In practical calculations of specialists in industrial automation, as well as in the design of automated control systems for technological processes, it is often required to calculate the surface area, and the volume of various geometric shapes.
These calculations are implemented in the Android application "Instrumentation & Automation". Below are the formulas by which the program calculates the area and volume of geometric shapes.
Table 1.
Calculating the lengths and areas of plane figures  
S  area  n  number of sides of a regular polygon 
p  semiperimeter  r  radius of inscribed circle 
P  perimeter  R  radius of the circumscribed circle 
h  height  α  angle in radians 
C  circumference  β  angle in degrees 
l  arc length  
Triangle  
S = (b h) / 2; S = (a b c) / (4 R); S = p r; S = √(p(pa)(pb)(pc)); p = (a + b + c) / 2; 

Parallelogram  
S = b h;  
Rhombus  
S = (D d) / 2;  
Rectangle  
S = a b; S = a√(d^{2}  a^{2}); S = b√(d^{2}  b^{2}); d = √(a^{2} + b^{2}); 

Trapezium  
S = ((a + b) / 2) h;  
Regular polygon  
S_{n} = (n a_{n} r) / 2; S_{n} = ((n a_{n}) / 2) √(R^{2}  (r^{2} / 4)); P_{n} = 2 n R Sin(π / n); 

Circle  
S = π r^{2}; S = (π d^{2}) / 4; C = 2 π r; C = π d; 

Sector  
l = α r; S = (r^{2} α) / 2; l = (π r β) / 180; S = (π r^{2} β) / 360; 

Segment  
c = 2 √(h (2 r  h)); S = ½ (r l  c (r  h)); 

Ring  
S = π (R^{2}  r^{2});  
Ring sector  
S = α (R^{2}  r^{2}) / 2; S = β π (R^{2}  r^{2}) / 360; 

Ellipse  
S = π a b;  
Calculation of areas of surfaces and volumes of geometric bodies  
S  surface area  r  radius of a circle 
S_{sid}  side surface area  R  ball radius 
S_{bs}  base area  D  ball diameter 
P_{bs}  base perimeter  H  height 
V  volume  a  apothem 
l  generatrix  
Rectangular parallelepiped  
S = 2 (ab + bc + ac); V = a b c; 

Cube  
S = 6 a^{2}; V = a^{3}; 

Regular pyramid  
S_{sid} = ½ P_{bs} a; V = (S_{bs} H) / 3; 

Regular truncated pyramid  
S_{sid} = ½ (P_{bs1} + P_{bs2}) a; V = H (S_{bs1}+S_{bs2} + √(S_{bs1}S_{bs2})) / 3; 

Cylinder  
S_{sid} = 2 π r H; S = 2 π r H + 2 π r^{2}; V = π r^{2} H; 

Hollow cylinder  
S_{sid} = 2 π H (r_{1} + r_{2}); V = π H (r_{2}^{2}  r_{1}^{2}), r_{2} > r_{1}; 

Cone  
S_{sid} = π r l; S_{sid} = π r √(r^{2} + H^{2}); V = (π r^{2} H) / 3; 

Truncated cone  
S_{sid} = π l (r_{1} + r_{2}); V = π H (r_{1}^{2} + r_{2}^{2} + r_{1} r_{2}) / 3; 

Ball  
S = 4 π R^{2}; S = π D^{2}; V = 4 π R^{3} / 3; V = π D^{3} / 6; 

Spherical sector  
S_{sid} = π R (r + 2H); V = (2 π R^{2} H) / 3; 

Spherical segment  
S_{sid} = 2 π R H; V = (π H (3 r^{2} + H^{2})) / 6; V = (π H^{2} (3 R  H)) / 3; 