Constants

• ans : the last calculated result
• PI : pi = 3.14159265...
• E : e = 2.71828182...
• LOG2E : log of e base 2
• LOG10E : log of e base 10
• LN2 : log of 2 base e
• LN10 : log of 10 base e
• SQRT2 : square root of 2
• SQRT1_2 : square root of 1/2

Functions

• abs(a) : the absolute value of a
• acos(a) : arc cosine of a
• asin(a) : arc sine of a
• atan(a) : arc tangent of a
• atan2(a,b) : arc tangent of a/b
• ceil(a) : integer closest to a and not less than a
• cos(a) : cosine of a
• exp(a) : exponent of a
• floor(a) : integer closest to and not greater than a
• log(a) : log of a base e
• max(a,b) : the maximum of a and b
• min(a,b) : the minimum of a and b
• pow(a,b) : a to the power b
• random() : pseudorandom number in the range 0 to 1
• round(a) : integer closest to a
• sin(a) : sine of a
• sqrt(a) : square root of a
• tan(a) : tangent of a

Examples

Some simple expressions:

• 2³ = pow(2,3) = 8
• log(3) = 1.099
• 2 * (3 + 2) = 10

A more complicated formula for calculating the pressure drop in compressed air pipe lines (using same values as the link)

• dp = 7.57 q^1.85 L 10⁴ / d⁵ p = 7.57 * pow(10,1.85) * 100e4 / (pow(52.501,5) * 7) = 0.19

Note that the expressions can be written in any editor and transferred to the calculator display by using "copy and paste".

Display

This calculator can handle input numbers in several different bases:

• Decimal (Base 10): Numbers that do not start with a zero like 15 or 3.14e15. Decimal numbers can contain digits 0-9, decimals, and scientific notation.
• Hexadecimal (Base 16): Integers that start with a zero x like 0x1a5. Hexadecimal numbers can contain digits 0-9 and a-f (or A-F) but no decimal or scientific notation.
• Octal (Base 8): Integers that start with a zero like 073. Octal numbers can contain digits 0-7 but no decimal or scientific notation.
• Binary (Base 2): Integers that start with a zero b like 0b101. Binary numbers can contain digits 0 and 1 but no decimal or scientific notation.

^ is a bitwise xor operation. To raise a number to a power use pow() function.

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Number	Square	Cube	Square Root	Cubic Root
1	1	1	1.000	1.000
2	4	8	1.414	1.260
3	9	27	1.732	1.442
4	16	64	2.000	1.587
5	25	125	2.236	1.710
6	36	216	2.449	1.817
7	49	343	2.646	1.913
8	64	512	2.828	2.000
9	81	729	3.000	2.080
10	100	1000	3.162	2.154
11	121	1331	3.317	2.224
12	144	1728	3.464	2.289
13	169	2197	3.606	2.351
14	196	2744	3.742	2.410
15	225	3375	3.873	2.466
16	256	4096	4.000	2.520
17	289	4913	4.123	2.571
18	324	5832	4.243	2.621
19	361	6859	4.359	2.668
20	400	8000	4.472	2.714
21	441	9261	4.583	2.759
22	484	10648	4.690	2.802
23	529	12167	4.796	2.844
24	576	13824	4.899	2.884
25	625	15625	5.000	2.924
26	676	17576	5.099	2.962
27	729	19683	5.196	3.000
28	784	21952	5.292	3.037
29	841	24389	5.385	3.072
30	900	27000	5.477	3.107
31	961	29791	5.568	3.141
32	1024	32768	5.657	3.175
33	1089	35937	5.745	3.208
34	1156	39304	5.831	3.240
35	1225	42875	5.916	3.271
36	1296	46656	6.000	3.302
37	1369	50653	6.083	3.332
38	1444	54872	6.164	3.362
39	1521	59319	6.245	3.391
40	1600	64000	6.325	3.420
41	1681	68921	6.403	3.448
42	1764	74088	6.481	3.476
43	1849	79507	6.557	3.503
44	1936	85184	6.633	3.530
45	2025	91125	6.708	3.557
46	2116	97336	6.782	3.583
47	2209	103823	6.856	3.609
48	2304	110592	6.928	3.634
49	2401	117649	7.000	3.659
50	2500	125000	7.071	3.684
51	2601	132651	7.141	3.708
52	2704	140608	7.211	3.733
53	2809	148877	7.280	3.756
54	2916	157464	7.348	3.780
55	3025	166375	7.416	3.803
56	3136	175616	7.483	3.826
57	3249	185193	7.550	3.849
58	3364	195112	7.616	3.871
59	3481	205379	7.681	3.893
60	3600	216000	7.746	3.915
61	3721	226981	7.810	3.936
62	3844	238328	7.874	3.958
63	3969	250047	7.937	3.979
64	4096	262144	8.000	4.000
65	4225	274625	8.062	4.021
66	4356	287496	8.124	4.041
67	4489	300763	8.185	4.062
68	4624	314432	8.246	4.082
69	4761	328509	8.307	4.102
70	4900	343000	8.367	4.121
71	5041	357911	8.426	4.141
72	5184	373248	8.485	4.160
73	5329	389017	8.544	4.179
74	5476	405224	8.602	4.198
75	5625	421875	8.660	4.217
76	5776	438976	8.718	4.236
77	5929	456533	8.775	4.254
78	6084	474552	8.832	4.273
79	6241	493039	8.888	4.291
80	6400	512000	8.944	4.309
81	6561	531441	9.000	4.327
82	6724	551368	9.055	4.344
83	6889	571787	9.110	4.362
84	7056	592704	9.165	4.380
85	7225	614125	9.220	4.397
86	7396	636056	9.274	4.414
87	7569	658503	9.327	4.431
88	7744	681472	9.381	4.448
89	7921	704969	9.434	4.465
90	8100	729000	9.487	4.481
91	8281	753571	9.539	4.498
92	8464	778688	9.592	4.514
93	8649	804357	9.644	4.531
94	8836	830584	9.695	4.547
95	9025	857375	9.747	4.563
96	9216	884736	9.798	4.579
97	9409	912673	9.849	4.595
98	9604	941192	9.899	4.610
99	9801	970299	9.950	4.626
100	10000	1000000	10.000	4.642

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Cube

• all figures made with Google SketchUp and the Engineering ToolBox SketchUp Plugin.
  

Volume

V = a³  (1)

where

V = volume (m³, ft³)

a = side (m, ft)

Surface Area

A₀ = 6 a²  (1b)

where

A₀ = surface area (m², ft²)

Diagonal

d = a 3^1/2  (1c)

where

d = innside diagonal (m, ft)

Cuboid

Volume

V = a b c         (2)
where
V = volume of solid (m³, ft³)
a = length of rectangular prism (m, ft)
b = width of rectangular prism (m, ft)
c = height of rectangular prism (m, ft)

Diagonal

d =  (a₂ + b₂ + c₂)^1/2         (2b)

Surface Area

A₀ = 2 (a b + a c + b c)         (2c)
where
A₀ = surface area of solid (m², ft²)

length
width
height

Volume:  
Surface:  

Parallelepiped

Volume

V = A₁ h  (3a)

where

A₁ = side area (m², ft²)

Related Sketchup Components from The Engineering ToolBox

• Geometric Figures - Cylinders, Boxes, Cones, Planes, Spheres, Lines, Curves and more..

- free Engineering ToolBox plugin for use with the amazing Sketchup 3D drawing application.

Cylinder

Volume

V = π/4 d² h = π r² h         (4a)
where
d = diameter of cylinder (m, ft)
r = radius of cylinder (m, ft)
h = height of cylinder (m, ft)

Surface

A = 2 π r h + 2 π r²         (4b)

radius
height

Volume:  
Surface:  

Hollow Cylinder

Volume

V = π/4 h (D² - d²)   (5)

Pyramid

Volume

V = 1/3 h A₁         (6)
where
A₁ = area of base (m², ft²)
h = perpendicular height of pyramid (m, ft)

Surface

A = ∑ sum of areas of triangles forming sides + A_b         (6b)
where
the surface areas of the triangular faces will have different formulas for different shaped bases

area of base
perpendicular height

Volume:  

Frustum of Pyramid

Volume

V = h/3 ( A₁ + A₂ + (A₁ A₂)^1/2) 

   ≈ h (A₁ + A₂)/2  (7)

Cone

Volume

V = 1/3 π r² h         (8a
where
r = radius of cone base (m, ft)
h = height of cone (m, ft)

Surface

A = π r l + π r²         (8b)
where
l = (r² + h²)^1/2 = length of cone side (m, ft)

radius
height

Volume:  
Surface:  

Side

m = (h² + r²)^1/2   (8c)

A₂ / A₁ = x² / h²   (8d)

Frustum of Cone

Volume

V = π/12 h (D² + D d + d²)   (9a)

m = ( ( (D - d) / 2 )² + h²)^1/2    (9c)

Sphere

Volume

V = 4/3 π r³  
= 1/6 π d³     (10a)
where
r = radius of sphere (m, ft)

Surface

A = 4 π r² 
= π d²     (10b)

radius

Volume:  
Surface:  

Zone of a Sphere

V = π/6 h (3a² + 3b² + h)    (11a)

A_m = 2 π r h    (11b)

A₀ = π (2 r h + a² + b²)   (11c)

Segment of a Sphere

V = π/6 h (3/4 s² + h²) 

=   π h² (r - h/3)    (12a)

A_m = 2 π r h  

=  π/4 (s² + 4 h²) (12b)

Sector of a Sphere

V = 2/3 π r² h    (13a)

A₀ = π/2 r (4 h + s)   (13b)

Sphere with Cylindrical Boring

V = π/6  h³    (14a)

A₀ = 4 π ((R + r)³ (R - r))^1/2  

= 2 π h (R + r)  (14b)

h = 2 (R² - r²)^1/2    (14c)

Sphere with Conical Boring

V = 2/3 π R² h   (15a)

A₀ = 2 π R (h + (R² - h²/4)^1/2)   (15b)

h = 2 (R² - r²)^1/2    (15c)

Torus

V = π²/4 D d²    (16a)

A₀ = π² D d   (16b)

Sliced Cylinder

V = π/4 d² h   (17a)

A_m = π d h (17b)

A₀ = π r (h₁ + h₂ + r + (r² + (h₁ - h₂)²/4)^1/2)   (17c)

Ungula

V = 2/3 r² h   (18a)

A_m = 2 r h (18b)

A₀ = A_m + π/2 r² + π/2 r (r² + h²)^1/2  (18c)

Barrel

V ≈ π/12 h (2 D² + d²)   (19a)

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Vector Addition

Online vector calculator - add vectors with different magnitude and direction

In mechanics there are two kind of quantities

• scalar quantities with magnitude - time, temperature, mass etc.
• vector quantities with magnitude and direction - velocity, force etc.

When adding vector quantities both magnitude and direction are important. Common methods adding coplanar vectors (vectors acting in the same plane) are

• the parallelogram law
• the triangle rule
• trigonometric calculation

The Parallelogram Law

The procedure of "the parallelogram of vectors addition method" is

• draw vector 1  using appropriate scale and in the direction of its action
• from the tail of vector 1 draw vector 2 using the same scale in the direction of its action
• complete the parallelogram by using vector 1 and 2 as sides of the parallelogram
• the resulting vector is represented in both magnitude and direction by the diagonal of the parallelogram

The Triangle Rule

The procedure of "the triangle of vectors addition method" is

• draw vector 1 using appropriate scale and in the direction of its action
• from the nose of the vector draw vector 2 using the same scale and in the direction of its action
• the resulting vector is represented in both magnitude and direction by the vector drawn from the tail of vector 1 to the nose of vector 2

Trigonometric Calculation

The resulting vector of two coplanar vector can be calculated by trigonometry using "the cosine rule" for a non-right-angled triangle.

F_R = [ F₁² + F₂² − 2 · F₁ · F₂ · cos(180^o - (α + β)) ]^1/2         (1)
where
F = the vector quantity - force, velocity etc.
α + β = angle between vector 1 and 2

The angle between the vector and the resulting vector can be calculated using "the sine rule" for a non-right-angled triangle.

α = sin^-1[ F_1 · sin(180^o - (α + β)) / F_R ]         (2)
where
α + β = the angle between vector 1 and 2 is known

Example - Calculating Vector Forces

A force 1 of magnitude 3 kN is acting in a direction 80^o from a force 2 of magnitude 8 kN.
The resulting force can be calculated as

F_R = [ (3 (kN))² + (8 (kN))² - 2 · 5 (kN) · 8 (kN) · cos(180^o - (80^o)) ]^1/2
    = 9 (kN)

The angle between vector 1 and the resulting vector can be calculated as

α = sin^-1[ 3 (kN) · sin(180^o - (80^o)) / 9 (kN) ]
    = 19.1^o

The angle between vector 2 and the resulting vector can be calculated as

α = sin^-1[ 8 (kN) · sin(180^o - (80^o)) / 9 (kN) ]
    = 60.9^o

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Elementary Curves

Ellipse, Circle, Hyperbola, Parabola - Parallel, Intersecting and Coincident Lines 

The simplest curve is the straight line. The next complexity are the second degree or conics curves.

Non-degenerate Conics

Ellipse

x²/a² + y²/b² = 1        (1)
where
a, b = intrinsic parameters

Circle

Circle, a special ellipse, can be expressed as

x² + y² = a²       (2)
where
a = radius

Hyperbola

x²/a² - y²/b² = 1        (3)
where
a, b = intrinsic parameters

Parabola

y² - 2 l x = 0       (4)
where
l = intrinsic parameter

Degenerate Conics

Parallel Lines

x² - a² = 0        (5)
where
a = half the distance between the parallel lines

Intersecting Lines

x²/a² - y²/b² = 0        (6)
where
tan^-1(b/a) = half the angle between the intersecting lines

Coincident Lines

x² = 0        (7)

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New York: Facebook has invited journalists to the unveiling of what it calls its "new home on Android." Next Thursday's event will take place at the company's Menlo Park, California, headquarters.

Facebook isn't providing further details. There has been speculation about a "Facebook phone" for a few years. Facebook has long said it would not make its own phone. Rather, such a phone would likely integrate Facebook deeper into the phone's software.

Citing unnamed sources, the tech blog TechCrunch says Facebook Inc. will launch a modified version of Android that embeds Facebook deeply into the operating system, on a phone made by HTC Corp.

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