Vachmi


Cuboid

A cuboid is a 3 dimensional shape comprising of 6 rectangles which are placed at right angles to each other. The rectangles opposite to each other are identical.
A cuboid is a 3D version of a rectangle.

The properties of cuboid are:

  • It has 6 faces (all rectangles).
  • It has 12 edges.
  • It has 8 corners or vertices.

Graphical representation of cuboid

Surface Area of a cuboid
= $2 ((l × b) + (l × h) + (b × h))$

Volume of a cuboid = $l × b × h$


Cube

A cube can be considered as a special case of cuboid. A cuboid in which all the faces are square is called a cube. A cube is a 3D version of a square.



Graphical representation of cube

Surface Area of a cube
= $6 × l^2$

Volume of a cube = $l × l × l = l^3$


Sphere

A sphere is a round solid object in three dimensional space with every point on its surface equidistant from its centre. A sphere is a 3D version of circle.



Graphical representation of Sphere

Surface Area of a sphere
$ = 4 × π × r^2$
$ = 4πr^2$

Volume of a sphere
$= \frac{4}{3} × π × r^3$
$= \frac{4}{3}πr^3$


Pyramid

A pyramid consists of four triangular lateral surfaces and a square or a rectangle as its base. In fact, base of a pyramid can take any shape - it can be a regular shape or an irregular shape. If it is a regular shape, it can be a triangle, square, ractangle, pentagon etc.

In our case, we will take rectangle as a base.
To calculate surface area of the pyramid we take the sum of the areas of the 4 triangles and the base rectangle.

The height of a triangle within a pyramid is called the slant height.



Graphical representation of pyramid

Surface Area of a pyramid
= Surface area of base + Surface area of lateral triangles
= $(l × w) + 4 × (\frac{1}{2} × w × sh)$

Volume of a pyramid
= $\frac{1}{3}$ × Area of base × height of pyramid
= $\frac{1}{3} × (l × w) × h$


Cylinder

A solid 3 dimensional shape bounded by a cylindrical surface and 2 parallel circular planes as its bases is called a cylinder. These circular planes are always parallel and congruent to one another.

The height (or altitude) of a cylinder is the perpendicular distance between its bases.



Graphical representation of cylinder

Surface Area of a cylinder
$=$ area of curved surface + (2 × area of circular base
$= (2πrh) + (2πr^2)$

Volume of a cylinder
$=$ area of circular base × height of cylinder
$= πr^2h$


Hollow Cylinder

A hollow cylinder is a solid 3 dimensional shape which is empty from inside and has some difference between internal and external radii. Examples of a hollow cylinders are tubes, hollow copper wires, straws etc These circular planes are always parallel and congruent to one another.

The height (or altitude) of a cylinder is the perpendicular distance between its bases.



Graphical representation of hollow cylinder

Let the inner radius be r and the outer radius be R

Lateral Surface Area of a hollow cylinder
$=$ area of outer curved surface + area of inner curved surface
$= 2πRh + 2πrh $

Total Surface Area of a hollow cylinder
$=$ area of outer curved surface + area of inner curved surface + (2 × area of circular base)
$= 2πRh + 2πrh + 2π(R^2 - r^2)$

Volume of a hollow cylinder
$=$ Volume of outer cylinder - Volume of inner cylinder
$= πR^2h - πr^2h$
$= πh(R^2 - r^2)$


Cone

A cone is a solid 3 dimensional object with circular base and which tapers smoothly from its base to a point called the apex or vertex. The shortest distance between the vertex and the base is called height. And, the distance from the vertex to a point on the circle is called the slant height.



Graphical representation of cone

Surface Area of a cone
$= π × r (r + \sqrt{h^2 + r^2})$
$= π r (r + \sqrt{h^2 + r^2})$

Volume of a cone
$= π × r^2 × \frac{h}{3}$
$= πr^2\frac{h}{3}$


Frustum of a Cone

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let s be the slant height and R and r the base and top radii.



Graphical representation of frustum of cone

Slant height s = $\sqrt{(R^2 - r^2) + h^2}$

Lateral Surface Area of frustum of cone
$= π × (R + r)s$
$= π × (R + r) \sqrt{(R^2 - r^2) + h^2}$

Total Surface Area of frustum of cone
= Area of base + Area of top + Lateral Surface Area $= π [R^2 + r^2 + (R + r)s]$

Volume of frustum of cone
$= \frac{1}{3} π h × (r^2 + R^2 + Rr)$


Triangular Prism

A triangular prism is a solid object that has 2 identical triangles at its ends and all flat sides as rectangles. The two triangles are parallel to each other.

It is also a polyhedron.



Graphical representation of triangular prism

Let each side of the triangular base be a and height of prism be h

Lateral Surface Area of this regular triangular prism
$=$ 3 × area of each rectangle
$= 3ah $

Total Surface Area of this triangular prism
$=$ 3 × area of each rectangle + 2 × area of triangular base
$= 3ah + \frac{\sqrt3}{2} a^2$

Volume of a triangular prism
$= \frac{\sqrt3}{4} a^2h$