Mountain View Guest
Happy people

Surface Area and Volume of 3 dimensional figures


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 a 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

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

Graphical representation of a cylinder

Surface Area of a cylinder
$= (2 × π × r × h) + (2 × π × r^2)$
$= (2πrh) + (2πr^2)$


Volume of a cylinder
$= π × r^2 × h$
$= πr^2h$




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.


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