Problems for composite Solids: Finding surface and volume by adding solids

Cone
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.

Cylinder
The surface of a cylindrical solid whose base is a circle

Cylindrical surface
The union of the bases and the lateral surface.

Hemisphere
Half of a sphere.

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

Lateral surface
The surface not included in the base (s).

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.

polyhedron
A polyhedron is a closed three-dimensional figure. All of the faces are made up of polygons.

Prism
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.

Pyramid
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.

Sphere
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.

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

Unit cube
Unit of measuring volume.

Some of the problems that you solve to find surface area in pyramids and cones

require that you use trigonometry to find the slant height, the height, or the apothem,

or radius in the base. This interactive applet allows you to review the trigonometric

ratios in context to their inverses.

Drag any of the acute angle vertices, and see how the values are updated in the table.

Even more common that using trigonometry in finding surface area of cones, and pyramids:

It is to use the Pythagorean Theorem. You may observe that the slant height, the height, and the

radius (Cones), or apothem (Pyramids) form a right triangle, thus this is why you may use the

Pythagorean Theorem. In the interactive applet below this text; you may drag any of the vertices

to verify how the Pythagorean Theorem works. Check the equations in the upper left corner of

the applet and see how the values are updated for the different triangles you generate.

Finding Surface Area, and Volume of Composite Solids by Adding Areas, and Volumes.

Look at your house. Is it a cube, or a cylinder? No, most buildings are a composition of several shapes. A church may have cylinders at the towers, a prism at the chapel, and the roof might be a triangular prism laying on the side. Would you be able to find the volume of that hypothetical church?

In this lesson, we will present you composite solids formed by cylinders, pyramids, spheres, and hemispheres. These solids are combined, and result in interesting shapes that will spark your creativity to find the volume, and/or surface area. Let's give some volume to our ideas!