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For every polyhedron there is a construction for a so called dual polyhedron by connecting together the centres of the faces. The diagram on the left below shows this construction for a cube that constructs an octahedron inside it. 2²³ B .O Choose an origin O to be the centre of the base of the cube and choose the i, j and k directions to be parallel to the faces of the cube. Let A be the centre of the face with positive i-coordinate and B be the centre of the face with positive j- coordinate, as shown in the right-hand diagram. Let the cube have a side length 2. (a) Write down OA and OB in terms of the unit vectors i, j and k. (b) Calculate OÀ OB and hence calculate the angle AOB. (c) Calculate OÀ X OB and hence calculate the ratio of the surface area of the octahedron to the surface area of the cube.

For every polyhedron there is a construction for a so called dual polyhedron by connecting together the centres of the faces. The diagram on the left below shows this construction for a cube that constructs an octahedron inside it. 2²³ B .O Choose an origin O to be the centre of the base of the cube and choose the i, j and k directions to be parallel to the faces of the cube. Let A be the centre of the face with positive i-coordinate and B be the centre of the face with positive j- coordinate, as shown in the right-hand diagram. Let the cube have a side length 2. (a) Write down OA and OB in terms of the unit vectors i, j and k. (b) Calculate OÀ OB and hence calculate the angle AOB. (c) Calculate OÀ X OB and hence calculate the ratio of the surface area of the octahedron to the surface area of the cube.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter9: Surfaces And Solids
Section9.2: Pyramids, Area, And Volume
Problem 6E
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For every polyhedron there is a construction for a so called dual polyhedron by connecting together the centres of the faces. The diagram on the left below shows this construction for a cube that constructs an octahedron inside it. j "K³ B i 0 Choose an origin O to be the centre of the base of the cube and choose the i, j and k directions to be parallel to the faces of the cube. Let A be the centre of the face with positive i-coordinate and B be the centre of the face with positive j- coordinate, as shown in the right-hand diagram. Let the cube have a side length 2. (a) Write down OA and OB in terms of the unit vectors i, j and k. (b) Calculate ŌÀ OB and hence calculate the angle AOB. (c) Calculate OÀ X OB and hence calculate the ratio of the surface area of the octahedron to the surface area of the cube.
Transcribed Image Text:For every polyhedron there is a construction for a so called dual polyhedron by connecting together the centres of the faces. The diagram on the left below shows this construction for a cube that constructs an octahedron inside it. j "K³ B i 0 Choose an origin O to be the centre of the base of the cube and choose the i, j and k directions to be parallel to the faces of the cube. Let A be the centre of the face with positive i-coordinate and B be the centre of the face with positive j- coordinate, as shown in the right-hand diagram. Let the cube have a side length 2. (a) Write down OA and OB in terms of the unit vectors i, j and k. (b) Calculate ŌÀ OB and hence calculate the angle AOB. (c) Calculate OÀ X OB and hence calculate the ratio of the surface area of the octahedron to the surface area of the cube.
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