About Area and Perimeter of Similar Figures
What are Similar Figures?
When two figures have the same shape but differ in size, they are said to be similar figures. In other words, two figures are said to be similar if they share many of the same characteristics but are not identical.
What is the Scale Factor?
The scale factor is the constant by which the size of a shape is increased or decreased. It is the ratio of the corresponding sides of the image created by the change in the original shape.
What is the Perimeter?
The path or boundary that surrounds a shape can be defined as the perimeter in geometry. The length of a shape’s outline is another way to define it.
The Relation between the Perimeter of the sides of Similar Figures
The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding side lengths.
\(\frac{\text {Perimeter of the}~\Delta PQR} {\text{Perimeter of the}~\Delta STU}=\frac{PQ}{ST}=\frac{QR}{TU}=\frac{PR}{SU}\)
Example:
Find the value of the ratio (red and blue) of the perimeters of similar figures.
Solution:
\(\frac{\text {Perimeter of the}~\Delta ~PQR} {\text{Perimeter of the}~\Delta~ STU}~=~\frac{PQ}{ST}~=~\frac{3}{5}\)
Therefore, the perimeter ratio is \(\frac{3}{5}\).
What do we Mean by Area?
In geometry, the area of a flat shape or an object’s surface can be defined as the space occupied by it. The number of unit squares that cover a closed figure’s surface is known as its area. The area of a shape is measured in square centimeters, square feet, square inches, and so on.
The Relation between the Area of Similar Figures
When two figures are similar, the square of the ratio of their corresponding side lengths equals the ratio of their area.
When the ratio of two corresponding sides (or other lengths) is expressed as \(\frac{a}{b}\), in similar figures, the ratio of the areas is expressed as \(\frac{a^2}{b^2}\)
\(\frac{\text{Area of the}~\Delta PQR} {\text{Area of the}~\Delta STU}~=~\left ( \frac{PQ}{ST}\right)^2~=~\left ( \frac{QR}{TU}\right)^2~=~\left ( \frac{PR}{SU}\right)^2\)
Example:
Calculate the ratio of the area (red and blue) of similar figures.
Solution:
\(\frac{\text{Area of the red triangle}} {\text{Area of the blue triangle}}~=~\left(\frac{5}{10}\right)^2~=~\left(\frac{1}{2}\right)^2\)
= \(\frac{1}{4}\)
Hence, the ratio of the area is
=\(\frac{1}{4}\).
Solved Examples on Area and Perimeter of Similar Figures
Example 1:
Find the ratio (yellow and pink) of the perimeters of the circles.
Solution:
All circles are similar. The perimeter of a circle is its circumference. Here, we have the diameter of each circle. So, the circumference is \(\pi d\).
\(\frac{\text{Perimeter of the yellow circle}} {\text{Perimeter of the pink circle}}~=~\frac{\pi ~\times~ 2}{\pi ~\times~ 4}~=~\frac{1}{2}\)
Therefore, the ratio of the perimeter is \(\frac{1}{2}\).
Example 2:
Calculate the ratio of the area (green to blue) of the similar figures.
Solution:
\(\frac{\text{Area of the green square}} {\text{Area of the blue square}}~=~\left(\frac{2}{3}\right)^2\)
\(~=~\frac{4}{9}\)
Hence, the ratio of the area is \(~=~\frac{4}{9}\).
Example 3:
The shape of a keyboard is similar to that of a rectangular piece of land pitch. Calculate the keyboard’s perimeter (P) and area (A).
Solution:
The rectangular piece of land and the rectangular keyboard are similar. To calculate the perimeter and area of the keyboard, use the ratio of corresponding side lengths to write and solve proportions.
Perimeter:
\(\frac{\text{Perimeter of the rectangular piece of land}} {\text{Perimeter of the keyboard}}~=~\frac{\text{Width of the rectangular piece of land}} {\text{Width of the keyboard}}\)
\(\frac{90}{P}~=~\frac{15}{10}\)
900 = 15 P
P = 60
Area:
\(\frac{\text{Perimeter of the rectangular piece of land}} {\text{Perimeter of the keyboard}}~=~\left(\frac{\text{Width of the rectangular piece of land}}{\text{Width of the keyboard}}\right)^2\)
\(\frac{200}{A}~=~\left(\frac{10}{15}\right)^2\)
\(\frac{200}{A}~=~\frac{100}{225}\)
200 × 225 = 100 A
\(\frac{200~\times~ 225}{100}~=~A\)
A = 450
So, the perimeter of the keyboard is 60 centimeters, and the area of the keyboard is 450 square centimeters.
Example 4:
A photograph is placed on a page of a photo album. The picture and the page are both rectangles.
What is the area of the picture as compared to the size of the page?
Solution:
\(\frac{\text{Area of the page}}{\text{Area of the picture}}~=~\left(\frac{\text{length of the page}}{\text{length of the picture}}\right)^2\)
\(~=~\left(\frac{8}{6}\right)^2~=~\left(\frac{4}{3}\right)^2=\frac{16}{9}\)
The area of the page is \(\frac{16}{9}\) times that of the image.
Instances of finding the area and perimeter often occur in everyday life. For example, determining the area of the floor of a house, the area of a footpath that will surround the ground, and the perimeter of a park fenced with wire, among other things.
The perimeter of a scaled object can be determined by multiplying the scale factor by its original perimeter. For example, if the scale factor is three, the new object’s perimeter will be three times the original. Similarly, the area of a scaled object is equal to the square of its scale factor.
Yes, all squares are similar. Although the size of each square may not be the same or equal, the ratios of their corresponding sides are the same and all their angles are equal.