## Mensuration of lines, surfaces, and volumes1873 |

### From inside the book

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**taking**the square root of the difference of the squares of the hypothenuse and given side . Interpreting formula ( 3 ) , we find the remaining side , by**taking**the square root of the product of the sum and difference of the given sides ... Page 25

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**Taking**the last found perimeters as given , we will then determine the perim- eters of polygons of double the number of sides by the same method , and so on . As the number of sides increases , it may be shewn that the perimeters ... Page 36

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**taking**half the square of the diagonal . - THE PARALLELOGRAM . PROP . II . To find the area of a parallelogram , having given its base and perpendicular height . A B Let AB a , DE = p . Draw through C , CF perpendicular to AB , and ... Page 38

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**taking**half the sum of the sides ; from this subtracting each side separately , multiplying half the sum and the three remainders together , and**taking**the square root of the product . Remark . It may here be noticed that each of the ... Page 39

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**taking**one - fourth of the square of the side , and multiplying it by the square root of 3 . Conversely , if S denotes the area of an equilateral triangle , we can find the length of its side , for a2 S 4S √3 = ... a = 2 √S 4/3 Cor ...### Other editions - View all

### Common terms and phrases

12 feet 20 feet 9 inches ABē ABCD altitude angle ABC angle AOB angle subtended arc equal area of ABC area of triangle axis base multiplied centre circle whose radius circular measure circular segment circumference circumscribing circle cube cylinder decagon diagonal divide edge equilateral triangle Euclid feet 6 inches feet 9 find the area find the length Find the number Find the radius find the side find the surface Find the volume foot 9 frustum generatrix half the sum Hence hypothenuse Interpreting this formula lateral surface lune magnitude number of sides parallel parallelogram plane prism PROP radii ratio rectangle rectangular parallelopiped regular hexagon regular polygon revolving round right angle right-angled triangle sector segment shew similar triangle slant height square feet square inches square yards total surface trapezoid triangle ABC triangular prism triangular pyramid unit angle vertex

### Popular passages

Page 1 - The area of the surface generated by a straight line revolving about an axis in its plane, is equal to the projection of the line on the axis multiplied by the circumference of the circle whose radius is the perpendicular erected at the middle of the line and terminated by the axis.

Page 59 - The volume of a triangular prism is equal to the product of its base by its altitude.

Page 40 - A Cylindrical surface is a curved surface generated by a moving straight line which continually touches a given curve and in all its positions is parallel to a given fixed straight line not in the plane of the curve.

Page 42 - A conical surface is a curved surface generated by a moving straight line which continually touches a given curve, and passes through a given fixed point not in the plane of the curve. Thus, if the straight line...

Page 46 - A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre.

Page 97 - From this it readily follows that all the three lines drawn from the angles of a triangle to the middle of the opposite sides, pass through one and the same point.

Page 46 - The axis of a circle of a sphere is the diameter of the sphere which is perpendicular to the plane of the circle. The ends of the axis are called the poles of the circle.

Page 98 - The area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

Page 40 - The areas of two circles are to each other as the squares of their radii, or as the squares of their diameters. S TrR2 R* If1' = ~R^ = "cT* = -D'*