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Common Core For Geometry

Worksheets for Geometry, Module 3, Lesson 7

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 π΅ in a plane πΈ and a point π not in πΈ, the rectangular pyramid with
base π΅ and vertex π is the collection of all segments ππ for any point π in π΅.

**GENERAL CONE**: Let π΅ be a region in a plane πΈ and π be a point not in πΈ. The cone with base π΅ and vertex π is the union
of all segments ππ for all points π in π΅ (See Figures 1 and 2).

**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 β³ π΄β²π΅β²πΆβ². If the area of β³ π΄π΅πΆ is 25 mm^{2}, what is the area of β³ π΄β²π΅β²πΆβ²?

**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 β³ π΄β²π΅β²πΆβ². The altitude from π is drawn; the intersection of the altitude with the base is π, and the
intersection of the altitude with the cross-section is πβ². If the distance from π to π is 18 mm, the distance from πβ² to π
is 12 mm, and the area of β³ π΄β²π΅β²πΆβ² is 28 mm^{2}, what is the area of β³ π΄π΅πΆ?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 β³ π΄β²π΅β²πΆβ². The altitude from π is drawn; the intersection of the altitude with the base is π, and the
intersection of the altitude with the cross-section is πβ². If the distance from π to π is 18 mm, the distance from πβ² to π
is 12 mm, and the area of β³ π΄β²π΅β²πΆβ² is 28 mm^{2}, what is the area of β³ π΄π΅πΆ?

**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 pyramidsβ cross sections are also coplanar. If π΅πΆ = 3β2 and π΅β²πΆβ² = 2β3, and the area of πππππππ is 30 units^{2}, what is the area of
cross-section π΄β²π΅β²πΆβ²π·β²?

**Lesson Summary**

**CONE:** Let π΅ be a region in a plane πΈ and π be a point not in πΈ. The cone with base π΅ and vertex π is the union of all
segments ππ for all points π in π΅.
If the base is a polygonal region, then the cone is usually called a pyramid.

**RECTANGULAR PYRAMID:** Given a rectangular region π΅ in a plane πΈ and a point π not in πΈ, the rectangular pyramid
with base π΅ and vertex π is the union of all segments ππ for points π in π΅.

**LATERAL EDGE AND FACE OF A PYRAMID:** Suppose the base π΅ of a pyramid with vertex π is a polygonal region, and ππ
is a
vertex of π΅. πππ is called a lateral edge of the pyramid. If ππππ+1
is a base edge of the base π΅ (a side of π΅), and πΉ is
the union of all segments ππ for π in ππππ+1, then πΉ 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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