Introduction:  Solid Geometry is about three dimensional (3D) objects like box, pyramid, topless pyramid, cylinder, cone, topless cone, and sphere. It is called three-dimensional, or 3D because there are three dimensions: width, depth and height. The Geometry of solid figures covers calculation of surface area and volume of selected solid figures, and sphere calculations using pi. It also covers shape diagrams and formulas for geometric solids. This may be used to check homework answers or practice problems.

 box Box  (Rectangular Prism or Cuboid)

Rectangular Prism
   A solid (3-dimensional) object which has six faces that are rectangles. It is a prism because it has the same cross-section along a length.
      Formulas for the rectangular prism:
   Volume = lwh
   Lateral Surface Area = 2(lh + wh)
Note: lw can be top or bottom. Bottom sometimes referred to as the base.
   Total Surface Area = 2(lw + lh + wh)

from User Manual:
   Several mathematical expressions can be written per row. Between each mathematical expression use a colon.
   RedCrab Calculator uses implied multiplication. That means you do not need to include the multiplication operator.
   RedCrab interprets a sequence of letters, for example, ab, as different variables.
      abc : a * b * c
      3ab : 3 * a * b
fig. 1  Box Volume and Surface Area fig. 2  Box Information and Reference
   Volume = lwh
   Lateral Surface Area = 2(lh + wh)
   Surface Area = 2(lw+lh+wh)
   Rectangular Prism
   Rectangular Prism

from User Manual:
  Several math expressions can be writtenin a row.
  Between each math expression, there must be a colon.
        Note. : l = 6: w = 5: h = 4
    The expression: c=√((w/2)²+h²):
  Enter: c = < Ctrl +1 > ((w/2) < Ctrl +2 > + h < Ctrl +2 > )
    The expression: d=√((l/2)²+h²):
  Enter: d = < Ctrl +1 > ((l/2) < Ctrl +2 > + h < Ctrl +2 > )

    l = length of base
    w = width of base
    h = height of pyramid

  c and d = slant height of pyramid faces (c= length face & d = width face)

   Reference: Pyramid Problems
fig. 3  Pyramid Volume and Surface Area fig. 4  Pyramid Information & Reference
   Volume = lwh/3    Reference: Online Rectangular Pyramid Calculator
   Surface Area = lc+wd+lw    Reference: Online Rectangular Pyramid Calculator
   c=√((w/2)²+h²)    d=√((l/2)²+h²)    Reference: Online Rectangular Pyramid Calculator

 Topless Pyramid Information and Reference
  A trunctated pyramid volume is one third the height times (A + B) + sqrt(A B) or h/3(ab+cd)+√(abcd) where A and B are the area of the top and bottom faces.
  A= area (AB) and B= area (CD)
from User Manual:
  RedCrab interprets a sequence of letters, for example, ab, as different variables. Excluding subscript letters, for example XL. Subscript letters always belong to the variable on the left.

It is not allowed to split an expression and continue in the next row.
Wrong: z = 12+14+15+20
Correct: X = 12+14+15+20
Z = X+5+10

  variables: Upper case and lower case letters are different variables.
    Example: v and V are different variables
fig. 5  Topless Pyramid Volume and Surface Area fig. 6  Topless Pyramid Information & Reference
   Volume = (h/3)((ab)+(cd)+√(abcd))    Reference: Online Topless Pyramid calculator
   r = (ab+cd)+(a+c)√((1/4)(b-d)²+h²)    Reference: Online Geametric shapes Calculator
   Surface Area = r+(b+d)√((1/4)(a-c)²+h²)    Reference: Online Truncated Pyramid Calculator

 Cylinder Information and Reference

  The expression: v = πr²h:
  Enter: v = < Ctrl + P > r < Ctrl + 2 > h

    r = radius of cylinder
    h = height of cylinder

Reference: cylinder volume
                 cylinder surface area cylinder calculator
fig. 7  Cylinder Volume & Surface Area fig. 8  Cylinder Information & Reference
   Volume = πr²h   Note: 2πr(r+h) = 2πr² + 2πrh
   Surface Area = 2πr(r+h)   Reference: cylinder formulas

 Cone Information and Reference

  The expression: v = πr²h/3
  Enter: v = < Ctrl + P > r < Ctrl + 2 > h/3

    r = radius of cone base
    h = height of cone
    l = slant height of cone (see diagram)
    l = √(r²+h²)

  Total Surface Area = Lateral Surface Area + Base Area
  Lateral Surface Area = πr√(r²+h²)
  Base Area = πr²
  Total Surface Area = πr²+πr√(r²+h²)
fig. 9  Cone Volume and Surface Area fig. 10  Cone Information and Reference
   Volume = πr²h/3    Reference: Cone formulas
   Surface Area = πr²+πR√(R²+h²)    Reference: Cone Calculator

 Topless Cone Information and Reference

 L= Lateral Surface Area : A= Total Surface Area
 V= Volume

 It is not allowed to split an expression and continue in the next row.
 Wrong: z= 12+14+15+20
 Correct: x= 12+14+15+20
         z= x+5+10

 Topless Cone = Cone Frustum = Truncated Cone

truncated-cone calculator
slant height:s   height:h   small radius:r   large radius:R   s=sqrt((R-r)^2+h^2)   v= volume   a= area
v=pi(R^2+rR+r^2)h/3   a=pi(r(r+s)+R(R+s))
fig. 11  Topless Cone Volume fig. 12  Topless Cone Information & Reference
   Volume = πh/3(R1²+R2²+R1R2)    Reference: Truncated Cone
   Surface Area = π(R1+R2)√((R1-R2)²+h²)+π(R1²+R2²)    Reference: Cone Frustum

 Sphere Information and Reference

  Error message:
  For error location RedCrab marks the cell where an error is detected with a blue frame. It also marks the incorrect formula with a red frame.


  The expression: v = 4/3πr³:
  Enter: v = 4/3 < Ctrl + P > r < Ctrl + 3 >
      r = radius of sphere
fig. 13  Sphere Volume and Surface Area fig. 14  Sphere Information and Reference
   Volume = 4/3πr³    Reference: Sphere Calculator
   Surface Area = 4πR²    Reference: Sphere Calculator

 Sphere 2 Base Segment Information and Reference

 You can write several mathematical expressions on one worksheet. The expression result displays only if terminated with an equal sign.
   a = 27+9 (result a= below)
    a = 36
   a= 27+9 = 36
   a = 27+9 (no result shown)

 a= large radius of segment
 b= small radius segment
 h= height of segment
 R= radius of sphere
fig. 15  Sphere 2 base segment Volume and Surface Area fig. 16  Sphere 2 base segment Information and Reference
   Volume = (π/3)h²(3R-h)    Reference: Volume of a Spherical Segment
   Surface Area = 2πRh    Reference: Surface Area of a Spherical Segment

 Spherical Lune/Wedge Volume and Surface Area
   A Spherical Lune is the surface area of a Spherical Wedge.

  θ = angle in degrees
  r = radius of sphere
     The keys < Ctrl + 2> write the exponent 2.
     The keys < Ctrl + 3> write the exponent 3.
     The keys < Ctrl + Q> write the Greek letter θ
   Reference: Sphere formulas
   Reference: Spherical wedge formulas
   Reference: Spherical wedge formulas
fig 17  Spherical Lune/Wedge Volume and Surface Area fig 18  Spherical Lune/Wedge Information and Reference
   Volume Spherical Wedge= (π/270)r³θ    Reference: Volume of a Spherical Wedge  in degrees
   Surface Area Spherical Lune= (π/90)r²θ    Reference: Surface Area of a Spherical Lune  in degrees

 Spherical Cap Information and Reference

  The expression: A = 2πrh:
  Enter: A = 2 < Ctrl + P > r h

   Implied multiplication means you do not need to include the multiplication operator.
   Example: A = 2πrh equals
          A = 2*π*r*h

     r = radius of sphere
     h = height of cap
     a = radius of cap
fig. 19  Spherical Cap Volume and Surface Area fig. 20  Spherical Cap Information and Reference
   Volume = πh²(3R-h)/3    Reference: Spherical Cap Calculator
   Surface Area = 2πRh    Reference: Spherical Cap

 Spherical Sector Information and Reference

 Save: If you are saving a file for the first time, use SaveAs on the File menu, it will prompt you for a file  name.
 If you are saving a changed file, click Save on the file menu

 Open: Click Open on the File menu. In the Navigation pane, click on the folder or drive that contains the file that you want to open. You can only load a file that is saved with RedCrab before, with the file extension *.rcc.
fig. 21  Spherical Sector Volume and Surface Area fig. 22  Spherical Sector Information and Reference
   Volume = (2/3)πR²h    Reference: Spherical Sector
   Surface Area = 2πRh    Reference: Spherical Sector

   √ = Crtl + 1
   x² (or exponent 2) = F3 + 2
   π = Ctrl + P
Mathematical References and Formulas:
   Kirt Blattenberger,Erie, PA, (814) 833-1967,
Dr. math forum
   Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, 215-895-2000
Handbook of Mathematics
   Calculator Operations, Calculator Usage, Special Keys, Algebra, Geometry, Trigonometry, Higher Concepts of Mathematics
Mathematical Software:
Maxima Software - Computer Algebra System
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Free Math eBooks:
The Handbook of Essential Mathematics
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Index for Geometry
   Math terminology from plane and solid geometry. This includes basic triangle trigonometry as well as a few facts not traditionally taught in basic geometry.
Problems in Plane and Solid Geometry
   Problems in Plane and Solid Geometry v.1 Plane Geometry. Viktor Prasolov translated and edited by Dimitry Leites
Geometry Formulas and Facts
   The present excerpt covers the area of Geometry (minus differential geometry). It was written by Silvio Levy and is reproduced here with permission. This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas
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