Volumes¶
CalcpadCE covers every common 3D body with a one-page worksheet that returns the volume and surface area from a few labelled dimensions.
Pick the body you need — cube, sphere, cone, cylinder, ellipsoid, rectangular or triangular pyramid, rectangular, triangular or right polygonal prism, or one of the spherical-cap and frustum variants — and override the inputs in the Desktop app; the result updates instantly.
Cube¶
Volume \(V = a^3\) and surface area \(S = 6 a^2\) of a cube from a single edge length.
'<div style = "max-width:90mm"><img style="height:65pt;" class="side" alt="cube.png" src = "../../Images/math/volume/cube.png"></div>
'Side length
a = ?
'Volume
V = a^3
'Surface area
S = 6*a^2
' 1
Side length
a = 1
Volume
V = a3 = 13 = 1
Surface area
S = 6 · a2 = 6 · 12 = 6
Sphere¶
Volume \(V = \tfrac{4}{3}\pi r^3\) and surface area \(S = 4\pi r^2\) of a sphere from its radius.
'<div style = "max-width:100mm"><img style="height:60pt;" class="side" alt="sphere.png" src = "../../Images/math/volume/sphere.png"></div>
'Radius
r = ?
'Volume
V = 4/3*π*r^3
'Surface area
S = 4*π*r^2
' 1
Radius
r = 1
Volume
V = 43 · π · r3 = 43 · 3.14 · 13 = 4.19
Surface area
S = 4 · π · r2 = 4 · 3.14 · 12 = 12.57
Cone¶
Volume \(V = \tfrac{1}{3}\pi r^2 h\) and lateral / total surface area of a right circular cone from radius and height.
'<div style = "max-width:90mm"><img style="height:55pt;" class="side" alt="cone.png" src = "../../Images/math/volume/cone.png"></div>
'Dimensions
'Base radius -'r = ?
'Height -'h = ?
'Volume
V = 1/3*π*r^2*h
'Area of side surface
S_o = π*r*sqr(h^2 + r^2)
'Area of base
A = π*r^2
'Total surface area
S = S_o + A
' 1 2
Dimensions
Base radius - r = 1
Height - h = 2
Volume
V = 13 · π · r2 · h = 13 · 3.14 · 12 · 2 = 2.09
Area of side surface
So = π · r · √ h2 + r2 = 3.14 · 1 · √ 22 + 12 = 7.02
Area of base
A = π · r2 = 3.14 · 12 = 3.14
Total surface area
S = So + A = 7.02 + 3.14 = 10.17
Ellipsoid¶
Volume \(V = \tfrac{4}{3}\pi\, a\, b\, c\) and approximate surface area of an ellipsoid from its three semi-axes.
'<div style = "max-width:110mm"><img style="height:70pt;" class="side" alt="ellipsoid.png" src = "../../Images/math/volume/ellipsoid.png"></div>
'Dimensions
a = ?
b = ?
c = ?
'Volume
V = 4/3*π*a*b*c
p = 1.6075
'Surface area (approx.)
S = 4*π*(((a*b)^p + (a*c)^p + (b*c)^p)/3)^(1/p)
' 3 2 1
Dimensions
a = 3
b = 2
c = 1
Volume
V = 43 · π · a · b · c = 43 · 3.14 · 3 · 2 · 1 = 25.13
p = 1.61
Surface area (approx.)
S = 4 · π · ( ( a · b ) p + ( a · c ) p + ( b · c ) p3)1p = 4 · 3.14 · ( ( 3 · 2 ) 1.61 + ( 3 · 1 ) 1.61 + ( 2 · 1 ) 1.613)11.61 = 48.97
Spherical Cap¶
Volume and surface area of a spherical cap from sphere radius and cap height.
'<div style = "max-width:120mm"><img style="height:55pt;" class="side" alt="cap.png" src = "../../Images/math/volume/cap.png"></div>
'Dimensions
'Base radius -'r = ?
'Height -'h = ?
'Radius of sphere
R = (r^2 + h^2)/(2*h)
'Volume
V = π*h/6*(3*r^2 + h^2)
'Area of curved surface
S_o = 2*π*R*h
'Area of base
A = π*r^2
'Total surface area
S = S_o + A
' 2 1
Dimensions
Base radius - r = 2
Height - h = 1
Radius of sphere
R = r2 + h22 · h = 22 + 122 · 1 = 2.5
Volume
V = π · h6 · ( 3 · r2 + h2 ) = 3.14 · 16 · ( 3 · 22 + 12 ) = 6.81
Area of curved surface
So = 2 · π · R · h = 2 · 3.14 · 2.5 · 1 = 15.71
Area of base
A = π · r2 = 3.14 · 22 = 12.57
Total surface area
S = So + A = 15.71 + 12.57 = 28.27
Spherical Segment¶
Volume and surface area of a spherical segment (the slab between two parallel planes cutting a sphere).
'<div style = "max-width:130mm"><img style="height:60pt;" class="side" alt="segment.png" src = "../../Images/math/volume/segment.png"></div>
'Raduses of bases
a = ?','b = ?
'Height
h = ?
'Radius of sphere
R = sqr(((a - b)^2 + h^2)*((a + b)^2 + h^2))/(2*h)
'Volume
V = π*h/6*(3*a^2 + 3*b^2 + h^2)
'Area of side surface
S_o = 2*π*R*h
'Base areas
A = π*a^2
B = π*b^2
'Total surface area
S = S_o + A + B
' 3 2 1
Raduses of bases
a = 3 , b = 2
Height
h = 1
Radius of sphere
R = √ ( ( a − b ) 2 + h2 ) · ( ( a + b ) 2 + h2 ) 2 · h = √ ( ( 3 − 2 ) 2 + 12 ) · ( ( 3 + 2 ) 2 + 12 ) 2 · 1 = 3.61
Volume
V = π · h6 · ( 3 · a2 + 3 · b2 + h2 ) = 3.14 · 16 · ( 3 · 32 + 3 · 22 + 12 ) = 20.94
Area of side surface
So = 2 · π · R · h = 2 · 3.14 · 3.61 · 1 = 22.65
Base areas
A = π · a2 = 3.14 · 32 = 28.27
B = π · b2 = 3.14 · 22 = 12.57
Total surface area
S = So + A + B = 22.65 + 28.27 + 12.57 = 63.5
Cylinder¶
Volume \(V = \pi r^2 h\) and surface area \(S = 2\pi r (r + h)\) of a right circular cylinder.
'<div style = "max-width:100mm"><img style="height:55pt;" class="side" alt="cylinder.png" src = "../../Images/math/volume/cylinder.png"></div>
'Dimensions
'Base radius -'r = ?
'Height -'h = ?
'Area of base
A = π*r^2
'Volume
V = A*h
'Area of side surface
S_o = 2*π*r*h
'Total surface area
S = S_o + 2*A
' 1 2
Dimensions
Base radius - r = 1
Height - h = 2
Area of base
A = π · r2 = 3.14 · 12 = 3.14
Volume
V = A · h = 3.14 · 2 = 6.28
Area of side surface
So = 2 · π · r · h = 2 · 3.14 · 1 · 2 = 12.57
Total surface area
S = So + 2 · A = 12.57 + 2 · 3.14 = 18.85
Rectangular Frustum¶
Volume and surface area of a rectangular frustum (obelisk) from the two parallel rectangular faces and height.
'<div style = "max-width:130mm"><img style="height:70pt;" class="side" alt="rectangular-frustum.png" src = "../../Images/math/volume/rectangular-frustum.png"></div>
'Dimensions
'<table>
'<tr><td>Bottom base -</td><td>'A = ?'</td><td>'B = ?'</td></tr>
'<tr><td>Top base -</td><td>'a = ?'</td><td>'b = ?'</td></tr>
'<tr><td><p>Height -</p></td><td>'h = ?'</td><td> </td></tr>
'</table>
'Heights of side faces
h_a = sqr(h^2 + (B - b)^2/4)
h_b = sqr(h^2 + (A - a)^2/4)
'Volume
V = h/6*(A*b + a*B + 2*(a*b + A*B))
'Area of side faces
S_o = h_a*(A + a) + h_b*(B + b)
'Base areas
A_1 = A*B
A_2 = a*b
'Total surface area
S = S_o + A_1 + A_2
' 8 2 6 4 5
Dimensions
| Bottom base - | A = 8 | B = 2 |
| Top base - | a = 6 | b = 4 |
Height - | h = 5 |
Heights of side faces
ha = h2 + ( B − b ) 24 = 52 + ( 2 − 4 ) 24 = 5.1
hb = h2 + ( A − a ) 24 = 52 + ( 8 − 6 ) 24 = 5.1
Volume
V = h6 · ( A · b + a · B + 2 · ( a · b + A · B ) ) = 56 · ( 8 · 4 + 6 · 2 + 2 · ( 6 · 4 + 8 · 2 ) ) = 103.33
Area of side faces
So = ha · ( A + a ) + hb · ( B + b ) = 5.1 · ( 8 + 6 ) + 5.1 · ( 2 + 4 ) = 101.98
Base areas
A1 = A · B = 8 · 2 = 16
A2 = a · b = 6 · 4 = 24
Total surface area
S = So + A1 + A2 = 101.98 + 16 + 24 = 141.98
Rectangular Pyramid¶
Volume \(V = \tfrac{1}{3} a\, b\, h\) and surface area of a rectangular-base pyramid.
'<div style = "max-width:130mm"><img style="height:65pt;" class="side" alt="rectangular-pyramid.png" src = "../../Images/math/volume/rectangular-pyramid.png"></div>
'Dimensions of base
a = ?','b = ?
'Height
h = ?
'Volume
V = 1/3*a*b*h
'Area of side faces
S_o = a*sqr(h^2 + b^2/4) + b*sqr(h^2 + a^2/4)
'Area of base
A = a*b
'Total surface area
S = S_o + A
' 3 2 1
Dimensions of base
a = 3 , b = 2
Height
h = 1
Volume
V = 13 · a · b · h = 13 · 3 · 2 · 1 = 2
Area of side faces
So = a · h2 + b24 + b · h2 + a24 = 3 · 12 + 224 + 2 · 12 + 324 = 7.85
Area of base
A = a · b = 3 · 2 = 6
Total surface area
S = So + A = 7.85 + 6 = 13.85
Right Polygonal Prism¶
Volume and surface area of a right prism with a regular \(n\)-gon base, from side length and height.
'<div style = "max-width:130mm"><img style="height:65pt;" class="side" alt="polygonal-prism.png" src = "../../Images/math/volume/polygonal-prism.png"></div>
'Dimensions
'Number of sides -'n = ?
'Side length -'a = ?
'Height -'h = ?
'<!--
#if n > 2
'<-->
'Area of base
A = a^2*n/4*cot(180/n)
'Volume
V = A*h
'Surface area
S = 2*A + n*a*h
'<!--
#else
'<-->
'It is required that n > 2
'<!--
#end if
'<--> 6 1 2
Dimensions
Number of sides - n = 6
Side length - a = 1
Height - h = 2
Area of base
A = a2 · n4 · cot(180n) = 12 · 64 · cot(1806) = 2.6
Volume
V = A · h = 2.6 · 2 = 5.2
Surface area
S = 2 · A + n · a · h = 2 · 2.6 + 6 · 1 · 2 = 17.2
Triangular Frustum¶
Volume of a triangular frustum from the two parallel triangular faces and height.
'<div style = "max-width:120mm"><img style="height:60pt;" class="side" alt="triangular-frustum.png" src = "../../Images/math/volume/triangular-frustum.png"></div>
'<p><b>Dimensions:</b></p>
'Bottom base
A = ?','B = ?','C = ?
'Top base
a = ?','b = B*a/A','c = C*a/A
'Height -'h = ?
'<!--
#if (A + B > C)*(B + C > A)*(C + A > B)
'<-->
'<p><b>Bottom base</b></p>
'Semiperimeter
p = 0.5*(A + B + C)
'Area (Heron equation)
A_1 = sqr(p*(p - A)*(p - B)*(p - C))
'<p><b>Top base</b></p>
'Semiperimeter
p = 0.5*(a + b + c)
'Area (Heron equation)
A_2 = sqr(p*(p - a)*(p - b)*(p - c))
'<p><b>Volume</b></p>
V = h/3*(A_1 + sqr(A_1*A_2) + A_2)
'<!--
#else
'<-->
'The sum of each two sides must be greater than the third.
'<!--
#end if
'<--> 10 8 6 5 2
Dimensions:
Bottom base
A = 10 , B = 8 , C = 6
Top base
a = 5 , b = B · aA = 8 · 510 = 4 , c = C · aA = 6 · 510 = 3
Height - h = 2
Bottom base
Semiperimeter
p = 0.5 · ( A + B + C ) = 0.5 · ( 10 + 8 + 6 ) = 12
Area (Heron equation)
A1 = √ p · ( p − A ) · ( p − B ) · ( p − C ) = √ 12 · ( 12 − 10 ) · ( 12 − 8 ) · ( 12 − 6 ) = 24
Top base
Semiperimeter
p = 0.5 · ( a + b + c ) = 0.5 · ( 5 + 4 + 3 ) = 6
Area (Heron equation)
A2 = √ p · ( p − a ) · ( p − b ) · ( p − c ) = √ 6 · ( 6 − 5 ) · ( 6 − 4 ) · ( 6 − 3 ) = 6
Volume
V = h3 · ( A1 + √ A1 · A2 + A2 ) = 23 · ( 24 + √ 24 · 6 + 6 ) = 28
Triangular Pyramid¶
Volume \(V = \tfrac{1}{3} A_b\, h\) of a pyramid with a general triangular base.
'<div style = "max-width:110mm"><img style="height:60pt;" class="side" alt="triangular-pyramid.png" src = "../../Images/math/volume/triangular-pyramid.png"></div>
'Dimensions of base
a = ?','b = ?','c = ?
'Height
h = ?
'<!--
#if (a + b > c)*(b + c > a)*(c + a > b)
'<-->
'Semiperimeter of base
p = 0.5*(a + b + c)
'Area of base (Heron equation)
A = sqr(p*(p - a)*(p - b)*(p - c))
'Volume
V = 1/3*A*h
'<!--
#else
'<-->
'The sum of each two sides must be greater than the third.
'<!--
#end if
'<--> 5 4 3 2
Dimensions of base
a = 5 , b = 4 , c = 3
Height
h = 2
Semiperimeter of base
p = 0.5 · ( a + b + c ) = 0.5 · ( 5 + 4 + 3 ) = 6
Area of base (Heron equation)
A = √ p · ( p − a ) · ( p − b ) · ( p − c ) = √ 6 · ( 6 − 5 ) · ( 6 − 4 ) · ( 6 − 3 ) = 6
Volume
V = 13 · A · h = 13 · 6 · 2 = 4
Triangular Prism¶
Volume and surface area of a right triangular prism, from triangle sides and prism length.
'<div style = "max-width:120mm"><img style="height:60pt;" class="side" alt="triangular-prism.png" src = "../../Images/math/volume/triangular-prism.png"></div>
'Dimensions of base
a = ?','b = ?','c = ?
'Height
h = ?
'<!--
#if (a + b > c)*(b + c > a)*(c + a > b)
'<-->
'Base perimeter
P = a + b + c
'Semiperimeter
p = 0.5*P
'Area of base (Heron equation)
A = sqr(p*(p - a)*(p - b)*(p - c))
'Volume
V = A*h
'Surface area
S = 2*A + h*P
'<!--
#else
'<-->
'The sum of each two sides must be greater than the third.
'<!--
#end if
'<--> 5 4 3 2
Dimensions of base
a = 5 , b = 4 , c = 3
Height
h = 2
Base perimeter
P = a + b + c = 5 + 4 + 3 = 12
Semiperimeter
p = 0.5 · P = 0.5 · 12 = 6
Area of base (Heron equation)
A = √ p · ( p − a ) · ( p − b ) · ( p − c ) = √ 6 · ( 6 − 5 ) · ( 6 − 4 ) · ( 6 − 3 ) = 6
Volume
V = A · h = 6 · 2 = 12
Surface area
S = 2 · A + h · P = 2 · 6 + 2 · 12 = 36
Tetrahedron¶
Volume \(V = \tfrac{a^3}{6\sqrt{2}}\) and surface area \(S = \sqrt{3}\, a^2\) of a regular tetrahedron from its edge length.
'<div style = "max-width:110mm"><img style="height:60pt;" class="side" alt="tetrahedron.png" src = "../../Images/math/volume/tetrahedron.png"></div>
'Side length
a = ?
'Volume
V = sqr(2)*a^3/12
'Surface area
S = sqr(3)*a^2
' 1
Side length
a = 1
Volume
V = √ 2 · a312 = √ 2 · 1312 = 0.118
Surface area
S = √ 3 · a2 = √ 3 · 12 = 1.73
Rectangular Prism¶
Volume \(V = a\, b\, c\) and surface area \(S = 2(a b + b c + c a)\) of a rectangular prism (box).
'<div style = "max-width:130mm"><img style="height:65pt;" class="side" alt="rectangular-prism.png" src = "../../Images/math/volume/rectangular-prism.png"></div>
'Dimensions of base
a = ?','b = ?
'Height
h = ?
'Volume
V = a*b*h
'Surface area
S = 2*(a*b + b*h + a*h)
' 3 2 1
Dimensions of base
a = 3 , b = 2
Height
h = 1
Volume
V = a · b · h = 3 · 2 · 1 = 6
Surface area
S = 2 · ( a · b + b · h + a · h ) = 2 · ( 3 · 2 + 2 · 1 + 3 · 1 ) = 22
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