# Formulas for Questions on Geometry

## Geometry related Formulas

In this page we are going to learn about various Formulas for Questions on Geometry which will include different types of  geometrical shapes and figures. For your knowledge a perimeter is the path that surrounds or encompasses a two-dimensional shape.

Definition for Geometry:

Geometry is all about shapes and their properties.

• Geometry can be divided into:-
Plane Geometry is about flat shapes like lines, circles and triangles … shapes that can be drawn on a piece of paper.
Solid Geometry is the geometry of three-dimensional space, the kind of space we live in.
• Geometry is generally to discover patterns, find areas, volumes, lengths and angles, and better understand the world around us.

### Formulas for Questions on Geometry

Largest Possible Sphere :

In this type you will be given a cube of “a” cm and will be asked the volume of largest possible sphere which can be chiseled out from it.

Diagonal of the Sphere: $\frac{a}{2} = \text{ Radius }$

Remaining Empty Space of the Cube: $a^3 = \frac{\pi a^3}{6}$

Largest possible cube:

In this type you will be given a sphere of  radius “A” cm and will be asked the volume of largest cube which can be chiseled out from it.

Here OA= radius of the sphere. So, diameter of the sphere = 2a

Diagonal of the cube = √3x

(If side of the Square is x)

$\Rightarrow \sqrt{3x} = 2a = \Rightarrow x = \frac{2a}{\sqrt3}$

The largest square that can be inscribed in a right angled triangle:

In this type you will be given a Square BDEF when one of its vertices coincide with the vertex of the right angle of the triangle ABC

Side of the Square: $\left ( \frac{ab}{a+b}\right )$

Area of the Square: $\left ( \frac{ab}{a+b}\right )^2$

The largest square that can be inscribed in a  semi circle:

In this type you will be asked the area of the largest Square which can be inscribed in a Semicircle of radius “r”

Area of the Square: $\frac{3}{5}r^2$

The largest square that can be inscribed in a  Quadrant:

In this type you can be asked the area or Side of the largest Square which can be inscribed in a Quadrant of Radius “r”

Side of the Square: $\frac{r}{\sqrt2}$

Area of Square: $\frac{r^2}{2}$

The largest square that can be inscribed in a right angled triangle:

In this type you will be given a Square DEGF when one of its vertices coincide with the hypotenuse of the right angle of the triangle ABC and you can be asked for the side of the square.

Side of the Square: $\frac{abc}{a^2 + b^2 + ab}$

The largest cube that can be chiseled out from a cone:

In this question you will be asked about the side of the largest cube which will be chiseled out of a cone of height ‘h’ cm and radius ‘r’ cm

Side of Cube: $\frac{\sqrt2rh}{h + \sqrt2r}$

The maximum volume of cylinder:

In this you will be asked about the volume of the largest cylinder which can be chiseled out from a right circular cone.

Maximum Volume of the Cylinder: $\pi \times \frac{2r^2}{3}\times \frac{h}{3}$