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Q: Two Pythagoras triangles with area equal to perimeter?
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Will the perimeter and the area of a triangle ever equal one another?

Consider any triangle with given angles. If you expand it by a linear scale factor x, then its perimeter is multiplied by x, and its area by x2. When x is big, x2 is bigger than x. The area thus grows relative to the perimeter; as x tends to infinity, the ratio area/perimeter (call it R) tends to infinity. When x is small, x2 is smaller than x. The area thus shrinks relative to the perimeter; as x tends to zero, the ratio R tends to zero. For perimeter to equal area, R must equal 1. Since R is continuously defined, and (as we have just seen) it varies between zero and infinity, there must be some value of x that renders R = 1. This proves that an infinite number of triangles have perimeter equal to area, since our reasoning applied to triangles of any shape. To give one example, we'll find the equilateral triangle with perimeter equal to area. Set the length of a side equal to 2y. area = height x base / 2 = y2sqrt3 perimeter = 6y So, solve 6y = y2sqrt3 6 = ysqrt3 y = 6/sqrt3 = 2sqrt3 One more trivial example: if perimeter equals zero, then it definitely equals area.


Are two triangles with the same area and perimeter always congruent?

I believe so, though I am not sure I can prove it.


What are the properties of isosceles triangles?

All isosceles triangles: - Have angles that add up to 180 degrees - Have two equal sides. The unequal side is called the base. - Have equal base angles. - Have areas and perimeters that can be found using the formulas Area=1/2 X (base X height) and Perimeter=side+side+side An equilateral triangle with a right angle is called a right isosceles triangle. Also, all equilateral triangles are isoceles triangles, but not all isosceles triangles are right triangles.


What is the perimeter of a square that has an area of 81 meters square?

36, area is equal to length X width, perimeter is equal to 2(length)+2(width) in a square length is equal to width, so we take the square root of the area to find that both are 9, which means the perimeter is 9 X 4, 36.


Is area on a triangle larger than perimeter?

Perimeter is length (units feet, centimeters, etc.) Area is length2 (square feet, square centimeters etc.). But if you want to disregard the units, you can find triangles which perimeter is larger, smaller or even 'equal' to area, depending on scale.Take a 3,4,5 right triangle. The perimeter = 3+4+5= 12 units. Area = 3*4/2 = 6 square units. Now double the sides.Perimeter = 6 + 8+ 10 = 24 units. Area = 6*8/2 = 24 square units (the numbers are equal). Scaling it larger, then the valueof the area (in square units) will be larger than the perimeter value (in straight units).

Related questions

If two triangles have the same perimeter will they have the same area?

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Why does the greatest area equal the greatest perimeter?

Because area is a function of perimeter.


What are four ways to describe Pythagoras?

Four triangles can be arranged in a square. Area of square built upon hypotenuse of right angle is equal to the sum of the area of the squares upon the remaining sides.


What is the area and perimeter of a square if the side equal 10cm?

Perimeter = 40cm. Area = 100cm2


Is area always more or equal to the perimeter in a square?

The area of a square is equal to twice the square's perimeter.


Do triangles with equal bases and heights have a equal area?

Yes.


How do you find perimeter and area of triangles?

Perimeter . . . add up the lengths of all three sides. Area . . . multiply (half the length of the base) by (the height).


How do you find the area of a equilateral triangle if you know the perimeter?

Divide the perimeter by 3 to find the length of each of its 3 equal sides Area = 0.5*side squared*sin(60 degrees) Alternatively use Pythagoras' theorem to find its height then area is: 0.5*base*height


How do you prove that triangles are equal?

The answer depends on what you mean by equal. Equal in area? Congruent?


What is the relation between perimeter and triangle area?

If you double (2 times) the perimeter the area will will be 4 times larger. Therefore the area is proportional to the square of the perimeter or the perimeter is proportional to the square root of area. The relationship as shown above applies only to triangles with similar proportions, that is when you scale up or down any triangle of fixed proportions. Other than that requirement, there is no relationship between perimeter and area of any shape of triangle except that it can be stated that the area will be maximum when the sides are of equal length (sides = 1/3 of perimeter).


Will the perimeter and the area of a triangle ever equal one another?

Consider any triangle with given angles. If you expand it by a linear scale factor x, then its perimeter is multiplied by x, and its area by x2. When x is big, x2 is bigger than x. The area thus grows relative to the perimeter; as x tends to infinity, the ratio area/perimeter (call it R) tends to infinity. When x is small, x2 is smaller than x. The area thus shrinks relative to the perimeter; as x tends to zero, the ratio R tends to zero. For perimeter to equal area, R must equal 1. Since R is continuously defined, and (as we have just seen) it varies between zero and infinity, there must be some value of x that renders R = 1. This proves that an infinite number of triangles have perimeter equal to area, since our reasoning applied to triangles of any shape. To give one example, we'll find the equilateral triangle with perimeter equal to area. Set the length of a side equal to 2y. area = height x base / 2 = y2sqrt3 perimeter = 6y So, solve 6y = y2sqrt3 6 = ysqrt3 y = 6/sqrt3 = 2sqrt3 One more trivial example: if perimeter equals zero, then it definitely equals area.


Who are the famous mathematicians in trigonometry?

Pythagoras was the most pivotal mathematician in the area of trigonometry. His pythagoras theorem literally redefined the way people studied right angled triangles.

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