General Pyramids and Cones and Their Cross-Sections

New York State Common Core Math Geometry, Module 3, Lesson 7

Worksheets for Geometry

Student Outcomes

• Students understand the definition of a general pyramid and cone and that their cross sections are similar to the base.
• Students show that if two cones have the same base area and the same height, then cross sections for the cones the same distance from the vertex have the same area.

General Pyramids and Cones and Their Cross-Sections

Classwork

Opening Exercise

Group the following images by shared properties. What defines each of the groups you have made?

RECTANGULAR PYRAMID: Given a rectangular region 𝐵 in a plane 𝐸 and a point 𝑉 not in 𝐸, the rectangular pyramid with base 𝐵 and vertex 𝑉 is the collection of all segments 𝑉𝑃 for any point 𝑃 in 𝐵.

GENERAL CONE: Let 𝐵 be a region in a plane 𝐸 and 𝑉 be a point not in 𝐸. The cone with base 𝐵 and vertex 𝑉 is the union of all segments 𝑉𝑃 for all points 𝑃 in 𝐵 (See Figures 1 and 2).

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Example 1

In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross-section △ 𝐴′𝐵′𝐶′. If the area of △ 𝐴𝐵𝐶 is 25 mm², what is the area of △ 𝐴′𝐵′𝐶′?

Example 2

In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross-section △ 𝐴′𝐵′𝐶′. The altitude from 𝑉 is drawn; the intersection of the altitude with the base is 𝑋, and the intersection of the altitude with the cross-section is 𝑋′. If the distance from 𝑋 to 𝑉 is 18 mm, the distance from 𝑋′ to 𝑉 is 12 mm, and the area of △ 𝐴′𝐵′𝐶′ is 28 mm², what is the area of △ 𝐴𝐵𝐶?In the following triangular pyramid, a plane passes through the pyramid so that it is parallel to the base and results in the cross-section △ 𝐴′𝐵′𝐶′. The altitude from 𝑉 is drawn; the intersection of the altitude with the base is 𝑋, and the intersection of the altitude with the cross-section is 𝑋′. If the distance from 𝑋 to 𝑉 is 18 mm, the distance from 𝑋′ to 𝑉 is 12 mm, and the area of △ 𝐴′𝐵′𝐶′ is 28 mm², what is the area of △ 𝐴𝐵𝐶?

Exercise 1

The area of the base of a cone is 16, and the height is 10. Find the area of a cross-section that is distance 5 from the vertex.

Example 3

GENERAL CONE CROSS-SECTION THEOREM: If two general cones have the same base area and the same height, then cross sections for the general cones the same distance from the vertex have the same area.
State the theorem in your own words.

Use the space below to prove the general cone cross-section theorem.

Exercise 2

The following pyramids have equal altitudes, and both bases are equal in area and are coplanar. Both pyramids’ cross sections are also coplanar. If 𝐵𝐶 = 3√2 and 𝐵′𝐶′ = 2√3, and the area of 𝑇𝑈𝑉𝑊𝑋𝑌𝑍 is 30 units², what is the area of cross-section 𝐴′𝐵′𝐶′𝐷′?

Lesson Summary

CONE: Let 𝐵 be a region in a plane 𝐸 and 𝑉 be a point not in 𝐸. The cone with base 𝐵 and vertex 𝑉 is the union of all segments 𝑉𝑃 for all points 𝑃 in 𝐵. If the base is a polygonal region, then the cone is usually called a pyramid.

RECTANGULAR PYRAMID: Given a rectangular region 𝐵 in a plane 𝐸 and a point 𝑉 not in 𝐸, the rectangular pyramid with base 𝐵 and vertex 𝑉 is the union of all segments 𝑉𝑃 for points 𝑃 in 𝐵.

LATERAL EDGE AND FACE OF A PYRAMID: Suppose the base 𝐵 of a pyramid with vertex 𝑉 is a polygonal region, and 𝑃𝑖 is a vertex of 𝐵. 𝑃𝑖𝑉 is called a lateral edge of the pyramid. If 𝑃𝑖𝑃𝑖+1 is a base edge of the base 𝐵 (a side of 𝐵), and 𝐹 is the union of all segments 𝑃𝑉 for 𝑃 in 𝑃𝑖𝑃𝑖+1, then 𝐹 is called a lateral face of the pyramid. It can be shown that the face of a pyramid is always a triangular region.

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