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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Derivation of the Formula for the Frustum [08/09/1999]

Using similar triangles to derive the formula of a frustum of a right circular cone, given the volume of a right circular cone.

Deriving the Area of a Sphere [10/21/2003]

I know the area of a sphere is 4phi(r^2), but I'm wondering how to derive that formula. I know it should be done in cylindrical coordinates, and I'm thinking that the arc of a circle is defined as rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi). Could you please help explain this?

Deriving the Volume of a Frustum [05/06/1998]

Can you derive the formula for the volume of a frustum of a cone?

Designing and Building a Cone Frustum [03/24/2004]

I need help making making a pattern for a frustum shape to protect a large piece of equipment. How can I find the distance between the top and bottom arcs in a straight line?

Determining an Angle From Side Views [07/24/2002]

In the end elevation of a drawing, a rod rises from point A at 32 degrees. In the side elevation, the same rod is seen rising from point A at 48 degrees. How do I work out at what angle to cut the end of the rod?

Determining Cone's Original Dimensions from a Slice [05/18/1999]

Given the inside and outside arc lengths and the thickness of a truncated cone, find the inner and outer radii and the angle.

Determining Distance between Two Cities [07/30/1997]

I need an equation to use in an application for a car dealership.

Diameter of a Ball [11/25/2001]

A child rolls a ball on a level floor 4.5m to another child. If the ball makes 15 revolutions, what is its diameter?

Diameter of Cone Frustum [05/24/2000]

Given the inner and outer diameters and the height of a cone frustum, how can I find the diameter when it is opened up flat?

Diameter of the Base of a Cone [08/12/1998]

How do you find the formula to calculate the diameter of the base of a cone of nine degrees at various lengths?

Differentiating and Integrating the Formula for Area of Circle [05/11/1998]

The formula for a circle's circumference is the derivative of the formula for its area. What is the significance of this?

Dimensions [08/30/2001]

What are 1-dimensional, 2-dimensional, and 3-dimensional? What's the difference?

Dimensions [02/15/1997]

Does 1-D space exist? How about 5-D space?

Dimensions of a Cardboard Box [09/28/1997]

A box with a square bottom and a volume of 2000 centimeters can be made by cutting 5-centimeter squares from the corners of a piece of cardboard...

Direction of Travel [10/18/1996]

If I know the latitudes and longitudes of where I am coming from and where I want to go, how do I figure out what compass heading to follow?

Distance Between 2 Lines: Vectors [8/19/1996]

What is the shortest distance between 2 lines?

Distance Between Two Points on the Earth [6/21/1996]

My latitude and longitude are in the form 40.266934, -74.204930 respectively, with negatives for South and West. How do I calculate the distance between them?

Distance From a Point to a Plane [03/31/1998]

Can you show me the proof of the formula for the distance between a point and a plane?

Distance using Latitude and Longitude [12/31/1997]

Is there a simple formula for calculating distance using known latitude and longitude?

Dividing a Cylinder [08/06/2001]

How can I divide a cylinder that is 4 by 9 inches into two smaller cylinders, each with the same volume?

Do Cones and Cylinders Have Faces or Surfaces? [03/16/2004]

Are the sides of a cone or cylinder called faces or are they curved surfaces? I've seen different books with different answers. Some also say that a cone has no vertex.

Drawing a Locus of Points [05/30/2008]

Draw the locus of all points in the plane that are equidistant from the rays of an angle and equidistant from two points on one side of the angle. I'm having trouble with this kind of locus problem. Can you explain how to think about and solve them?

Earth's Curvature [01/27/1997]

How do you figure out the degree of curvature of the earth's surface?

Einstein, Curved Space, and Pi [10/09/1999]

If space is curved and Euclidean geometry doesn't apply, doesn't that mean that the value of pi changes and can sometimes be rational?

Equation of a Line in 3-D Space [03/18/2004]

Given two points in 3-D space, such as A(x1,y1,z1) and B(x2,y2,z2), what would be the equation of the line that connects those points? I know that in the 2-D plane the equation of a line in slope-intercept form is y = mx + b. Is there something similar in 3-D?

Equation of a Line in Three or More Dimensions [05/18/2000]

Can the equation y = mx + b be used to define a line in three dimensions? What about four or more dimensions?

Etymologies of Algebra, Geometry, Trigonometry [10/22/2001]

What are the origins and roots of the words geometry, algebra, and trigonometry?

Etymology of the Word Tessellation [11/05/2001]

Are tessellations related to the tesseract in Madeleine L'Engle's _Wrinkle in Time_ series?

Euclid's Parallel Postulate [2/14/1995]

We are studying Euclid's Parallel Postulate. Why did mathematicians disagree with him? What other geometries resulted from this disagreement? What postulate replaced the Parallel Postulate?

Evaluating Axioms [07/09/2002]

Given points {1,2,3,4,5}; lines {1,2,3,4}, {1,5}, {2,5}, {3,5}, and {4,5}; planes {1,4,5}, {2,4,5}, and {3,4,5}; explain why the following axioms are inconsistent with the model...

Explanation of the Formula for the Volume of a Pyramid [06/16/2005]

Can you explain WHY the formula for the volume of a pyramid is (base*height)/3?

Figuring Out Formulas for Area [08/24/1999]

How can I remember formulas for squares, triangles, cylinders, etc?

Filling a Garden with Topsoil [2/5/1996]

I have a garden that is 10' x 10' (100 square feet). I want to add 6" of topsoil to my garden. Topsoil is sold by the cubic yard. How many cubic yards of topsoil will I need for my project?

Filling Cylindrical Holes [05/18/1999]

What volume of cement, in cubic yards, will fill 42 holes that have a 9" diameter and are 4' deep?

Finding a Best-Fit Regression Plane [12/15/2005]

I have a height map of a terrain where the x and y values are fixed. I can calculate best-fit slopes in the x and y directions, but I can't figure out how to combine them into a best-fit regression plane.

Finding a Plane Shape for a Truncated Cone [05/10/2000]

What is the best way to cut a flat two-dimensional piece of sheet metal into a three-dimensional truncated cone?

Finding Dimensions of Elliptical Shadow Cast on Plane [01/19/2005]

A sphere is positioned above a plane and light is shown on it from one side, resulting in an elliptical shadow on the plane. How can you determine the lengths of the axes of the resulting ellipse?

Finding Distance Using the Earth's Grid System [08/05/1997]

How do I find the shortest distance between two points on the earth using degrees of latitude and longitude?

Finding Points on the Earth [06/08/2001]

Find the point that has latitude and longitude five miles north of a given point, and the other three points to the south, east, and west.

Finding the Angle of Solar Collectors on a Sloped Roof [08/30/2008]

I have solar collectors on my roof. They are mounted so that the base of each panel runs up the slope of the roof, and the panels themselves are mounted at an angle. I'd like to know how to determine the various angles created by this situation.

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