What is the area of a right angle triangle when its perimeter is 18 cm?

Answer 1

It can have any area greater than 0 cm2 and less than or equal to 81(3-√8) ≈ 13.897 cm2.

Without further information that provides the length of (at least) one side it is impossible to say which of these areas the triangle has.

For example:

base = 9 cm, hypotenuse = 9 cm

⇒ height = 18 - 9 - 9 cm = 0 cm

⇒ area = 9x0/2 = 0 cm2

This triangle does not actually exist and represents the minimum of the area.

base = 6 cm, hypotenuse = 7.5 cm

⇒ height = 18 - 6 - 7.5 cm = 4.5 cm

⇒ area = 6x4.5/2 = 12.5 cm2

Note that base2 + height2 = 36 + 20.25 = 56.25 = 7.52 = hypotenuse2 and so this is a right angle triangle.

base = 9(2-√2) cm ≈ 5.272 cm, hypotenuse = 18(√2 -1) cm ≈ 7.456 cm

⇒ height = 18 - 9(2-√2) - 18(√2 -1) cm = 9(2-√2) cm ≈ 5.272 cm

⇒ area = 9(2-√2) x 9(2-√2)/2 = 81(3-√8) cm2 ≈ 13.897 cm2

This triangle is a right angle triangle (check out Pythagoras on the lengths) and is the right angle triangle of largest area with perimeter 18 cm.