'n Keël met radius *r*, hoogte *h* en lengte van skuinssy *l*.
'n **Keël** is 'n driedimensionele geometriese vorm wat deur twee parameters beskryf word. Dit kan vergelyk word met die vorm van 'n (afgeslote) heksehoed.

Oppervlak en Volume
-------------------

Indien {\displaystyle r} die straal van die sirkelvormige basis van die keël is, {\displaystyle h} die hoogte is en {\displaystyle l} die lengte van die skuinste van die punt van die keël tot die sirkelvormige rand, dan is:

{\displaystyle {\text{Oppervlak kegel}}=\pi lr=\pi r{\sqrt {(r^{2}+h^{2})}}}
{\displaystyle {\text{Oppervlak basis}}=\pi r^{2}}
{\displaystyle {\text{Totale oppervlak}}=\pi lr+\pi r^{2}=\pi r\left(r+{\sqrt {(r^{2}+h^{2})}}\,\right)}
{\displaystyle {\text{Volume}}={\frac {1}{3}}\pi r^{2}h}

Gedeeltelike volume
-------------------

Keël.
Hier volg die oppervlak en volume van 'n keël.

**Totale volume**

{\displaystyle {\text{Totale volume}}={\frac {1}{3}}\pi R^{2}H}
**Gedeeltelike volume**

* {\displaystyle R}, {\displaystyle H}: Radius en hoogte van groot keël
* {\displaystyle r}: Klein radius
* {\displaystyle h}: Afstand tussen groot en klein radius (sien skets)

In terme van R, H en h: {\displaystyle {\text{Gedeeltelike volume}}=\pi {\frac {h}{H}}R^{2}\left(H-h+{\frac {h^{2}}{3H}}\right)}

In terme van R, H en r: {\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}

In terme van R, r en h: {\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{3}{\frac {h}{R-r}}\left(1-{\frac {r^{3}}{R^{3}}}\right)}

Afleiding van gedeeltelike volume
---------------------------------

### In terme van R, H en h

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi r^{2}\left(H-h\right)}

{\displaystyle ={\frac {1}{3}}\pi \left[R^{2}H-r^{2}\left(H-h\right)\right]}
{\displaystyle r} kan soos volg geskryf word in terme van {\displaystyle R}, {\displaystyle H} en {\displaystyle h}:

{\displaystyle {\frac {r}{H-h}}={\frac {R}{H}}}
{\displaystyle r={\frac {R}{H}}\left(H-h\right)}
Hierdie kan weer terug vervang word in die oorspronklike formule:

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi \left[R^{2}H-{\frac {R^{2}}{H^{2}}}\left(H-h\right)^{3}\right]}

{\displaystyle \left(H-h\right)^{3}=H^{3}-3H^{2}h+3Hh^{2}-h^{3}}
Vervang weer terug in hoof formule:

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}\left[H-{\frac {1}{H^{2}}}\left(H^{3}-3H^{2}h+3Hh^{2}-h^{3}\right)\right]}

{\displaystyle ={\frac {1}{3}}\pi R^{2}\left[H-H+3h-3{\frac {h^{2}}{H}}+{\frac {h^{3}}{H^{2}}}\right]}
{\displaystyle ={\frac {1}{3}}\pi hR^{2}\left[3-3{\frac {h}{H}}+{\frac {h^{2}}{H^{2}}}\right]}
{\displaystyle =\pi hR^{2}\left[1-{\frac {h}{H}}+{\frac {h^{2}}{3H^{2}}}\right]}
{\displaystyle =\pi {\frac {h}{H}}R^{2}\left[H-h+{\frac {h^{2}}{3H}}\right]}
### In terme van R, H en r

{\displaystyle {\text{Gedeeltelike volume}}={\text{groot kegel}}-{\text{klein kegel}}}

Stel:

* {\displaystyle x} = hoogte van klein keël
* {\displaystyle r} = radius van klein keël

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi r^{2}x}

Die verhouding tussen {\displaystyle x}, {\displaystyle r}, {\displaystyle H} en {\displaystyle R} is:

{\displaystyle {\frac {x}{r}}={\frac {H}{R}}\qquad \Rightarrow \qquad x={\frac {rH}{R}}}
Dus:

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi {\frac {r^{3}H}{R}}}

{\displaystyle ={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}

  

### In terme van R, r en h

Kry eers H in terme van R, r en h:

{\displaystyle {\frac {H}{R}}={\frac {H-h}{r}}}
{\displaystyle {\frac {H}{R}}={\frac {H}{r}}-{\frac {h}{r}}}
{\displaystyle {\frac {H}{r}}-{\frac {H}{R}}={\frac {h}{r}}}
{\displaystyle H\left({\frac {1}{r}}-{\frac {1}{R}}\right)={\frac {h}{r}}}
{\displaystyle H\left({\frac {R-r}{Rr}}\right)={\frac {h}{r}}}
{\displaystyle H={\frac {Rrh}{r\left(R-r\right)}}}
{\displaystyle H={\frac {Rh}{R-r}}}
Vervang hierdie nou in die formule wat in terme van R, H en r is:

{\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}

{\displaystyle ={\frac {1}{3}}\pi R^{3}{\frac {h}{R-r}}\left(1-{\frac {r^{3}}{R^{3}}}\right)}

Kyk ook
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* Omtrek, oppervlakte en volume