Explanation Given: Each edge of a regular tetrahedron has length e .as below, Properties Used: The lateral area L of a regular pyramid with slant height of length l and perimeter P of the base is given by L = 1 2 l P . In a regular pyramid, the lengths of the apothem a of the base, the altitude h , and the slant height l satisfy the Pythagorean Theorem; that is, l 2 = a 2 + h 2 . According to the Pythagorean theorem, in a right-angled triangle h y p o t e n u s e 2 = b a s e 2 + p e r p e n d i c u l a r 2 . Area of a right-angled triangle is 1 2 × b a s e × h e i g h t . Approach Used: i) Using the property l 2 = a 2 + h 2 , find the height or altitude h by substituting the values of l and a . ii) Substitute the values of base and height h in equation 1 2 × b a s e × h e i g h t to get the area of each lateral face since it is a triangular pyramid. iii) Compare the given value with the value from step (ii). Calculation: From the given figure the sides of the square base and lateral edge all are e . Considering the right-angled triangle with triangle base the sides of the triangular base measure e each and the angle at the center for each base is = 360 ° 3 = 120 ° which is bisected by the perpendicular so the hypotenuses in the right-angled triangle is O A ¯ = s i d e 2 sin 60 ° To determine (b) To show: The volume of the tetrahedron is V = 2 12 e 3 . To determine (c) To find: The volume of the tetrahedron.

Elementary Geometry for College Students

6th Edition

ISBN: 9781285195698

Author: Daniel C. Alexander, Geralyn M. Koeberlein

Publisher: Cengage Learning

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Question

Chapter 9.2, Problem 38E

To determine

(a)

To show:

The altitude of the tetrahedron measures h=23e.

To determine

(b)

To show:

The volume of the tetrahedron is V=212e3.

To determine

(c)

To find:

The volume of the tetrahedron.

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Chapter 9.2, Problem 37E

Chapter 9.2, Problem 39E

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