Cubes | Math Olympiad for Class 5 PDF Download

Basic Structure of a Cube

A cube is a 3-dimensional structure with three sides (length, width, and height) where all the sides equal (length = width= height). The word cube is derived from the Arabic word ”Kaba” a large cube-shaped structure.

A cube has 6 faces: The faces appear at front, back, right, left, top and bottom.

Cube has 8 vertices: The corner points are called vertices. There are four vertices on the top face and four more at the bottom face.

Cube has 12 Edges: The line which connects the two vertices is called edges. There are twelve edges in a cube. Four on the top surface and four at the bottom and four more verticle lines connecting the opposite vertices of top and bottom face.

Summary:

Surface Area of Cube

A cube is a 3-dimensional representation of a square. Since dimensions of all the three sides, i.e. length, breadth, and height of a cube are equal, they are referred to as sides and is indicated by a symbol ‘s’.
The total surface area of a cube is:

• The surface area of a cube = (area of one square) *6
• The surface area of a cube = (s*s)*6
• The surface area of a cube = 6
• The sum of areas of 4 constituting squares (faces) gives the lateral surface area of the cube.
• The lateral surface area of a cube = 4*(area of one square)
• The lateral surface area of a cube = 4*(s*s)
• The lateral surface area of a cube = 4

Creation of Rubik’s cube

If we divide a cube into the size part of its side, we get smaller cubes. A cube Shown below, which is painted on all the sides and then cut into 1/4th of its original face. Each small cube is known as “unit cube”.
The total number of smaller(unit) cubes generated due to the division of a cube to its.
Example: If we cut a cube to form three rows and three columns on each face then the total number of unit cubes will be = 3 * 3 * 3 = 27.
Following is the list of unit cube generation based on the division of a bigger cube:

Evaluation of Cube with ‘n’ sides painted

Observe the cube shown. It has been divided into ¼ of its original side length. Hence the total number of smaller or unit cubes formed will be 64.

Important Facts

The most important thing to consider while solving the questions of such types is to visualizing the cube in your mind. By looking at the cube you can clearly identify the basic terminologies such as the face, vertex, and edge of a cube.

Rules

• When a cube have its side measuring unit as ‘a’ and is painted on every face, and then it is reduced into smaller parts with measuring unit of sides as ‘b. Then you are expected to answer the quantity of cubes with ‘n’ faces painted. Reasonably if you see one specific edge of any big cube and visualize the smaller parts of that by making a/b then the number of smaller cubes will be calculated by (a/b)³.
• As we know that all the reduced cube parts will always have at least one face in the inner side, which means not on the exterior side; therefore, all the reduced cubes will have faces that are not painted. Also as the larger cubes meet at the corner points, i.e. 3, hence, the lesser cubes will be having a limit of 3 painted faces. Therefore, the smaller cubes with 3 faces painted = Number of large cube’s corners = every time 8 cubes. The only condition is that all the faces of the larger cubes are painted.
• To get the total number of smaller cubes having 2 faces painted only, we need to check the cube edges points. These are the points where 2 faces of the larger cubes meet. To solve these questions, you always need to include the corner cubes also, hence if you will remove 2 cubes from the total number of cubes on each edge, then you will easily get the answer.
• The cubes with one face painted can only be at the cubes at the face of the bigger cubes.