A cylinder is a solid figure with straight parallel sides and a circular or oval cross section with a radius (r) and a height (h). The volume of a cylinder is π r²h and the surface area is 2(π r²) + 2π rh.

The perimeter of a triangle is the sum of the lengths of its sides:

p = x + y + z
p = 9cm + 9cm + 9cm = 27cm

Perimeter is equal to the sum of the four sides:

p = a + b + c + d
p = 2 + 9 + 7 + 6
p = 24

The first step to solve a quadratic expression that's not set to zero is to solve the equation so that it is set to zero:

c² - 6c - 35 = -2c - 3
c² - 6c - 35 + 3 = -2c
c² - 6c + 2c - 32 = 0
c² - 4c - 32 = 0

Next, factor the quadratic equation:

c² - 4c - 32 = 0
(c + 4)(c - 8) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (c + 4) or (c - 8) must equal zero:

If (c + 4) = 0, c must equal -4
If (c - 8) = 0, c must equal 8

So the solution is that c = -4 or 8

To find the diagonal we need to know the length of one of the square's sides. We know the area and the area of a square is the length of one side squared:

a = s²

so the length of one side of the square is:

s = \( \sqrt{a} \) = \( \sqrt{64} \) = 8

The Pythagorean theorem defines the square of the hypotenuse (diagonal) of a triangle with a right angle as the sum of the squares of the other two sides:

c² = a² + b²
c² = 8² + 8²
c² = 128
c = \( \sqrt{128} \) = \( \sqrt{64 x 2} \) = \( \sqrt{64} \) \( \sqrt{2} \)
c = 8\( \sqrt{2} \)