Solid Shapes and Their Nets

Solid-Shapes

Solid shapes are there dimensional. Solids can be described in various types. Some have a flat surface, some have curved surface and some have both flat and curved surface.

Nets for Solid figure

A pattern that you can cut and fold to make a model of a solid shape are a net of a solid figure.

Cube and Cube net

A cube is an object which looks like solid box-shaped that has six identical square faces.

A cube is a special case of a cuboid in which all six faces are squares i. e. lenght = breadth = height

Let each side of cube be a units

Then, total surface area of a cube =2(lb+lh+bh)

= 2(a x a + a x a + a x a)

= 2(a² +a² +a²)

= 2 x 3a²

= 6a²

= 6(side)²

Similarly, lateral surface area of a cube = 2h(l +b)

= 2a(a +a)

= 2a x 2a

= 4a²

= 4(side)²

and volume of a cube = l x b x h

= a x a x a

= a³

= (side)³

Cuboid and Cuboid net

Cuboid is a solid figure bounded by six faces. A cuboid has 6 rectangular faces. The opposite rectangular plane surfaces are identical. So, it is also called rectangular prism.

The surface of a cuboid consists of six faces which are rectangular in shape. We can categories them in pairs of opposite faces as follows:

1. Front face ABGH and back face EFCD
2. Top face EFGH and bottom face ABCD
3. Side faces ADEH and BCFG
   These opposite faces are congruent.

Let l, b and h represent the length, breadth and height of the cuboid respectively. To calculate the surface area of the cuboid we need to first calculate the area of each face and then add up all the areas to get the total surface area.

Total area of top and bottom surfaces is lb + lb =2lb

Total area in front and back surfaces is lh + lh = 2lh

Total area of two side surface is bh + bh =2bh

Thus, total surface area of cuboid = 2lb+2lh+2bh

=2(lb + lh +bh)

The lateral surface area of a cuboid includes a front face, back face, and two side faces.

Therefore, lateral surface is of a cuboid = 2lh + 2bh

= 2h(l+b)

Also, volume of a cuboid = l x b x h

Cylinder and Net of Cylinder

A cylinder has two circular plane surfaces, one at its base and another at its top. It has a curved surface in the middle.

To find the surface area of the cylinder, add the surface area of each end plus the surface area of the side. Each end is the circle so the surface area of each end isπr², where r is the radius of the end.

There are two ends so their combined surface area is 2πr². The surface area of the side is the circumference times the height or 2πr x h where r is the radius of the end and h is the height of the cylinder.

Therefore, curved surface area = 2πrh

and total surface area = 2πr² = 2πrh = 2πr(r+h)

The base of a cylinder is a circle, with area πr²

The volume of a cylinder is, therefore,

πr² x height = πr²h 

Cone and Net of Cone

A cone is a closed figure with a plane surface and a curved surface.

Pyramid and Its Net

There are three different solids that you can make with triangles. Mathematicians use the word tetrahedron to describe a triangular pyramid. Since it uses squares as well as triangles.

Tetrahedron

Tetrahedron is a triangular pyramid having congruent equilateral triangular for each of its faces.

The total surface area of a tetrahedron

= 4 x area of an equilateral triangle

= 4 x \(\frac{√3}{4}\) side²

=√3 side²

Volume of a tetrahedron = \(\frac{1}{3}\) area of base x height of pyramid

Net of tetrahedron

Square pyramid

A square pyramid is a pyramid having a square base.

In the figure, ABCD is a square base PO is the height of the pyramid. M is the mid-point of BC.

PM is the slant height of the pyramid.

Lateral surface area of a square pyramid = area of triangular faces

= 4 x area of Δ PBC

= 4 X \(\frac{1}{2}\) x BC X PM

= 2 X BC X PM

Total surface area of the square pyramid = LSA + area of base.

Also, volume of the square pyramid = \(\frac{1}{3}\) area of base x height

= \(\frac{1}{3}\) A x h

Net of square pyramid