Tiangles and volume can you help?

Question

If there are three perfect triangles, and they get smaller by size at the increment of forty percent per triangle, what is the size of the inner triangle, if the outer triangle has 4 inch sides, and this is what can help me most the volumes of area between large triangle and middle, large and small, middle and small, and the volume of the small triangle. no not home work but I am curious as to the formula.

sorry about the typographical error.

Answer 1

I interpret this as follows:

three perfect triangles → three equilateral triangles

volumes of area → ratio of areas

AND they get smaller by size at the increment of forty percent per triangle linearly

---------------------- --------------------- ---------------------- -------------------

So let outer triangle have sides of length x"

Thus next triangle has sides (100 - 40)% of x" = 0.6x"

And the innermost triangle has sides (100 - 40)% of 0.6x" = 0.36x"

So linear scale factor = 0.6

Area of outer triangle = A = b * h / 2
So area of next triangle = 0.6b * 0.6h /2
= 0.36 * b * h / 2
= 0.36 * A

And area of inner triangle = 0.36 * (0.36 * A)
= 0.1296 *A

So area scale factor = 0.36 = (0.6)² = (linear scale factor)²

Ratio of areas (large : middle :small) = 1 : 0.36 : 0.1296
(= 1 : (0.6)² : [(0.6)²]²)
= 10000 : 3600 : 1296
= 625 : 225 : 81 (in simplest form)

Area of large triangle = Base * height / 2
Base = 4"; height = 2√3" (Cut triangle in half And you get 2 right angled triangles with hypotenuse 4" one side 2" and height to be calculated)
So Area = 4√3 in²

Area of middle triangle = 0.36 * 4√3 in²
= 1.44√3 in²

Thus area of small triangle = 0.1296 * 4√3 in²
= 0.5184√3 in²

So Area between large and small triangle = 4√3 in² - 0.5184√3 in²
= 3.4816√3 in²
Area between large and middle triangles = 4√3 in² - 1.44√3 in²
= 2.56√3 in²
and Area between middle and small triangles = 1.44√3 in² - 0.5184√3in²
= 0.9216√3in²

Answer 2

Equilateral triangle = all 3 sides the same.
In this case for the largest triangle sides are 4 inches
Forty % of 4 is 1.6 so for middle triangle 4 - 1.6 = 2.4 inches
the middle triangle has 3 sides each 2.4 inches long.
Forty % of 2.4 is .96 so for smallest triangle 2.4 - .96 = 1.44inches
Smallest triangle has 3 sides each 1.44 inches long.

Now the formula for area = 1/4*( a^2)*(3^-2)
For the largest triangle (.25 * 16* 1.73) = 6.92
For the middle triangle (.25 * 5.76 * 1.73) = 2.49
For the small triangle (.25 * 2.07 * 1.73) = 0.89

Now just subtract the smaller from the larger out of any two and you get the area of the space between them.

Answer 3

By "perfect triangles", I assume you mean "isosceles triangles". If so, then the area of the first would be:

½ * base * height = ½ * 4 * 2*sqrt(3) = 4 * sqrt(3)

If you take 40% of that and 40% again, you get your answer:

area of the small triangle = 0.64 * sqrt(3)

Answer 4

triangle 1 has 4" sides
triangle 2 has 4*.6" sides=2.4"sides
triangle 3 has 2.4*.6=1.44" sides

Area of an equilateral triangle is s^2*sqrt(3)/4
t1 A=16/4*sqrt(3)4sqrt(3)=6.928in^2
t2 A=5.76/4*sqrt(*3)=1.44sqrt(3)=2.494 in^2
t3 A=1.44^2/4*sqrt(3)=2.0736/4*sqrt(3)=.8979in^2

t3-t2=6.928-2.494=4.434in^2
t3-t1=6.928-.898=6.030in^2
t2-t1=2.494-.898=1.596in^2