Cubes

A box-shaped solid object that has six identical square faces.

Examples of cubes and cuboid

The concept of a cube was first conceived by the ancient smurfs, and is recorded in their primitive writings as "Kube" (sic; this was before the invention of spelling). The figure was imagined yet not visualized, and mathematicians for thousands of years could only speculate as to its shape. Gauss's guess was that its shape resembled a short rolling pin, flattened in places. The first correct visualization of the cube was made by the first female mathematician, Renée Descartes, who applied her Descartesian coordinate system with the parameters (0, 0, 0, 9am) and (1, 1, 1, noon), where the first three are spacial coordinates and the last is the Time Cube index for the implicit earth rotation. This produced a foundation that, with some rough sketching using a broken charcoal pencil, created the ideal cube we know today.

What items in every day life have the shapes of a cylinder, cube, cuboid?

• Cube: sugar cube, child's blocks set, dice

Cuboid: gift box, package of chewing gum, stick of butter

CUBE:

1. It is square in shape. 2. It is a three dimensional solid figure. 3.Three of its sides meet at each vertex. 4.Formula to find the volume of a cube is a x a x a,when the edge length is a. 5.Examples of cubes are playing dice.

Can You say me Five properties of cube ?

CUBOID:

1.It is rectangular in shape. 2.It is three dimensional and solid. 3.It is also called a rectangular prism. 4.All its angles are right angles (they measure up to 90 degrees) 5.The opposite faces of a cuboid are equal.

Can You say me Five properties of cuboid ?

Observe the cuboid and find how many faces ithas? How many corners and how many edges it has?See the faces are surfaces? Which pair of faces areequal in size? Do you get any idea to find the surfacearea of the cuboid?

• A cube is just a special case of a square prism, and

• Note: The name "cuboid" comes from "cube" and -oid (which means "similar to, or resembling") and so says "it is like a cube".

• Another use of -oid is when we talk about the Earth being a spheroid (not exactly a sphere, but close).

A cuboid is a box-shaped object. It has six flat sides and all angles are right angles.And all of its faces are rectangles.It is also a prism because it has the same cross-section along a length. In fact it is a rectangular prism.

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A cube has 6 equal and plane surfaces. All the faces of a cube are square in shape. In a cube there are 6 plane surfaces. There are 8 vertices and 12 edges.

Parts of a cube:(i) Face: Face is also known as sides. A cube has six faces and all the faces of a cube are square in shapes. Each face has four equal sides.(ii) Edge: When two edges meet each other a line segment formed. There are 12 edges in a cube. All the 12 edges are equal in length because all faces are squares. These edges are straight edges.(iii) Vertex: When three edges meet each other a point formed. There are 8 vertices in a cube.(iv) Face Diagonals: Face Diagonals of a cube is the line segment that joins the opposite vertices of a face. There are 2 diagonals in each face so altogether there are 12 diagonals in the cube.(v) Space Diagonals: Space diagonals of a cube are the line segment that joins the opposite vertices of a cube, cutting through its interior. There are 4 space diagonals in a cube.

Properties of a cube:Volume: The volume of a cube is s3 where s is the length of one edge.

Surface Area: The surface area of a cube is 6s2, where s is the length of one edge.

Definition of cuboid:The cuboid has 6 rectangular faces. The opposite rectangular plane surfaces are identical (equal in all respects).It has 8 vertices and 12 edges.In a cuboid there are 6 rectangular plane surfaces. There are 8 vertices and 12 edges.

PROPERTIES OF A CUBOID:Formulas for the above rectangular box:Volume: The volume of a cuboid is lwh, where l is the length, w is the width and h is the height.Lateral Surface Area: The lateral surface area of a cuboid is 2lh + 2wh, where l is the length, w is the width and h is the height.Surface Area: The surface area of a cuboid is 2lw + 2lh + 2wh, where l is the length, w is the width and h is the height.

Examples of Cubes

Take a cuboid shaped box made up of thick paper or cardboard for example toothpaste box.Cut and open it as shown in figure. Observe its shape of the faces. How many sets of identicalfaces are found?Look at the figure, if length ‘l’, breadth ‘b’, height ‘h’ are its dimensions, then you can find threepairs of identical faces.

14.1 Cuboid

Now we can see that the total surface area of a cuboid isArea I + Area II + Area III + Area IV + Area V + Area VI= h l + l b + b h + l h + b h + l bSo total surface area = 2 (h l + b h + b l)= 2 (lb + bh + hl)

Lateral Surface Area:• The lateral faces (the faces excluding the top andbottom) make the lateral surface area of thecuboid. For example, the total area of all the fourwalls of the cuboidal room in which you are sittingis the lateral surface area of the room.Hence, the Lateral Surface Area of a cuboid(L.S.A.) = (l × h) + (b × h) + (l × h) + (b × h)= 2lh + 2bh= 2h (l + b)

CubeDraw the net Fig. (i) given below, on a graph paper and cut it out. Fold it along the lines asshown in Fig. (i) and joined the edges as shown in Fig(ii) and Fig. (iii). What is the shape of it?Examine its faces and its dimensions.

(a) How many faces does a cube have ? Are all faces equal?(b) If each side of the cube is l, what will be the area of each face?(c) What is the total surface area of the cube.(d) What is the lateral surface area of cube?

Observe the cube and its net diagramIn the figures (i) and (ii). Do all the faces of a cube are square in shape? Do the length,height and width of a cube are equal?

Volume of Cube and CuboidAmount of space occupied by a three dimensional object is called its volume. Try to compare thevolume of objects arround you. For example, volume of a room is greater than the volume of analmirah kept in the room. Similarly, volume of yourpencil box is greater than the volume of the pen andthe eraser kept inside it. Do you measure volume ofeither of these objects?Remember, we use square units to find the area of aregion. How will we find the volume. Here we willuse cubic units to find the volume of a solid, as cube isthe most convenient solid shape (just as square is themost convenient shape to measure this area).To measure the area we divide the area into square units, similarly, to find the volume of a solidwe need to divide the space into cubical units. Unit cube is a cube of unit length . Observe that thevolume of each of the solids which are arranged in different forms are of 8 cubic units (as in Figabove).We can say that the volume of a solid is measured by counting the number of unit cubes itcontains. Cubic units which we generally use to measure the volume are1 cubic cm = 1 cm × 1 cm × 1 cm = 1 cm3

= 10 mm × 10 mm × 10 mm = ____________________ mm31 cubic m = 1 m × 1 m × 1 m = 1 m3= 100 cm × 100cm × 100cm = ____________________ cm31 cubic mm = 1 mm × 1 mm × 1 mm = 1 mm3= 0.1 cm × 0.1 cm × 0.1 cm = ____________________ cm3

Vollume of a CubeA Cube is a cuboid whose length, breadth and height are same,So Volume of a cube = side side side= (side)3 = a3

Where a is the side of the cube.Length of Cube Volume of the Cube10mm = 1cm 1000 mm3 = 1cm310cm = 1dm 1000 c m3 = 1dm310dm = 1m 1000 d m3 = 1 m3100cm = 1m 1000000 c m3 = m31000m = 1km 1000000000 m3 = 1km3

Find the surface area of a cuboid whose length, breadth and height are 15cm,12cm and 10cm respectively.Solution: Length of the cuboid (l) = 15cmBreadth of the cuboid (b) = 12cmHeight of the cuboid (h) = 10cmSurface area of a cuboid = 2 (lb + bh + hl)= 2 (15 12 + 12 10 + 10 15) cm2= 2 (180 + 120 + 150) cm2= 2 (450) cm2= 900 cm2

Find the volume of a block of wood whose length is 20cm, breadth is 10 cm andheight is 8 cm.Solution: The block of wood is a cuboid and the volume of a cuboid = l × b × hHere, length (l) = 20 cm, breath (b) = 10 cm, and height (h) = 8 cmVolume of the block = 20 cm × 10 cm × 8 cm = 1600 cm3

Find the side of a cube whose surface area is 600 cm2.

Do you knowCapacity:There is not much difference between the two words.i.e volumeand capacity(a)Volume refers to the amount of space occupied by an object.(b)Capacity refers to the quantity that a container holds.If a water tin holds 100 cm3 of water then the capacity of the watertin is 100cm3. Capacity can also measure in terms of litres.

What we have discussed1. If l, b, h are the dimensions of cuboid , then:(i) its lateral surface area is 2h (l + b)(ii) its total surface area is 2 (lb + bh + hl)2. Lateral Surface area of a cube is 4a²3. Total Surface area of a cube is 6a²4. Volume of a cuboid is l × b × h5. Volume of a cube is side × side × side = a³6. 1 cm³ = 1 ml1 l = 1000 cm³1 m³ = 1000000 cm³ = 1000 l= 1 kl(kilolitre)