XXIX. Mensuration.

MENSURATIÓN COMPREHENDS THE ADMEASUREMENT OF SURFACES, AND THE SOLID CONTENTS OF BODIES.

TO FIND THE AREA OF ANY PARALLELOGRAM, WHETHER IT BE A SQUARE, A RECTANGLE, A RHOMBUS, OR A RHOMBOID.

RULE.-Multiply the length by the perpendicular breadth or height, and the product will be the area.

What is the area of a rhomboid, the length being 7, and the perpendicular being 4?

TO FIND THE AREA OF A TRIANGLE.

Ans. 28.

RULE.-Multiply the base by the perpendicular height, and take half the product for the area.

What is the area of a triangle, the base being 9, and the perpendicular 81? Ans. 381.

TO MEASURE ANY REGULAR POLYGON.

RULE.-Multiply the length of one of the sides by the number of sides; then multiply this product by the half of a perpendicular let fall from the centre of the figure to the middle of one of the sides, and the product will be the area of the polygon.

1. What is the area of a regular pentagon, each of its sides being 95, and its perpendicular 65,367 Ans. 15523

2. What is the area of a regular nonagon, each of its sides being 5, and its perpendicular 6,8686935?

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Ans. 154,54+.

THE DIAMETER OF A CIRCLE BEING GIVEN, TO FIND THE

CIRCUMFERENCE.

RULE.—As 7 are to 22, so is the diameter of a circle to the circumference. Or, as 113 are to 355, or 1 is to 3,14159, so is the diameter to the circumference.

1. What is the circumference of a circle, whose diameter is 12? Ans. 37,69908.

2. What is the circumference of a circle, its diameter being 6? Ans. 18,8496.

THE CIRCUMFERENCE OF A CIRCLE BEING GIVEN, TO FIND THE DIAMETER.

RULE. AS 22 are to 7; or 355 to 113; or as 1 is to ,31831, so is the circumference of a circle to its diameter.

THE DIAMETER OF A CIRCLE BEING GIVEN, TO FIND ITS

AREA.

RULE.-Multiply the square of the diameter by ,7854.

1. What is the area of a circle, the diameter of which is 12? Ans. 113,0976. 2. What is the area of a circle, the diameter being 9? Ans. 63,6174.

THE CIRCUMFERENCE OF A CIRCLE BEING GIVEN, TO FIND THE AREA.

RULE.-Multiply the square of the circumference by,07958.

1. What is the area of a circle, the circumference of which is 37,7? Ans. 113,1062582.

2. What is the area of a circle, the circumference of which is 28,27?

Ans. 63,599.

THE AREA OF A CIRCLE BEING GIVEN, TO FIND THE DIAMETER.

RULE.-Multiply the given area by 1,2732, and the product will be the square of the diameter; extract the square of this product, and you will have the diameter.

What is the diameter of a circle, the area of which is 113,09? Ans. 11,999.

THE AREA OF A CIRCLE BEING GIVEN, TO FIND THE CIRCUMFERENCE.

RULE.-Multiply the given area by 12,566, and extract root of the product, and the root will be the cir

the square cumference.

What is the area of a circle, the circumference of which is 113,03? Ans. 37,68.

THE SIDE OF A SQUARE BEING GIVEN, TO FIND THE DIAMETER OF A CIRCLE EQUAL TO THE SQUARE THE SIDE OF WHICH IS GIVEN.

RULE.-Maltiply the given side by 1,128, and the product will be the diameter of a circle, the area of which is equal to the area of the given square.

What is the diameter of a circle equal to a square, one side of which is 10,635? Ans. 11,99628.

What is the diameter of a circle equal to a square, one side of which is 83,33?

Ans. 94.

TO FIND THE AREA OF AN ELLIPSIS, OR OVAL.

RULE.-Multiply the longest diameter by the shortest; then multiply the product by the decimal,7854.

What is the area of an oval, its diameters being 7 and 51

Ans. 27,489.

-TO FIND THE SURFACE OF A PRISM, OR CYLINDER.

RULE.-Multiply the perimeter of one end of the prism by the length of the solid, and the product will be the surface of all its sides. To which add also the area of the two ends of the prism.

What is the surface of the sides of an equilateral triangular prism, its length being 9, and side 3? Ans. 81.

TO FIND THE SURFACE OF A PYRAMID, OR CONE.

RULE.-Multiply the perimeter of the base by the slant height or length of the side, and half the product will be the surface of the sides, or the sum of the areas of all the triangles which form it. To which add the area of the end or base.

What is the surface of a pyramid, its slant height being 20, and the perimeter of its base 15? Ans. 150.

TO FIND THE SURFACE OF THE FRUSTUM OF A PYRAMID.

RULE. Add the perimeters of the two ends, and multiply their sum by the slant height, taking half the product for the answer.

What is the surface of the frustum of a pyramid, the slant height being 12, the diameters 8 and 6?

Ans. 263,88.

TO MEASURE A CYLINDER.

RULE. The diameter of the base being given, find the area of the end; then multiply the area of the base by the length.

What are the solid contents of a cylinder, 1 foot, 9 inches in diameter, and 12 feet, 6 inches long? Ans. 30,0625.

TO MEASURE A PRISM.

RULE.-Find the surface of one end, and multiply that

by the length.

What are the contents of a prism, each of its sides being 10 inches, and the length 12 feet?

Ans. 3,6.

TO MEASURE A PYRAMID.

RULE.-Find the surface of the base, and multiply that by one third of the height.

What are the solid contents of a triangular pyramid, each of its sides being 13, and the height 48?

Ans. 1144.

TO MEASURE A SPHERE, OR GLobe.

RULE.-Multiply the cube of the diameter by ,5236.

What are the solid contents of a globe, its diameter being

4,5?

Ans. 47,71305.

XXX. Calculation of Speeds.

TO FIND THE SPEED OF ANY GIVEN SHAFT.

RULE.-Begin at the water wheel, and trace out all the driving and driven wheels separately, from the first driver to the last driven on the first end of the given shaft. Multiply the number of teeth in all the driving wheels together, and the product by the revolutions of the water wheel per minute; then multiply all the driven wheels together, and divide the product of the former by the product of the latter; the quotient will express the revolutions per minute of the given shaft.