# Geometry, properties, mensuration formula 3D

## Three-dimensional shapes (Solid shapes)

Three-dimensional shapes have length, width and height. Three-dimensional shapes can only be defined in a 3D Co-ordinate geometry. The properties and formulas of all three-dimensional shapes in mathematics are given below.

### Cube

The cube is made up of 6 squares. its length, width and height are equal.

Properties-

Surface- 6

Edges- 12

Vertices- 8

Formulas-

Total surface area = 6(side)2

Lateral surface area = 4(side)2

Volume = (side)3

Diagonal = $$side\sqrt {3}$$

### Cuboid

It is made up of 6 rectangles. its length, width and height are different.

Properties-

Surface- 6

Edges- 12

Vertices- 8

Formulas-

Total surface area = 2(L✕W + W✕H + H✕L)

Lateral surface area = 2(W✕H + H✕L)

Volume = Leight✕Width(breath)✕Height =L✕W✕H

Diagonal = $$\sqrt {L^2 + W^2 + H^2}$$

#### Euler's Formula

Euler's Formula for polyhedral figures like cubes and cuboids.

V + F - E = 2

F- Surfaces/Faces

E- Edges

V- Vertices

### Cylinder

The top and bottom surfaces of the cylinder are circular and the side is covered with a curved surface.

Formulas-

Curved surface area = 2πrh

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

Volume = πr2h

r ⟶ Radius of the cylinder

h ⟶ Height of cylinder

### Hollow cylinder

The hollow cylinder's top and bottom surfaces are ring surfaces with two curved surfaces. Outer curved surface and inner-curved surface.

Formulas-

Curved surface area = 2πRh + 2πrh = 2πrh(R + r)

Total surface area  = 2πrh(R + r) + 2(πR2 - πr2) =  2π(R + r) (h + R - r)

Volume = πR2h - πr2h = πh(R2 - r2)

R ⟶ Radius of the outer cylinder

r ⟶ Radius of the inner cylinder

h ⟶ Height of cylinder

### Cone

Cone has a flat circular base and a pointed top. The pointed top of the cone is called the 'Apex' and the side is covered with a curved surface.

Formulas-

Curved surface area = $$\frac {1}{2} ✕ perimeter \space of \space base ✕ slant \space height$$

= $$\frac {1}{2} ✕ 2πr ✕ l$$ = πrl

Total surface area  = πrl + πr2 =  πr(l + r)

Volume = $$\frac {1}{3} ✕ πr^2h$$

l = $$\sqrt {r^2 + h^2}$$

h ⟶ Height of the cone

r ⟶ Radius of the cone

l ⟶ Slant height of the cone

### Frustum

If we cut off the top end of the cone, we get a frustum.

Formulas-

Curved surface area = πrl + πRl

Total surface area  = πl (r + R) + πr2 + πR2 =  πl(r + R) + π(r2 + R2)

Volume = $$\frac {πh}{3} ✕ (r^2 + R^2 + Rr)$$

l = $$\sqrt {(R - r)^2 + h^2}$$

h ⟶ Height of the cone

R ⟶ Radius of the base

r ⟶ Radius of the top

l ⟶ Slant height of the frustum

### Sphere

The sphere has no vertex or edge. It looks similar from all sides just like a ball.

Formulas-

Curved surface area or Total surface area = 4πr2

Volume = $$\frac {4}{3} ✕ πr^3$$

#### Semisphere

If the sphere is cut from the middle, then a semisphere is obtained.

Formulas-

Curved surface area = 2πr2

Total surface area  = 2πr2 + πr2 = 3πr2

Volume = $$\frac {2}{3} ✕ πr^3$$

### Prism

The prism consists of identical polygons at the top and bottom.

Formulas-

Lateral surface area = Perimeter of base ✕ Height

Total surface area  = Perimeter of base ✕ Height + 2✕area of the triangle

Volume = Area of base ✕ height

### Pyramid

It has a square at its base but has triangles of equal size on all four sides.

Lateral surface area = $$\frac {1}{2}✕Perimeter \space of \space base ✕ slant \space Height$$

Total surface area  = $$\frac {1}{2}✕Perimeter \space of \space base ✕ slant \space Height + area \space of \space base$$

Volume = $$\frac {1}{3}✕area \space of \space base ✕ Height$$

l = $$\sqrt {h^2 + a^2}$$

h ⟶ Height of the pyramid

l ⟶ Slant height of the pyramid

### Tetrahedron

Its four faces are of the same triangle(equilateral triangle) shape.

Lateral surface area = $$\frac {3}{4}✕\sqrt {3} (side)^2$$

Total surface area  = $$\sqrt {3} (side)^2$$

Volume = $$\frac {\sqrt 2}{12}✕(side)^3$$