See also the
Dr. Math FAQ:
Browse High School Polyhedra
Stars indicate particularly interesting answers or
good places to begin browsing.
- Definitions as a Tool of Mathematics [03/04/2003]
Defining polyhedra: regarding unusual polyhedra, how can we exclude
'doubled' shapes, e.g. two tetrahedra that join at a single vertex
(sort of like an hourglass)?
- How Many Cube Faces Were Painted? [02/23/2003]
Some unit cubes are put together to form a larger cube. Some of the
larger cube's faces are painted and then it is taken apart. 45 small
cubes are found to have no paint on them. How many faces of the large
cube were painted?
- Only Five Platonic Solids [03/05/1998]
Why there are only five platonic solids.
- Painting Cubes [01/06/2003]
How many unit cubes would be needed to create a cube with a side
length of 3 units? If you painted the larger cube and broke it up
into unit cubes, how many faces of each unit cube would be painted?
- Polyhedra: Solids or Surfaces? [06/06/2003]
Is a polyhedron always a solid figure?
- Angles in a Hexagonal Pyramid [07/25/2002]
In a gazebo whose roof is a hexagonal pyramid with a particular pitch,
how can I find the angles at which the roof panels meet?
- Angles of an Octahedron [07/20/1998]
What is the angle between two adjacent faces of an octahedron?
- The Angles of a Tetrahedron [07/08/1998]
Why is the angle from one vertex to the exact center of the tetrahedron
around 109.5 degrees?
- Bases and Faces [12/05/2001]
I can't figure out the difference between a base and a face on the shapes
we are learning.
- Biggest Cuboid in a Sphere [03/20/2003]
I have a sphere with radius of 10cm. I have to find the biggest cuboid
that fits in that sphere.
- Calculating Angles Between Faces of a Solid [09/15/2003]
How can I compute the dihedral angles for a Great Rhombicosidodecahedron?
- Classifying Quadrilaterals [01/25/2006]
How can a square also be a rectangle?
- Constructing a Pyramid [05/28/1999]
How can I calculate the sides and angles needed to construct a pyramid
with an 8-foot-square base and a height of 6 feet?
- Constructing a Sphere out of Paper [02/08/2003]
I'm working on a school project and can't seem to find the answer
anywhere. The assignment is to construct a 3D 9'-diameter sphere out
- Construction of Cone Surface [10/12/2002]
What is the relation between the angle of a cone at the vertex and the
flat angle (theta) of the developed surface on a plane?
- Cubes in a Big Cube [03/11/2002]
Is there a formula for the number of cubes in an n*n*n cube?
- Cups and Volume [12/06/2001]
How can I calculate the volume of a box, if I know how many cups of rice
fill it? And how can 2 cups be a volume measure?
- Cutting a Cube [07/05/1999]
How many pieces will there be if you make every (flat) slice through a
cube that goes through exactly three of the cube's corners - no more and
- Cutting a Square into Five Equal Pieces [07/12/1999]
How can you divide a square cake into five equal parts, cutting through
the center point?
- Cutting Cake with Geometry [2/4/1996]
A cake is square when viewed from the top. Height is unspecified. It is
iced on top and the four vertical sides. How can the cake be divided in 5
pieces such that each piece has the same amount of cake and the same
amount of icing? How can you minimize the number of cuts you have to make
in the cake and still meet this target? Alternatively, how can you
minimize the total length of the cuts you make in the cake?
- Cylinders and Euler's Rule [09/06/2002]
How and why does Euler's rule work for cylinders?
- Defining and Counting Faces, Edges, and Vertices of Shapes [04/06/2004]
How do you find the faces, edges, and vertices of any shape? I just
go blank, because I really don't understand it.
- Definitions of Cones and Cylinders [03/02/2004]
Are cones and cylinders pyramids, prisms or neither? My 5th grade
geometry class cannot reach an agreement on this subject!
- Derivation of Sphere Volume and Surface Area Formulas [03/14/2005]
Currently in math class we are discussing surface areas and volumes of
solids. I would like to know why the volume formula for the sphere is
(4/3)*pi*r^3 and why the surface area formula is 4*pi*r^2.
- Determining Polyhedron Name from Given Description [11/16/2003]
What is the name of the geometric solid having 7 faces, 10 vertices,
and 15 edges? Thank you!
- Diagonals and Symmetry in Polyhedra [09/15/2002]
I would like a formula for finding the number of diagonals in a
- Diagonals in 3D Figures [06/21/1999]
Could you help me develop a formula for determining the number of
diagonals in various 3D figures, especially pyramids and prisms?
- 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
- 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.
- Do Pyramids Really Exist? [02/27/2002]
If the base of an isosceles triangle is 4, and the height is 5, then the
sides are equal to the square root of 21. How can this triangle exist
(except in theory) if you can never measure or draw the square root of
- Edges, Vertices, Surfaces [04/22/2003]
How can a shape have 8 edges and 5 vertices? How can a shape have 12
edges, all the same length? How can a shape have 3 surfaces, one
- Eleven Nets of a Cube [11/15/2001]
My teacher says that there are 11 combinations to make a cube without
reversing them, but I can only find 6.
- Euler's Formula [11/26/2001]
I have to find Euler's formula for two-dimensional figures and explain it
at a university level and at an elementary-school level.
- Euler's Formula for Polyhedra [08/12/1997]
How would you prove Euler's formula V-E+F = 2 for all polyhedra of genus
- Faces, Vertices, and Edges of Cylinders, Cones, and Spheres [12/28/2003]
Do cylinders, cones and spheres have faces, vertices, and edges? I'm
not sure how they would fit into Euler's formula of v - e + f = 2.
- Find Depth of Water in a Tank [08/02/2003]
A rectangular tank measures 4m long, 2m wide and 4.8m high. Initially
it is half full of water. Find the depth of water in the tank after
4000 litres of water have been added to it.
- Finding Area and Volume [04/12/2001]
When working with area and volume of triangular shapes, how do I know
when to divide the base by 2 and when to divide it by 3?
- Finding Surface Areas [06/17/2002]
Can you explain how to find the surface areas of solids?
- Finding the Area of an Irregular Polygon [02/23/2008]
What is the formula for finding the area of an irregular polygon?
- Finding the Dimensions of a Box [10/21/2001]
You want to construct a cardboard box from a cardboard strip that is 8
inches wide. The dimensions of the box are 8"x8"x4". How long does the
strip need to be?