From Handbook of Machining and Metalworking Calculations


Mensuration is the mathematical name for calculating the areas, volumes, length of sides, and other geometric parts of standard geometric shapes such as circles, spheres, polygons, prisms, cylinders, cones, etc., through the use of mathematical equations or formulas. Included here are the most frequently used and important mensuration formulas for the common geometric figures, both plane and solid. (See Figs. 2.1 through 2.36.)




a, b, etc.


A, B, C



height perpendicular to base b


length of side or edge




number of sides






surface area

Figure 2.1: Oblique triangle.

Figure 2.2: Oblique triangle.

Figure 2.3: Rectangle.

Figure 2.4: Parallelogram.

Figure 2.5: Rhombus.

Figure 2.6: Trapezoid.

Figure 2.7: Polyhedra.

Figure 2.8: Trapezium.

Figure 2.9: Regular polygon.

Figure 2.10: Inscribed circle.

Figure 2.11: Circumscribed circle.

Figure 2.12: Inscribed polygon.

Figure 2.13: Circumscribed polygon.

Figure 2.14: Circle circumference.

Figure 2.15: Circle area.

Figure 2.16: Length of arc.

Figure 2.17: Chord and sector.

Figure 2.18: Segment of a circle.

Figure 2.19: Ring.

Figure 2.20: Ellipse.

Figure 2.21: Pyramid.

Figure 2.22: Sphere.

Figure 2.23: Cylinder.

Figure 2.24: Cone.

Figure 2.25: Spherical segment.

Figure 2.26: Frustum of a cone.

Figure 2.27: Truncated cylinder.

Figure 2.28: Spherical zone.

Figure 2.29: Spherical wedge.

Figure 2.30: Paraboloid.

Figure 2.31: Spherical sector.

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Topics of Interest

3.1 GEOMETRIC CONSTRUCTION The following figures show the methods used to perform most of the basic geometric constructions used in standard drawing and layout practices. Many of these constructions...

3.1 MENSURATION Plane areas, surface areas, and volumes of the regular geometric shapes and solids illustrated herein can be calculated by using the formulas which appear next to the respective...

Learning Objectives After completing this lesson, you will be able to: Draw an Inscribed or Circumscribed Polygon Create an Ellipse using two different methods Define an Elliptical Arc Create...

Chapter 1: Overview Figure 1.1: Hierarchy of modeling and simulation. Chapter 2: Mathematical Concepts in Modeling Figure 2.1: Frame A and coordinate system ] A. Figure 2.2: Vector triangle.

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