Introduction: We examine 3-dimensional objects known as pyramids and cones, defining them, and measuring their surface areas and volumes.

Definitions:

Pyramid: A 3-dimensional solid in which the base is a polygon and the sides are triangles which meet in one point called the vertex. We shall examine regular pyramids in which the base is a regular polygon and the sides are congruent triangles.

A Right Circular Cone: A 3-dimensional solid in which the base is a circle. The side of a cone is formed by straight lines which connect the circular base to a vertex. The height is the perpendicular distance from the vertex to the base and meets the base in the center of the circle.

The Lesson:

The diagrams below show a pyramid and a cone. Both have a height of h and radius of r. In the pyramid at left, r is the radius of the regular hexagon that is the base of the pyramid. In the cone at right, r is the radius of the circular base. The slant height is s in both diagrams.

Pyramid:

The perimeter of the base of the pyramid, which is a regular hexagon, is 6r since r is the same length as the side of a regular hexagon.

The area of the base of the pyramid is given by

.

The area of the sides (lateral area), which are congruent triangles, is given by

because in this case n = number of sides = 6, the base is equal to r, and the slant height of the triangles is s.

Notice that this can be rewritten as:

.

where P is the perimeter of the base.

Overview

In general, calculating the surface area of a pyramid requires finding the sum of the areas of the base and the triangular sides. The lateral area is given by where s is the slant height of a triangular side and P is the perimeter of the base. Remember that this is only true because we are assuming that the triangular sides are congruent; that is, we are working with a regular pyramid. Note that in a regular pyramid it is also true that (1) each triangle makes the same angle with the base and (2) the height connects the vertex to the center of the regular polygonal base.

Pyramid Summary: Surface area of a regular pyramid = area of base + . Volume of a pyramid =
Cone Summary: The area of the base of the cone is . The lateral area is given by . The volume of a cone is given by .

Let's Practice:

1. In Egypt, the Great Pyramid of Giza is 145.75 m in height and has a square base of 229 m on a side. The triangular sides are congruent and form an angle of 51º with the square base. What are the surface area and the volume of this pyramid?

The base area is

229² = 52441 m².

The lateral area is

where s is the slant height of a triangular side.

To calculate the slant height, we use the height and the slant height (as a hypotenuse of a right triangle) as in the diagram shown above. Using the trig function sine, we get

.

This gives us

.

The lateral area is

458s = 85875 m².

The total surface area is

52441 + 85875 = 138,316 m².

The volume is area of base x height =

x 229² x 145.75 = x 7,643,275.75 m³ = 2,547,758.583 m³.

Despite the fact that well over 1,000,000 stone blocks weighing between 2 and 150 tons were manually installed, the exact measurements of the sides of the base and the angles of the triangles forming the sides are off by less than 0.1% from that of a perfectly regular pyramid.

2. A cone has height 5 feet and the radius of the base is 2 feet. What are the surface area and the volume?

To find the volume we use

cubic feet.

To find the surface area we use

.

This means we must calculate the slant height s.

In order to do this, we note that the height is perpendicular to the base of the cone at the center. We use the height, radius, and slant height s to form a right triangle. A diagram is shown below.

We now use the Pythagorean Theorem.

s² = 5² + 2² = 25 + 4 = 29

Now solve for s:

s =

This gives us a surface area of

ft².