Solid Mensuration

Solid Mensuration Solid Mensuration (also known as Solid Geometry) Geometry) is the study of various solids. It is the study of the measure of volume, area, height, length, and many more. This subject is used extensively in the practice of engineering. The knowledge of this subject is a necessity to engineers in any project construction. This subject will cover the following: Cube, Rectangular Parallelepiped, Prism, Cylinder, Cone, Frustums, Spheres, Solids of Revolution, Prismatoid, etcetera.

Solids in which V = A b h

This is a group of solids in which the volume is equal to the product of the area of the base and the altitude. Two types of solids belong to this group namely, prisms and cylinders. cylinders.

Prism and Cylinder

A solid is said to be prismatic or cylindrical if every cutting plane parallel p arallel to base are the same s ame in both shape and size. If the base is a closed polygon the solid is a prism, whereas, the solid is a cylinder if the base is a closed loop of a curve line. The name of the prism and cylinder is according to the shape of its base. A prism with a pentagon base is called a pentagonal prism and a cylinder of circular base is called circular cylinder. The volume of these two solids is given by the formula

V = A_b h

where V = volume, Ab = area of the base, and h = altitude.

Solids for which Volume = 1/3 Area of Base times Altitude This is a group of solids in which the volume is equal to one-third of the product of base area and altitude. There are two solids that belong to this group; the pyramid and the cone.

Like cylinder and prism, the name of pyramid and cone is according to the shape of the base. If  the base is pentagon, the pyramid is called pentagonal pyramid, and if the base is circle, the cone is called circular cone. The formula for the volume of these solids is

V=31Abh

Where Ab = area of the base, and h = altitude.

Frustum Frustum

Frustum of a pyramid (cone) is a portion of pyramid (cone) included between the base and the section parallel to the base not passing through the vertex.

Formula for Volume of a Frustum

The volume of a frustum is equal to one-third the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the bases. In symbols

V=31 A1+A2+

A1A2 h

Similar Figures Two surfaces or solids are similar if any two corresponding sides or planes are proportional.

In similar figures of any kind, pairs of corresponding line segments such as x1, x2 and y1, y2 have the same ratio.

x2x1=y2y1

The areas of similar surfaces A1 and A2 have the same ratio as the squares of any two corresponding lines x1 and x2.

A2A1=x22x12

The volumes of similar solids V1 and V2 have the same ratio as the cubes of two corresponding lines x1 and x2.

V2V1=x23x13

The Sphere Sphere is a solid bounded by closed surface every point of which is equidistant from a fixed point called the center.

Properties of a Sphere 

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Every section in the sphere made by a cutting plane is a circle. If the cutting plane passes through the center of the sphere, the section made is a great circle; otherwise the section is a small circle. For a particular circle of a sphere, the axis is the diameter of the sphere perpendicular to the plane of the circle. The ends of the axis of the circle of a sphere are called poles.

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The nearer the circle to the center of the sphere, the greater is its area. The largest circle in the sphere is the great circle. The radius (diameter) of the great circle is the radius (diameter) of the sphere. All great circles of a sphere are equal. Every great circle divides the sphere into two equal parts called hemispheres. The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line  joining the centers of the surfaces and whose center is on that line. (See figure to the right.) A plane perpendicular to a radius at its extremity is tangent to the sphere.

Formulas for a Sphere

Surface Area, A The surface area of a sphere is equal to the area four great circles. A=4 R2

A= D2

Volume, V V=34 R3

V=81 D3

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