the cuboids?

Let’s have a look at the cuboids here:

Use the sliders to open the nets of the cuboids.

INTERACTIVE-2.13: solid <-> net

Drag the area calculations on the corresponding regions.

Since nets are composed of plane figures that form the faces of a 3D Solid, we can always use them to calculate the surface area of even more complex solids.

In the next chapter, we are going to look at the nets, surface area, and volume of different types of polyhedra.