Refresher - Three-Dimensional Geometry
A cube is a three-dimensional figure with six congruent square sides.
If s is the length of one of its sides, the volume of the
cube is given by V(s)=s3.
Since the cube has six square-shape sides, the surface area of a cube is given by F(s)=6s2.
In a rectangular solid, the length, width, and height may all be different.
The volume of the rectangular solid below would be the product of the length, width, and height.
The following are the formulas for volume of a rectangular solid and the surface area of a rectangular solid:
V = lwh
S = 2(lw + wh + lh).
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism and the length of the edge joining the two bases is called the height.
The above diagrams show two prisms: one with a triangle-shaped base called a triangular prism and another with a pentagon-shaped base called a pentagonal prism.
A rectangular solid is a prism with a rectangle-shaped base. A cylinder is another example of a prism -- one that has a circular base. As was stated earlier, the volume of a prism is given by the product of the area of its base and its height.
Volume of prism = area of base * height
A cylinder is formed with two congruent circles joined by a curved surface. If you roll up a rectangular sheet of paper you will get the sides of a cylinder.
In the figure, the radius of the circular base is r and the height is h. The total surface area is made up of a circular top and bottom, which are congruent circles, and the side surface, which is a rectangle.
To find the surface area, we need to find the area of the bases (circles)
Area of bases = 2(π)(r2).
The area of the side is a rectangle. Imagine the cylinder to be a simple tin can. If the top and bottom of the can were removed, what is left is an open ended tube. The length of one side would be the circumference of the base circles, and the other side would be the height between them.
Area of the side = (h)(2πr).
Therefore, the total surface area of the cylinder is equal to the area of the bases plus the area of the sides.
Total Surface Area = 2πr2+(2πr)h.
A sphere is a solid with all its points the same distance from the center.
The formula for the volume of a sphere is:
43πr3
The formula for the surface area is:
4πr2
A circular cone has a circular base, which is connected by a curved surface to its vertex.
A cone is called a right circular cone if the line from the vertex of the cone to the center of its base is perpendicular to the base.
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex.
A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.
The formula for the volume of a pyramid is:
13 * area of base * height