Compute the exact volume, area, or length as directed. Square roots do not need to be simplified, but arithmetic does.

A circle's perimeter and area have the same numeric value. What is the diameter of this circle?
\(
2 \pi r = \pi r^2 \\
r = 2 \\
d = 4
\)

A ice cream cone consists of a hemisphere atop the base of a cone. The width is 6 and the height (of cone and hemisphere together) is 7. Find all measurements (length, area, volume) that you can. \( \text{hemisphere (and cone) radius} = 3 \\
\text{cone height} = 4 \\
\text{cone slant height} = 5 \\
\text{surface area} = 18 \pi + 15 \pi = 33 \pi \\
\text{volume} = 18\pi + 12\pi = 30\pi
\)

The radius and height of a cylinder are equal. Further, the surface area and volume are also the same number. What is the radius of this cylinder? \(
2 \pi r h + 2 \pi r^2 = \pi r^2 h; r = h \\
2 \pi r^2 + 2 \pi r^2 = \pi r^3 \\
4 \pi r^2 = \pi r^3 \\
4 r^2 = r^3 \\
4 = r
\)

A shape is made up of an isosceles right triangle and a semicircle. The hypotenuse of this triangle is the diameter of a semicircle. The length of the triangle's legs are 2. What is the area and perimeter of this shape? \(
\text{hypotenuse} = \sqrt{8} = 2 \sqrt{2} \\
\text{radius} = \frac{\sqrt{8}}{2} = \sqrt{2} \\
\text{area} = 2 + \pi \\
\text{perimeter} = 4 + \pi \frac{\sqrt{8}}{2} = 4 + \pi \sqrt{2}
\)

Challenge

The surface area and volume of a cylinder are the same number. Multiple pairs of radius and height can achieve this, e.g., \(r = 4, h = 4\) from above. Give two other pairs. \(
2 \pi r h + 2 \pi r^2 = \pi r^2 h \\
2 r h + 2 r^2 = r^2 h \\
2 h + 2 r = r h \\
2 r = r h - 2 h \\
2 r = (r - 2)h \\
\frac{2r}{r - 2} = h \\
(r, h) = (3, 6), (4, 4), (5, 10/3), (6, 3), (7, 14/5), (8, 16/6), etc.
\)