Calculating the Area of Three-Dimensional Objects

# Calculating the Area of Three-Dimensional Objects

In practical situations, it is necessary to understand how much area is covered when completing activities such as tiling a floor or carpeting a room. In other situations, it is just as important to have a strategy to determine how much of a liquid a specific can hold. Still in another situation, one might want to understand how much of an item a container can hold without overflowing. Specifically, an example which comes to mind is a canister which will be filled with sugar. Sugar is purchased in a five pound bag and before the owner pours the sugar, it would be helpful to understand if all of the sugar will fit into the canister to avoid spilling. In order to find the answers to these questions, it is necessary to find the area of a three dimensional container.

As previously discussed, finding the area of two-dimensional shapes can involve counting the squares the shape covers or be found using a mathematical formula such as length * width. Calculating the area for three dimensional shapes brings a more complex picture into view. There are two different types of area which can be calculated for three-dimensional shapes. The first type of area is known as surface area. Surface area is the amount of space all of the outside surfaces of a three dimensional shape cover. This information would be most helpful if you needed to cover the outside of a three dimensional object, such as wrapping a box or painting a cylinder. The second type is known as volume. Volume is all of the space which can be found inside a three-dimensional shape. This information is most helpful when you want to fill a three-dimensional shape, such as filling a glass with water or the canister described previously.

Surface area is most easily found by determining the individual areas of each two-dimensional side of a three dimensional shape. If you were able to cut apart a three-dimensional shape and lay it flat, the surface area would be the space the shape covers; this is typically referred to as a net. Since it is rarely possible to actually cut apart a three-dimensional shape, another way must be considered to calculate the surface area. For polyhedrons (a three-dimensional shape where all of the surfaces are made up of polygons) the surface area can be found by calculating the area of each face and adding all of these areas together. For example:

Consider a cube where each side is 6 inches in length. The area of one face of the cube would be 6 inches (length) * 6 inches (width) = 36 square inches. Since a cube has six faces (sides) you would need to add the 36 square inches area of one face 6 times, which could be written as 36 + 36 + 36 + 36 + 36 + 36 = 216 square inches. Remember that repeated addition, such as 36 + 36 + 36 + 36 + 36 + 36, can also be written as a multiplication problem. In this case, the multiplication problem would be 36 (surface area of one side) * 6 (number of faces).

To calculate the surface area of a cylinder you must use the mathematical formula below:

Surface area of a cylinder= 2 Π r2 + 2 Π r h

In the above formula r refers to the radius (a line segment which goes from the center of a circle to the outside edge of the circle) of the top of the cylinder, h refers to the height of the cylinder, and Π refers to the value of pi, typically represented as 3.14. Look at the example below for further clarification:

Consider cylinder X where the height is 7 inches and the radius of the top is 3 inches. Applying the formula you would have the following equation:

Surface area of cylinder X= 2 Π 32 + 2 Π 3(7)

Surface area of cylinder X= 2 Π (9) + 2 Π (21)

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Surface area of cylinder X= 2 (28.26) + 2 (65.94)

Surface area of cylinder X= 56.52 + 131.88

Surface area of cylinder X = 188.40

The volume of a three-dimensional polyhedron is the amount of cubes which you could fit inside the shape. Volume is measured in cubic inches. In the case of a rectangular solid such as a die, volume can be found by multiplying the length, by the width, by the height of a shape. Considering the cube from the previous example with a side length of 6 inches, the equation to find volume would be:

6 inches * 6 inches * 6 inches = 216 cubic inches

As with finding the area of a two-dimensional square, since a cube is created using only regular squares it can be found by taking one side to the third power (length3). When raising a number to the third power, it is the same as multiplying the number by itself three times (7*7*7 = 73).

The volume of a cylinder can be found by squaring the product of the height and radius. Consider that a cylinder, if taken apart, consists of a rectangle and two circles. It may be hard to imagine the rectangle portion of a cylinder, however, consider the outside of the cylinder. If you cut of the top circle and bottom circle and unrolled the part left, it would make a rectangle. In other words to find the volume of a cylinder, you would use the following formula:

Volume = Π * radius2* height

In the example of the cylinder above, where the height was 7 inches and the radius was 3 inches, the volume could be found in the following manner:

Volume = Π * 3 inches2 * 7 inches

Volume = Π * 9 square inches* 7 inches

Volume = Π * 63 cubic inches

Volume = 197.82 cubic inches

To find the surface area of the same cylinder, a different formula is applied. In this case, it is necessary to first find the surface area of the top and bottom circles, using the two-dimensional formula for determining the area of a circle. In order to find the area of the rectangle which forms the area between the two circles, the length of the rectangle needs to be determined. As, it can not be simply measured due to the bends and curves, the circumference is found. Circumference allows you to find the length of the rectangle. The formula for finding the circumference is:

Circumference = Π * Diameter of one of the circles

In this formula, diameter refers to the distance across the entire circle. To complete the surface area calculation, it would be necessary to add together the surface areas of the rectangle and the two circles.