# When the length of the three sides of a triangle are increased by a factor n, by what factor is the area of the triangle increased.

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The lengths of the sides of a triangle are increased by a factor of n. What is the factor of increase of the area of the new triangle?

Note that the transformed triangle is similar to the original with a scale factor of n.

Then by Galileo's square-cube law, the ratio of the areas is the square of the scale factor.

The scale factor is 1:n, thus the area is in a ratio of `1^2:n^2 ` .

**The new area is increased by a factor of `n^2 ` .**

(If we had increased in three dimensions by a factor of n, the volume of the transformed figure would be increased from the original by a factor of `n^3 ` )

**Sources:**

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My teacher says that increase by a factor of n means that the final value is n times greater than the original. Eg. 10 increased by 10% makes it 11. The length of each of the new sides is therefore 1 + n times the original. Is she wrong in saying that.

Another way to think about by what factor the area of a triangle will change by when increasing the length of each side by a factor of n, is to think about the two resulting triangles being *similar* triangles.

Because they are similar, the resulting area will increase by a factor of `n^2` . I'm not sure what you are studying right now, but if you are studying similarities, congruency and other concepts like that, this might be the route they are looking for.

If you draw a square or right triangle, and increase the length of each side by a factor of *n*, it is easy to draw in the number of original copies needed to fill in the new scaled figure.

If you increase the length of each side of a square by a factor of 2, you can fit 4 of your original squares into the new scaled square. Since a right triangle is just half the area of a rectangle, you can see that this relationship holds up as well.

Therefore if you increase the length of each side by a factor of *n*, the area will increase by a factor of `n^2 ` .

Let the length of the sides of the original triangle be a, b and c. When the length of the three is increased by a factor n each of them is now (1 + n) times the original length. The length of the sides of the new triangle are (1+n)*a, (1+n)*b and (1+n)*c

The area of a triangle with sides a, b and c is given by the formula `A = sqrt(S*(S - a)*(S - b)*(S - c))` where S is the semi-perimeter `S = (a + b + c)/2`

Using this formula, the area of the original triangle is `A = sqrt(S*(S - a)*(S - b)*(S - c))` where `S = (a + b + c)/2` . The semi-perimeter of the enlarged triangle is `S' = ((1+n)*a + (1 +n)*b + (1 + n)*c)/2 = (1 + n)*(a + b + c)/2`

The area of the triangle is `A' = sqrt(S'*(S' - (1+n)*a)*(S' - (1+n)*b)*(S' - (1+n)*c))`

= `sqrt(((1 + n)*(a + b + c))/2*(((1 + n)*(a + b + c))/2 - (1+n)*a)*(((1 + n)*(a + b + c))/2 - (1+n)*b)*(((1 + n)*(a + b + c))/2 - (1+n)*c))`

= `sqrt((1 + n)^4*(a + b + c)/2*((a + b + c)/2 - a)*((a + b + c)/2 - b)*((a + b + c)/2 - c))`

= `(1 + n)^2*sqrt((a + b + c)/2*((a + b + c)/2 - a)*((a + b + c)/2 - b)*((a + b + c)/2 - c))`

= `(1 + n)^2*A`

If each of the sides of a triangle is increased by a factor n, the area of the triangle is increased to `(1 + n)^2` the original area.

There appears to be some confusion on terminology.

(1) To increase by a factor of n means to multiply each length by n. Thus if n=2, then each length is doubled. If n=10, each length is multiplied by 10. If n=.1, then each length is multiplied by .1. Here n is a dilation factor.

(2) When talking about exponential growth, the growth rate is r, while the growth factor is (1+r). So if the growth factor is 1.1, the growth rate is r=.1.

Your original question gives a change in the length by a factor of n. This is a dilation of factor n. You would multiply each side length by the given n.

If however, you are asked to increase each side by n%, then you would multiply each side by (1+n).

Even though an example cannot prove the statement, we can look at an example to see what is happening.

(a) If we accept n as a dilation factor: Let the triangle be equilateral with side length 10. Then if n is 1.1 then each side of the transformed triangle is 11. The area of the original triangle is `25 sqrt(3) ` . The area of the transformed triangle is `(121 sqrt(3))/4 ` . Note thatĀ `(25 sqrt(3))*(121/100)=(121 sqrt(3))/4 ` where `n^2=1.1^2=(121)/100 ` . So the area increased by a factor of `n^2 ` .

(b) If we have n as a percent increase: Let the triangle be equilateral with side length 10 and an increase of 10%. Then each side of the transformed triangle is 10(1+.1)=11. These are the same triangles as above, so the increase in area is `1.1^2=(1+.1)^2 ` . This would work for any n.

As the original question is worded, I would choose (a).

For a specific example, I would use (a) if n is a dilation factor and (b) if n is a % increase. I.e. if the instructions say to double each side I use n=2 in (a), even though I could use n=1 in (b). If the instructions say to increase each side by 5% I choose (b) with n=.05, even though I could use (a) with n=1.05.

The example you give from your teacher only works in specific cases. If n is 10% but the side length is 5, then each side is increased to 5.5 which is not 1+n. The first sentence is correct: My teacher says that increase by a factor of n means that the final value is n times greater than the original. This means you take n times each side.

The increase will be n^2 according to Galileo's law.

I feel like the easiest way to think of this is drawing a picture and using the area formula. The area of a triangle is `A=(1)/(2)(bh)`. The b stands for base and h stands for height. So, draw yourself a triangle and label the horizontal leg b and the vertical leg h.

Now to show you are multiplying both sides by n you would simple put b times n and h times n in the formula. Now you have `A=(1)/(2)(b*n*h*n)`. You can now combine like terms giving you `A=(1)/(2)(b*h*n^(2))`. Your increase would just beĀ `n^(2)`.

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I was just wondering if it would be more complete if you showed through the pythagorean theorem that the hypotenuse would also increase by a factor of `n `? The problem was concerned about if all three sides increased by a factor of ` n`, so there might be a need to show that in this case (a right triangle), that all three sides would have increased by a factor of `n ` .