Measurement problems

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

  1. A circle's perimeter and area have the same numeric value. What is the diameter of this circle?
  2. 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.
  3. 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?
  4. 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?

Challenge

  1. 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.