Section 12.3 Notes

Pyramids

Pyramids In geometry, a pyramid is a three-dimensional figure with one base that is a polygon. The other faces of a pyramid, called the lateral faces, are triangles that connect the base to the vertex. The pyramid is a regular pyramid if the base is a regular polygon and the lateral faces are congruent isosceles triangles.

The height, h, of a regular pyramid is the distance from the vertex to the center of the base. The slant height, s, of a regular pyramid is the altitude of a lateral face.

Parts of a Pyramid vertex lateral face h s a base

Three Right Triangles Found in a Pyramid

#1 – A lateral face lateral edge s base edge

#2 – The triangle made by the height, apothem, and slant height.

#3 – The triangle made by the height, radius, and lateral edge.

Surface Area of Pyramids The surface area of a pyramid is the sum of the lateral area and the area of the base.

Net LA = ½ps s base edge

Surface Area Formula SApyramid = ½ps + B where p is the perimeter of the base, s is the slant height and B is the area of the base.

Example 1 Find the surface area of the following square pyramid. Round your slant height to the nearest whole number. 471 ft. 707.75 ft.

s 471 ft. a 471 ft. 707.75 ft.

Example 2 Find the surface area of a right pyramid whose base is an equilateral triangle with side lengths of 4 cm. And the slant height of the pyramid is 10 cm.

s = 10 cm 4 cm.

Cones A cone is a three-dimensional figure with one circular base and a vertex. In this unit, you will learn about right cones.

The vertex of a right cone is directly above the center of the base. The height, h, of a right cone is the segment that connects the vertex with the center of the base. The radius, r, of a right cone is the radius of the base. The slant height, s, of a right cone is the segment from the vertex to any point on the base.

vertex h s r

Surface Area of Cones The surface area of a cone is the sum of the lateral area and the area of the base.

Surface Area Formula for a Cone SA = rs + r2 where r is the radius of the base and s is the slant height.

Example 3 Find the surface area of the following cone. 6 ft 2 ft