Semicircle perimeters and cone-cylinder surface proportions

The semicircle's perimeter equals a quarter-circle's circumference

The perimeter of the semicircle equals the circumference of a circle worth half of that semicircle, and hence one-quarter of the whole circle, invoking the rule that circle is to circle as the square of one diameter to the square of the other. It follows that the circumferences of two circles, one quadruple the other, are as double to one another.

Surfaces of cone and cylinder on the same base

When a cone and a cylinder share the same circular base and have sides of equal length, the cylinder's surface is double the cone's, quadruple the base, and double the cone's lateral surface. The circumference is taken as the diameter multiplied by 3 and 1/7, and the largest triangle drawn from a square holds half of that square.

The largest cylinder drawn from a cube

The largest cylinder that can be taken from the cube has a rectangular (lateral) surface equal to four circles similar to its face. The note stands in the left margin among the solid-geometry statements.

Fraction reductions and a subtraction table

Leonardo runs a series subtracting 4 from successive even numbers and reducing the remainders to lowest terms: take 4 from 10 leaves 6 (6/10 = 3/5), from 14 leaves 10/14 = 5/7, from 18 gives 14/18 = 7/9, and so on. Related rows reduce fractions like 32/36 to 16/18 to 8/9, under the heading 'four are always taken away.'

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Due figure con lettere.

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Nel margine sinistro del foglio:

Figura con lettere.

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Nella metà inferiore del foglio:

Al centro, capovolte, figure con scritte e numeri:

Nella metà superiore del foglio, capovolto:

Due figure con lettere:

Two figures with letters.

Beside the figures:

In the left margin of the sheet:

Figure with letters.

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In the center, upside down, figures with writing and numbers:

In the upper half of the sheet, upside down:

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Semicircle perimeters and cone-cylinder surface proportions

On this page

The semicircle's perimeter equals a quarter-circle's circumference

Surfaces of cone and cylinder on the same base

The largest cylinder drawn from a cube

Fraction reductions and a subtraction table