- #1
IniquiTrance
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Homework Statement
If I have similar pyramids each with similar triangular bases, how can I prove that the ratio of the area of the bases = the ratio of the heights of the pyramids squared?
Thanks!
Proving Ratio of Areas/Heights of Similar Pyramids
- Thread starter IniquiTrance
- Start date
In summary, the conversation discusses how to prove that the ratio of the area of similar pyramids' bases is equal to the ratio of their heights squared. It is mentioned that if the pyramids are similar, the ratio of their heights is the same as the ratio of the sides of the base triangle. It is then suggested to consider how the area changes when all sides of the triangle are scaled by a constant ratio. It is then understood that the scaling factor is equal to the ratio of the sides of the bases, which is also equal to the ratio of the heights. Thus, when squared, this ratio is equal to the ratio of the areas.
If I have similar pyramids each with similar triangular bases, how can I prove that the ratio of the area of the bases = the ratio of the heights of the pyramids squared?
Thanks!
- #2
Dick
Science Advisor
Homework Helper
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If the pyramids are similar then the ratio of the heights is the same as the ratio of the sides of the base triangle. If you change all of the sides of the triangle by a constant ratio, how does the area change?
- #3
IniquiTrance
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I'm not sure how to proceed... 
- #4
IniquiTrance
- 190
- 0
I see that if the scaling factor is k, then the ratio of the areas is k2. But how does that relate to the height of the pyramids?
- #5
IniquiTrance
- 190
- 0
Hmm, I think I see it.
k equals the ratio of the sides of the bases which = the ratio of the heights. Squaring this equals the ratio of the areas.
Thanks!
k equals the ratio of the sides of the bases which = the ratio of the heights. Squaring this equals the ratio of the areas.
Thanks!
1. How do you prove the ratio of areas/heights of similar pyramids?
The ratio of areas/heights of similar pyramids can be proved by using the formula A₁/A₂ = (h₁/h₂)², where A₁ and A₂ are the areas of the bases and h₁ and h₂ are the heights of the pyramids. This formula is based on the fact that similar pyramids have proportional dimensions.
2. What is the importance of proving the ratio of areas/heights of similar pyramids?
Proving the ratio of areas/heights of similar pyramids is important because it helps us understand the relationship between the dimensions of similar pyramids. This knowledge can be applied in various fields such as architecture, engineering, and geometry.
3. Can you provide an example of how to prove the ratio of areas/heights of similar pyramids?
Sure, let's say we have two pyramids with base areas of 25 cm² and 36 cm² and heights of 10 cm and 12 cm respectively. To prove that they are similar, we can use the formula A₁/A₂ = (h₁/h₂)². Substituting the values, we get (25/36) = (10/12)², which simplifies to 5/6 = 5/6. This proves that the pyramids are similar.
4. What other methods can be used to prove the similarity of pyramids?
Apart from using the ratio of areas/heights, we can also prove the similarity of pyramids by comparing their angles and corresponding side lengths. If the angles are equal and the sides are in proportion, then the pyramids are similar.
5. Are there any real-world applications of proving the ratio of areas/heights of similar pyramids?
Yes, proving the ratio of areas/heights of similar pyramids has various real-world applications. For example, in architecture, this knowledge is used to design structures with similar proportions. It is also used in engineering to determine the scale of models and in geometry to calculate unknown dimensions of similar figures.
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