Measurements - Scaling
- Scaling
- Scaling Factor
- Types of Scaling
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Scaling
Scaling is used to measure the size of an object or to enhance the visibility of an object. When we enlarge or shrink a figure such that the figure retains its original shape, we say that we have scaled the image.
Look at the given images:
What did you observe from the above images?
In image (a), size of the object has been decreased (reduced) and in image (b), the size of the object has been increased or enlarged. After decreasing or increasing the objects still look similar, which means we can call them similar objects with different dimensions. This process is called scaling.
Scaling Factor
The scaling factor is defined as the ratio between the size (dimensions) of the new object and the original object. We use a colon (`:`) to denote the scaling factor. For example `5:2`.
We can calculate the scale factor if the size of an object is increased or decreased.
Here is the formula to calculate the scale factor of an object which has been increased or decreased.
\[\text{Scale factor} = \frac{\text{New dimension of the object}}{\text{Original dimension of the object}}\]
Types of Scaling
There are two types of scaling, which are as follows:
- Scale up
- Scale down
Let’s discuss them one by one.
Scale-up: If you want to increase the size of an object, then we need to scale-up. To scale up the size of the object we multiply the dimension of the object by any number greater than `1`. The scale-up factor can be calculated by dividing the increased dimension by the original dimension of an object.
\[\text{Scale-up} =\frac{\text{Increased dimension of the object}}{\text{Original dimension of the object}}\]
For example:
Scale factor `= 5:10 = 1:2`
Scale down: When we reduce the size of an object, we scale down that object.
\[\text{Scale-down} = \frac{\text{Decreased dimension of the object}}{\text{Original dimension of the object}}\]
For example: The original image of a pentagon has been reduced or decreased into a new image of a pentagon.
Scale factor `= 7:3`
Solved Examples
Example `1`: Find the scale factor of the two similar images given below.
Solution:
`\text{Scale factor} = \frac{\text{New dimension of the base}}{\text{Original dimension of the base}}`
Scale factor `= 18/6 = 3/1` or `3:1`
Therefore, the new size of the object is three times of the original object.
Example `2`. What is the length of the new object that has been scaled up, if the length of the original object was `15` units and the scale factor is `2`?
Solution:
Given: Length of the original object `= 15` units
Scalar factor `= 2`
Now we need to find the length of the new object.
Let’s apply scale factor formula:
\(\begin{align*}
\text{Scale factor} & = \frac{\text{New length of the object}}{\text{Original length of the object}} \\
2 & = \frac{\text{New length of the object}}{15} \\
2 \times 15 & = \text{New length of the object} \\
30 & = \text{New length of the object}
\end{align*}\)
Therefore, the length of the new object `= 30` units.
Example `3`. What is the scalar factor of the circle whose radius has increased as shown below?
Solution: Given, the radius of the original circle `= 3` cm
The radius of the new circle `= 7` cm
Note: the radius (size) has been increased which means the circle has been scaled up.
Formula: `\text{Scale-up} = \frac{\text{Increased radius of the circle}}{\text{Original radius of the circle}}`
Scale factor `= 7/3` or `7:3`
Example `4`. What is the height of the enlarged cylinder if the original height of the cylinder is `125` cm and the scale factor is `5:7`?
Solution:
Given, scale factor `= 5/7` and the original height of the cylinder is `125` cm.
\(\begin{align*}
\frac{5}{7} & = \frac{\text{Original height of the cylinder}}{\text{Height of the enlarged cylinder}} \\
\frac{5}{7} \times \text{Height of the enlarged cylinder} & = \text{Original height of the cylinder} \\
\text{Height of the enlarged cylinder} & = \text{Original height of the cylinder} \times \frac{7}{5} \\
\text{Height of the enlarged cylinder} & = 125 \times \frac{7}{5} = 25 \times 7 = 175
\end{align*}\)
Therefore, the height of the enlarged cylinder is `175` cm.
Practice Problems
Q`1`. Find the scale factor of the given geometrical figures.
- `1:3`
- `1:4`
- `2:4`
- `3:1`
Answer: b
Q`2`. What is the reduced length of the given figure rounded to the nearest whole number?
- `6` units
- `15` units
- `18` units
- `20` units
Answer: a
Q`3`. What is the new base of a triangle whose original base is `4` units and whose scale factor is `4:9`?
- `9` units
- `8` units
- `4` units
- `36` units
Answer: a
Q`4`. If the scalar factor is `1`. Which of the following options is true?
- The actual size of the object is more than the new size of the object
- The actual size of the object is less than the new size of the object
- The actual size has not changed.
- The actual size of the object is more than `2`.
Answer: c
Frequently Asked Questions
Q`1`. What is scaling in geometry?
Answer: In geometry, scaling refers to the transformation of a shape, object, or figure by changing its size, either enlarging or reducing, while preserving its shape and proportions.
Q`2`. How is the scale factor determined in a scaling process?
Answer: The scale factor is determined by comparing the corresponding measurements (lengths, areas, volumes, etc.) of the original and scaled objects. It is the ratio of the new dimension to the original dimension.
Q`3`. What are the practical applications of scaling in real life?
Answer: Scaling is commonly used in map-making, architectural design, model building, and graphics. It is also applied in resizing images, adjusting dimensions in construction, and simulating proportions in various fields.
Q`4`. How does scaling affect different geometric properties?
Answer: Scaling affects properties such as perimeter, area, and volume. For example, if an object is scaled by a factor of `2`, its perimeter will be doubled, and its area will be quadrupled.
Q`5`. Can scaling result in negative scale factors?
Answer: No, scaling typically involves positive scale factors. The scale factor represents the ratio of the new size to the original size, and it is conventionally expressed as a positive value. Negative scaling would imply a transformation that includes a reflection or inversion, which is distinct from a simple change in size.