Linking Area and Similarity
Solids with similar shapes and proportional corresponding dimensions are referred to as similar solids.
When two solids are similar, the square of the ratio of their linear measures is equal to the ratio of their surface areas.
List of Surface Area of Similar Solids Formulas
The list above gives us the formulas that are applied to calculate the surface area of similar solids. These formulas will be elaborated upon further as we move further.
The surface area of similar solids can be calculated using the following formula:
Surface area of similar solids = \(\frac{\text{Surface Area of solid A}}{\text{Surface Area of solid B}}\)
Surface Area of Similar Solids formula
To calculate the surface area of similar solids, we need the radius, height and surface area of the solids.
Mathematically it can be represented as:
Surface area of similar solids = \(\frac{\text{Surface Area of solid A}}{\text{Surface Area of solid B}}\)
Solved Examples
Example 1: Is there a similarity between the two rectangular prisms shown below? If yes, how can we figure this out?
Solution:
If the heights, widths, and lengths of the two prisms are all in proportion, they are similar.
\(\frac{\text{small prism}}{\text{large prism}}=\frac{2}{4}=\frac{2~\div~2}{4~\div~2}=\frac{1}{2}\) simplify
\(\frac{\text{small prism}}{\text{large prism}}=\frac{4}{8}=\frac{4~\div~4}{8~\div~4}=\frac{1}{2}\) simplify
\(\frac{\text{small prism}}{\text{large prism}}=\frac{5}{10}=\frac{5~\div~5}{10~\div~5}=\frac{1}{2}\) simplify
All three corresponding dimensions have the same ratio of \(\frac{1}{2}\). As they all have common ratios the two rectangular prisms are proportional.
Example 2: The following figures are similar. What is the numerical value of x?
Solution:
\(\frac{\text{Height of smaller triangular prism}}{\text{Height of larger triangular prism}}=\frac{\text{Length of smaller triangular prism}}{\text{Length of smaller triangular prism}}\)
\(\frac{4}{8}=\frac{6}{x}\) Substitute
\(4 \times x=6 \times 8\) Cross Multiply
\(4x = 48\) Simplify
\(4x~\div~4=48~\div~4\) Divide both sides by 4
\(x=12\)
The length that is missing is 12 centimeters.
Example 3: These two cylinders are similar in shape. If the smaller cylinder has a volume of 50 cubic feet, what is the volume of the larger cylinder?
Solution:
\(\frac{\text{Volume of the larger cylinder}}{\text{Volume of the smaller cylinder}}=\frac{\text{(Radius of the larger cylinder)}^3}{\text{(Radius of the larger cylinder)}^3}\)
\(\frac{V}{50}=\frac{3^3}{2^3}\) Substitute
\(\frac{V}{50}=\frac{27}{8}\) Evaluate
\(\frac{V}{50}\times 50=\frac{27}{8}\times 50\) Multiply each side by 50 and simplify
\(V=\frac{1350}{8}\)
\(V=168.75\)
Hence, the volume of the larger cylinder is 168.75 cubic feet.
Example 4: The volume of the cylindrical pitcher below is approximately 90 cubic inches. The pitcher and the creamer are made of similar materials. Calculate the creamer’s volume.
Solution:
\(\frac{\text{ Volume of the creamer}}{\text{Volume of the pitcher}}=\left ( \frac{\text{Diameter of the creamer}}{\text{Diameter of the pitcher}} \right )^3\)
\(\frac{V}{90}=\left ( \frac{3}{6} \right )^3\) Substitute
\(\frac{V}{90}=\left ( \frac{27}{216} \right )\) Evaluate
\(V=11.25\) Multiply each side by 90 and simplify
Hence, the volume of the creamer is 11.25 cubic inches.
Example 5: The following pyramids are similar and the larger pyramid covers 200 square centimeters in surface area. What is the smaller pyramid’s surface area?
Solution:
\(\frac{\text{Surface area of the smaller pyramid}}{\text{Surface area of the larger pyramid}}=\left ( \frac{\text{Base of the smaller pyramid}}{\text{Base of the larger pyramid}} \right )^2\)
\(\frac{S}{200}=\left ( \frac{10}{14} \right )^2\) Substitute
\(\frac{S}{200}=\frac{100}{196} \) Evaluate
\(S=\) 102.04 Multiply each side by 200 and simplify
Hence, the surface area of smaller pyramid is 102.04 square centimeters.
The square of the ratio of their corresponding linear measures is equal to the ratio of their surface areas when the two solids are similar.
Similar solids are those that have the same shape but not the same size, implying proportional segments and similar polygons for corresponding faces.
The sum of the areas of all faces (or surfaces) of a 3D shape is known as surface area. For instance, there are six rectangular faces on a cuboid. Add the areas of all six faces to get the surface area of a cuboid. We can also label the prism’s length (l), width (w), and height (h) and calculate the surface area using the following formula: SA = 2lw + 2lh + 2hw.
When two solids are similar their respective dimensions are proportional. The ratio of the dimensions, when squared, is equal to the ratio of their surface areas. Note that the ratios do not show the actual surface areas as they are simplified and common multiples are cancelled off. The ratio of the dimensions, when cubed, is equal to the ratio of volumes of the two solids.
The total surface area of a solid combination is equal to the sum of the total surface areas of each individual solid, excluding the overlapping part.
A solid object’s surface area is a measurement of the total area occupied by the object’s surface.