Circular truncated cone

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A circular truncated cone (えんすいだい British: circular truncated cone) is the frustum which assumed Japanese yen a base. In other words, I cut a cone on a parallel plane in a base and am the stereogram except the part of the conelet.

The form of the pudding is generally a circular truncated cone. It is a figure made questions for by examination mathematics particularly a junior high school entrance examination of Japan well.

Table of contents

Volume

Beginning class-like derivation

You know the petrosal volume formula, but should think the integral calculus calculation to have removed conelet from the cone to find a volume when I do not know it (many primary and secondary students of Japan are so). When assume r₂, height h at the radius of the r₁, lower base at the radius of the upper side of the circular truncated cone; height H of the original big cone

${\displaystyle {\frac {r_{1}}{r_{2}}}={\frac {H-h}{H}}}$ 

I satisfy を. When I solve this for H,

${\displaystyle H={\frac {r_{2}h}{r_{2}-r_{1}}}}$ 

となる. Volume of the circular truncated cone V

${\displaystyle V={\frac {1}{3}}\pi r_{2}^{2}H-{\frac {1}{3}}\pi r_{1}^{2}(H-h)}$ 

When I substitute であるから, H like the point and arrange it,

${\displaystyle V={\frac {\pi h}{3}}(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2})}$ 

となる. When this assumes the area of the S₁, lower base S₂ at the area of the upper side

${\displaystyle V={\frac {h}{3}}{\Bigl (}S_{1}+{\sqrt {S_{1}S_{2}}}+S_{2}{\Bigr)}}$ 

I can express とも.

Integral calculus

You may integrate an area of Japanese yen to become the base to find a volume.

${\displaystyle V=\int _{0}^{h}\pi \left({\frac {r_{1}-r_{2}}{h}}x+r_{2}\right)^{2}\,dx={\frac {\pi h}{3}}(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}).}$ 

Or I can consider it to be a body of rotation which turned a trapezoid. The distance from an axis to the trapezoidal center of gravity

${\displaystyle {\frac {r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}}{3(r_{1}+r_{2})}}}$ 

When であることに is careful and uses a theorem of cataplasms – Guldin,

${\displaystyle V=2\pi {\frac {r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}}{3(r_{1}+r_{2})}}{\frac {h(r_{1}+r_{2})}{2}}={\frac {\pi h}{3}}(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2})}$ 

となる.

Allied item

• Body of rotation
• Scape
• Rotary hyperboloid
• Elizabeth color

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