# Formulas For Geometry

## Formulas & 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 To Solve Geometry

1. Largest possible sphere that can be chiseled out from a cube of side “a” cm.

Diagonal of sphere is a, so radius= $\frac{a}{2} \$
Remaining empty space in the cube = a3$\frac{π a^3}{6} \$
2. Largest possible cube that can be chiseled out from a sphere of radius “a” cm.

Here OA= radius of the sphere. So, diameter of the sphere = 2a
Let the side of the square = x, then the diagonal of the cube = √3x
⇒√3x = 2a ⇒ x =$\frac{2a}{√3} \$
3. The largest square that can be inscribed in a right angled triangle ABC when one of its vertices
coincide with the vertex of right of the triangle

Side of square = $\frac{ab}{a+b} \$ and are of the square = $(\frac{ab}{a+b})^2 \$
4. The largest square that can be inscribed in a semi circle of radius ‘r’ units

Area of the square =$\frac{3}{5} \$$r^2$
5. The largest square that can be inscribed in a quadrant of radius ‘r’ cm.

Side of the square = $\frac{r}{√2} \$, and area of the square = $\frac{r^2}{2} \$
6. The largest square that can be inscribed in a right angled triangle ABC when one of its vertices lies on the hypotenuse of the triangle

side of square =$\frac{abc}{a^2+b^2+ab} \$
7. The largest cube that can be chiseled out from a cone of height ‘h’ cm and radius of ‘r’ cm.
Square side = $\frac{√2rh}{h+√2r} \$
8. Find the maximum volume of cylinder that can be made out of a cone of radius ‘r’ and height ‘h

Maximum volume of the cylinder= $π$ $\frac{2r}{3}^2 \$ $\frac{h}{3} \$