Spheres
What is the formula for the volume of a sphere?
4/3πr2
What is the formula for the volume of a pyramid?
1/3 lwh
1/3 Bh
What is the formula for the volume of a cone?
V = 1/3πr2h
What is the formula for the volume of a cylinder?
V = πr2h
What is the formula for the volume of a prism?
V = Bh (or lwh)
What is the formula for the surface area of a sphere?
4πr2
What is the surface area of an equilateral triangle pyramid with side length 2?
(A pyramid formed by 4 equilateral triangles)
4sqrt(3)
What is the formula for the surface area of a cone?
A = πr2+πrl
What is the formula for the surface area of a cylinder?
A = 2πrh + 2πr2
The dimensions of a rectangular solid are x, y, and xy.
If the interior diagonal of this solid has length xy + 1, find all possible expressions for y in terms of x.
(MML 2016 R1 Q2)
y = x+1, y = x-1
The volume of a sphere is 9π/2 cubic units. Compute its surface area.
(MML 2017 R1 Q2)
9π
A pyramid has a square base with area of A2 , and congruent lateral faces, each with area A2/3. The total surface area of the pyramid is 84.
Compute the height of this pyramid.
(MML 2019 R1 Q2)
sqrt(7)
A water tank, in the shape of an inverted right circular cone, has a diameter of 12 meters and a height of 10 meters. If the height of the water in the tank is 3 meters, compute the exact amount of water (in m3) in the tank.
(MML 2018 R1 Q1)
81π/25
The height of a cylinder is 6 units. Compute the radius of the base, if the lateral surface area equals the area of one of its bases.
(MML 2017 R1 Q1)
r = 12
What is the volume of the largest cube that fits within a sphere with radius 1?
8sqrt(3)/9
A sphere with integer radius r has a volume greater than a cube with a side of length 8. Determine the smallest possible value of r.
Assume π ≈ 3.1416 .
(MML 2015 R1 Q1)
r = 5
A pyramid has its vertex at the center of a face of a cube, and its base coincides with the opposite face of the cube. If the volume of the region inside the cube and outside the pyramid is 1152 units3, compute the edge of the cube.
(MML 2018 R1 Q2)
12
Suppose we have right triangle ABC, where AB is the hypotenuse. We also have point D, where CD is the altitude from C to AB. Then, point C is rotated about AB, creating two cones. If AC = 4 and BC = 3, the positive difference between the volumes of the two cones is kπ. Compute k.
(MML 2019 R1 Q3)
k = 336/125
Assume an underground storage tank is a cylinder 72 inches long and 36 inches wide, capped on each end by a hemisphere of radius 18 inches. Assume there are exactly 7.5 gallons of water in a cubic foot. In terms of π, compute the maximum number of gallons of water this tank will hold. Leave your answer in terms of π.
(MML 2015 R1 Q3)
135π gallons
Three faces of a rectangular solid have areas of 864, 1440 and 2160 units2, respectively. This solid is packed with k congruent cubes with edge E, where k is as small as possible. Each of the k cubes is inscribed in a sphere. The total volume of the E spheres, in simplified form, is Aπ sqrt(B) . Compute the ordered pair (A,B).
(MML 2016 R1 Q3)