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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Perimeters and Areas of Similar Figures

Master Geometry with Hands-On Similar Figures Problems

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting the Scaling Shapes Showdown, a geometry quiz for middle school students.

If a shape is scaled by a factor of 3, what happens to its perimeter?
It is increased by 3 units
It is multiplied by 9
It remains unchanged
It is multiplied by 3
Scaling a shape by a factor of 3 means every linear measurement, such as the perimeter, is multiplied by 3. This direct multiplication applies only to linear dimensions, unlike areas that multiply by the square of the factor.
A shape has an area of 10 cm². If it is enlarged with a scale factor of 2, what is its new area?
25 cm²
20 cm²
40 cm²
30 cm²
When scaling a shape, the area increases by the square of the scale factor. With a scale factor of 2, the area becomes 2² times the original, which is 4 x 10 = 40 cm².
Two similar figures have corresponding side lengths in the ratio 1:2. What is the ratio of their perimeters?
1:2
1:4
2:1
1:8
Since the perimeters of similar figures are directly proportional to the side lengths, a ratio of 1:2 for their sides implies a ratio of 1:2 for their perimeters. This direct relation holds true in linear measurements.
A square has a side length of 5 cm. If the square is scaled so that its side length is doubled, what is its new perimeter?
20 cm
10 cm
30 cm
40 cm
Doubling the side length of a square doubles its perimeter. The original perimeter is 4 x 5 = 20 cm, so the new perimeter becomes 4 x 10 = 40 cm.
A triangle is scaled by a factor of 4. How is its area affected?
It is multiplied by 4
It is multiplied by 8
It is multiplied by 16
It increases by 4 square units
When a shape is scaled by a factor, its area increases by the square of that factor. So, a scaling factor of 4 means the area is multiplied by 4², which equals 16.
In similar figures, if the ratio of the corresponding sides is r, what is the ratio of their areas?
r²
r/2
r
2r
The area of similar figures scales with the square of the scale factor, so if the side lengths are in a ratio of r, the areas will be in a ratio of r². This property stems from the fact that area is a two-dimensional measurement.
A rectangle has an area of 24 cm² and a perimeter of 20 cm. If a similar rectangle is scaled by a factor of 1/2, what is its new area?
6 cm²
8 cm²
12 cm²
24 cm²
Scaling by a factor of 1/2 changes the area by (1/2)² = 1/4. So, the new area is 24 divided by 4, which is 6 cm². This demonstrates the quadratic effect of scaling on area.
If a circle's radius is tripled, by what factor does its area increase?
12 times
3 times
6 times
9 times
The area of a circle is proportional to the square of its radius. Tripling the radius increases the area by 3², which is 9 times. This is a standard result in scaling circular areas.
Two similar hexagons have a scale factor of 5. What is the ratio of their areas?
5:1
125:1
10:1
25:1
The ratio of areas for similar figures is the square of the scale factor. Given a scale factor of 5, the area ratio is 5²:1, which simplifies to 25:1. This outcome is consistent with the principle of area scaling.
A shape has a perimeter of 30 units. When scaled by a factor of 3, what is its new perimeter?
30 units
90 units
120 units
60 units
When a shape's linear dimensions are scaled by a factor of 3, the perimeter is also multiplied by 3. Therefore, the new perimeter is 3 x 30 = 90 units.
If two similar figures have areas in the ratio 16:25, what is the ratio of their corresponding side lengths?
2:3
1:1
4:5
16:25
The side lengths scale by the square root of the ratio of areas. Since sqrt(16)=4 and sqrt(25)=5, the ratio of corresponding side lengths is 4:5. This is an essential property of similar figures.
Consider two similar parallelograms where the longer base of the first is 8 cm and the corresponding base of the second is 12 cm. What is the scale factor from the first to the second?
0.67
4
2
1.5
The scale factor is determined by dividing the corresponding side lengths. Here, 12 cm ÷ 8 cm equals 1.5. This constant multiplier applies to all dimensions in similar figures.
A square has an area of 49 cm². If a similar square is scaled by a factor of 3, what is its new area?
441 cm²
343 cm²
147 cm²
98 cm²
When a square is enlarged by a scale factor, its area increases by the square of that factor. Therefore, the new area is 49 × 3² = 49 × 9 = 441 cm².
A rectangle with dimensions 3 cm by 7 cm is enlarged with a scale factor of 2. What are the new dimensions of the rectangle?
6 cm by 14 cm
7 cm by 21 cm
5 cm by 10 cm
3 cm by 7 cm
Multiplying the original dimensions by the scale factor gives the new dimensions. Hence, 3 cm becomes 6 cm and 7 cm becomes 14 cm, resulting in dimensions of 6 cm by 14 cm.
Two similar rectangles have perimeters in the ratio 2:3. What is the ratio of their areas?
2:3
4:9
3:2
1:2
Since the perimeter ratio reflects the scale factor for linear dimensions, the side lengths are in a ratio of 2:3. Consequently, the areas, being two-dimensional, will be in the ratio of 2²:3², which simplifies to 4:9.
You have two similar triangles. The smaller triangle has sides 3, 4, 5 and an area of 6 cm². If the larger triangle's shortest side is 9 cm, what is its area?
54 cm²
24 cm²
18 cm²
36 cm²
The scale factor is determined by the ratio of corresponding sides, which is 9/3 = 3. Since the area scales by the square of the scale factor, the new area is 6 × 3² = 54 cm². This demonstrates the quadratic effect of scaling on area.
A trapezoid and its scaled version are similar. The original trapezoid has bases of 5 cm and 9 cm with an area of 35 cm². If the scale factor is 2, what is the area of the larger trapezoid?
105 cm²
175 cm²
140 cm²
70 cm²
Scaling a shape by a factor of 2 increases the area by 2², which is 4. Multiplying the original area by 4 gives an area of 35 × 4 = 140 cm². This calculation illustrates the squared relationship between linear scaling and area.
Given two similar circles, if one circle's circumference is 10π and the other's circumference is 15π, what is the ratio of their areas?
4:9
3:2
9:4
2:3
The ratio of the circumferences is 10π:15π = 2:3, so the scale factor for the radius is 2:3. Since area depends on the square of the radius, the ratio of areas is (2²):(3²), or 4:9. This highlights the squaring effect in area calculations.
A regular pentagon has a perimeter of 25 units. If each side is increased by 40%, what is the new perimeter?
45 units
40 units
35 units
30 units
An increase of 40% means each side becomes 1.4 times its original length, so the perimeter is also multiplied by 1.4. Thus, 25 × 1.4 = 35 units is the new perimeter.
The sides of a square are increased by 150%. What is the percent increase in the area of the square?
525%
300%
225%
150%
Increasing the sides by 150% implies that the new side length is 250% of the original (or a factor of 2.5). The area scales by the square of this factor, resulting in 2.5² = 6.25 times the original area. This represents a 525% increase (625% - 100% = 525%), showcasing the exponential impact on area.
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Study Outcomes

  1. Apply geometric scaling principles to compute perimeters of similar figures.
  2. Determine how scale factors affect area calculations in geometric shapes.
  3. Analyze the relationship between dimensions and scaling in similar figures.
  4. Synthesize methods for solving problems involving scaled perimeters and areas.

Perimeters & Areas of Similar Figures Cheat Sheet

  1. Understanding Similar Figures - Imagine being a shape detective - similar figures are like twins with the same angles and proportional sides. Mastering this concept helps you spot patterns and confidently solve scaling problems in geometry. Read more about Similar Figures
  2. Defining the Scale Factor - The scale factor is simply the ratio of corresponding side lengths between two similar figures. If one side is twice as long as its counterpart, you know the scale factor is 2, making calculations a breeze. Explore Scale Factors
  3. Perimeter Proportionality - Perimeters of similar figures grow or shrink exactly with the scale factor, so a scale factor of 3 means the big figure's perimeter is three times the small one's. This linear relationship keeps your perimeter calculations straightforward. CliffsNotes on Perimeters
  4. Area and the Square of the Scale Factor - Areas of similar shapes expand or contract with the square of the scale factor, so doubling a figure makes its area four times bigger. Knowing this squared relationship prevents nasty surprises in your area answers. MathBits on Areas
  5. Calculating Area Ratios - To find the area ratio, just square the scale factor: a factor of 3 gives you 9, a factor of 4 gives 16, and so on. This shortcut is a lifesaver when you need quick results on exams. Per Area Similar Figures
  6. Spotting the Key Difference - Remember the magic words: "Perimeter Proportional, Area Squared." Perimeters scale linearly, while areas go by the square - this mantra will stick with you during crunch time. Quick Guide to Perimeters & Areas
  7. Scaling Up Figures - When you want a bigger version, multiply every side length by the scale factor to get your new dimensions. This keeps every angle intact and ensures the shape stays true to the original. Scale Up Methods
  8. Scaling Down Figures - Shrinking a shape is just as simple - divide each side by the scale factor to maintain perfect proportions. It's like creating a miniature without losing any of the original's flair. Scale Down Techniques
  9. Real‑World Practice - Grab a map or build a model to see scaling in action - real examples make abstract ratios clear and boost your confidence. Hands‑on practice is the fastest way to seal your understanding. Geogebra Practice
  10. Using Mnemonics - Employ catchy tricks like "Perimeter Proportional, Area Squared" to lock in the difference between linear and squared relationships. A solid mnemonic can turn tricky formulas into instant recall gold. Mnemonic on Perimeter & Area
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