Rotating Areas to Generate Solids of Revolution: Surface Area.

Have you tried to figure out how much paper you need to cover an irregular shape like the pieces that connect the handrail on a staircase to the floor and have a circular transversal section but irregular profile along the length. If you slice it and get a polynomial function from the profile. You may use this to find out the surface area with accuracy. In calculus this is done using integrals. The intent of this lesson is to give a glimpse of this process but using basic plane geometry and illustrating the solutions in calculus using a graphing calculator.

In the industry, the way complicated pipe connectors are obtained is using this process. This process was analyzed in the same type of lesson but for volumes. Now we are going to do it for one application requiring to know with accuracy the surface area. Have a surfing adventure over the calculus surface...just on the surface by now...!

Lesson's Content

Study this lesson and the corresponding for Volume for Solids of Revolution. Then attempt this problem. Scroll down the page for a partial solution after you attempted.

Real World Follow up application. Scroll down the page for a partial solution after you attempted.


Lesson's Glossary

It is a 3-D figure with circular base, a vertex not in the plane of the circle, and has a curved surface connecting the base with the vertex. The altitude of the cone is the perpendicular segment that goes from the vertex to the plane containing the circle at the base. The height is the length of the altitude. The slant height is the length of the distance from the vertex to the edge of the base. For a right cone it is necessary that the altitude contains the center of the circle at the base.

Conic solid
The set of points between a point (the vertex) and a non-coplanar region (the base), including the point and the region.

The union of the bases and the lateral surface.

Cylindrical surface
The surface of a cylindrical solid whose base is a circle.

Half of a sphere.

Lateral area.

Lateral area
The area of the lateral surface of a solid.

Platonic solid
A Platonic solid is defined as a solid in which all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.

It is a polyhedron that has two congruent and parallel faces, called bases. The remaining faces, which are parallelograms are called lateral faces. The altitude is the perpendicular segment whose endpoints are at the bases. The length of the altitude is the height of the prism. A right prism is one with all lateral faces rectangles. An oblique rectangle has some to be nonrectangular. 

It is a polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex called the vertex of the pyramid. The perpendicular segment whose endpoints are at the base and at the vertex of the pyramid is the altitude, which is the height of the pyramid. A regular pyramid has congruent isosceles triangles in the lateral faces and a regular polygon in the base. The length of the height for the lateral triangles is the slant height for the pyramid.

Regular pyramid
A pyramid whose base is a regular polygon and whose vertex forms a segment with the center of the polygon perpendicular to its plane.

Slant height
The length of a lateral edge of a conic solid.

Right prism
A prism whose direction of sliding is perpendicular to the plane of the base.

A sphere is properly defined as the set of points in the space that are equidistant of one point called center. Planes that go through the center of the sphere, intersect GREAT CIRCLES. The circumference of the sphere is the perimeter of any of its great circles.

The boundary of a 3-D figure.

Surface area
The total area of the surface of a solid.

Unit cube
Unit of measuring volume.



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Partial solution to the featured problem. Showing first a non-accurate logitudinal cut of the bottle, and a second solution with a more accurate longitudinal cut. You may find useful to pause the video from time to time to study particular frames.