Ask Dr. Math

High School Archive

Browse High School Higher-Dimensional Geometry
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Do cones or cylinders have edges?
    Latitude and longitude.
    MaximizIng the volume of a cylinder.

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Nearest Point on a Great Circle [05/27/2002]

Given points A, B, and C on the surface of a unit sphere, find the point P on the great circle defined by A and B that is nearest to C.

Number of Equations Needed in a Simultaneous Linear System [10/29/2003]

Could you tell me why we need the same number of equations as variables in order to get a unique solution to a system of simultaneous linear equations?

Number of Faces of a Cylinder and a Cone [1/29/1995]

If you have a cylinder, how many faces does it have? What about a cone?

Numerically Equal Volumes and Surface Areas [06/04/2001]

Find all rectangular solids with integral dimensions, the volumes and surface areas of which are numerically equal.

Obtaining Bearing from a Velocity Vector [08/01/2001]

I have the x, y, and z components of a velocity vector of an airplane, and must use this vector to calculate the bearing of the plane.

Oil Can Dimensions [12/11/2001]

What are the dimensions of an oil can with a one-liter capacity that uses the least amount of tin?

One Degree Latitude, Longitude: How Many Miles? [04/02/2002]

Approximately how many miles are there in one degree of longitude and one degree of latitude in the states of Kansas and Oklahoma?

Optimization [11/26/1996]

To make a funnel, we take a circular piece of metal, cut out a sector, and connect the two radial edges together to make an open cone. What should the angle of the sector be to maximize the volume of the cone?

Order of a 3D Triangle [08/29/2001]

If I visit the vertices of a 3D triangle in order going from a to b to c, am I going clockwise or anticlockwise?

Packing 4 Spheres Into a Tetrahedron [09/03/99]

How can I find the dimensions of the smallest tetrahedron that can serve as a container for 4 spheres packed as snugly as possible?

Packing Pennies in a Jar [06/08/1999]

If a jar has a height of 11" and a radius of 7" and is full of pennies evenly to the top, how many pennies can fit in it?

Packing Spheres [08/16/1999]

How many 1.68-inch-diameter spheres fit into a 96.3-cubic-foot space? How many 1.68-inch-diameter spheres would fit into a 96.3-foot- diameter sphere?

Painting a Column [07/13/1997]

If I have a column 9.5' long and 8'' in diameter, how do I figure out how much paint I need to cover the outside area?

Painting Bell Towers [6/3/1996]

I need to know the total surface area of three domes with assorted radii.

Paper Patterns for Building Cones [11/03/2003]

How do you create a paper pattern which can be rolled up into a cone that is 72 inches long and 26 inches in diameter at the open end?

Pappus' Centroid Theorem [06/05/2003]

What is the formula for the volume of a circular torus?

Parallel Lines [12/31/1998]

What are some ways of proving lines parallel - geometrically and algebraically?

Parallel Lines, Concentric Circles [04/24/2003]

If parallel means that two lines never intersect, would the lines that form a circle drawn inside another circle be considered parallel?

Parametrics [12/18/1996]

You're in 3-D space at point A, you want to get to point B, and you know the coordinates to point B from point C (but B is moving). What heading do you need to set in order to meet point B?

Parts of a Cone [04/18/2001]

Does a solid cone have any edges?

Passing a Larger Cube Through a Smaller One [09/25/2003]

How is it possible to cut a hole through a solid cube so that a cube, larger than the original, can be passed in one end and out the other?

Pattern for Lampshade [02/17/2001]

I would like to know the formula for computing the surface area of a lamp shade so that I can make a pattern from it.

Planar Approximation: Latitude and Longitude [04/18/2003]

I am trying to calculate the midpoint between cases of legionella and their nearest neighbor. How can I calculate the distance between two points below which they can be treated as if they were in a plane rather than on a sphere?

Planes between Planes [04/02/2001]

How can I find if a plane lies between two other planes?

Planes Intersecting Space [11/24/2001]

Can we say that n planes divide space into at most 2^n regions?

Polyhedron inside Sphere [5/24/1996]

How long do the sides of a dodecahedron have to be to fit into a sphere of diameter 2.9 m?

Proof for Volume of a Segment of a Sphere [11/19/2001]

I am in need of assistance in proving the volume of a truncated spherical cap (or a segment of a sphere I think it is also called).

Proportions of Exact Enlargements [03/18/1998]

How are two objects related if one is an "exact enlargement of the other"?

Pyramid Construction [05/24/1997]

How do you figure out the angles necessary to construct a pyramid out of plywood?

A 'Pyramiddle' Tent Problem [07/12/1999]

Figure out an equation that yields d when values for h and r are inserted.

A Pyramid of Layered Marbles [06/02/1999]

How can I find the number of layers, the number of marbles, and the size of a container containing a pyramid of layered marbles?

Pyramids and Triangular Prisms [05/09/2000]

What's the difference between a pyramid and a triangular prism?

Radius of a Sphere [01/29/1998]

What is the ratio of the areas and volumes of two spheres, one with radius 3 times the other? What possible theorems are suggested?

Radius of a Tennis Ball [04/03/2001]

How can we find the radius of a tennis ball without cutting it open?

Radius of the Earth as an Ellipsoid [06/26/2000]

I have been given two equations to determine the radius of the earth for a given latitude, based on ellipsoid model WGS84. I get different answers...

Ratio of Volume to Surface Area in Humans [12/09/1999]

What is the healthy ratio of volume to surface area for humans?

Ratios and Geometry [10/29/1998]

An airplane flying at 33,000 feet has a visibility of 100 miles. What percent of the total land area to the horizon is visible?

Rectangle Needed To Make Cone [9/5/1996]

I want to make a lampshade by rolling a piece of paper into a non-closed cone shape...

Rectangles Inscribed in a Dodecahedron [06/19/1999]

What is the equation for determining the size of the rectangles inscribed within a regular dodecahedron?

A Rectangular Prism [11/26/1995]

Is it possible to have a rectangular prism that has a volume greater than its surface area?

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