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#### Page No 215:

#### Question 1:

Write down the number of faces of each of the following figures:

(i) Cuboid

(ii) Cube

(iii) Triangular prism

(iv) Square pyramid

(v) Tetrahedron

#### Answer:

(i) A cuboid has 6 faces, namely* ABCD, EFGH, HDAE, GCBF, HDCG* and *EABF.
*

(ii) A cube has 6 faces, namely

*IJKL, MNOP, PLIM, OKJN, LKOP*and

*IJNM.*

(iii) A triangular prism has 5 faces (3 rectangular faces and 2 triangular faces), namely

*QRUT, QTVS, RUVS, QRS*and

*TUV.*

(iv) A square pyramid has 5 faces (4 triangular faces and 1 square face), namely

*OWZ, OWX, OXY, OYZ*and

*WXYZ.*

(v) A tetrahedron has 4 triangular faces, namely

*KLM, KLN, LMN*and

*KMN.*

#### Page No 215:

#### Question 2:

Write down the number of edges of each of the following figures:

(i) Tetrahedron

(ii) Rectangular pyramid

(iii) cube

(iv) Triangular prism

#### Answer:

(i) A tetrahedron has 6 edges, namely *KL, LM, LN, MN, KN* and *KM*.

(ii) A rectangular pyramid has 8 edges, namely* AB, AE, AD, AC, EB, ED, DC* and* CB*.

(iii) A cube has 12 edges, namely *PL, LK, KO, OP, MN, NJ, JI, IM, PM, LI, ON* and *KJ. *

(iv) A triangular prism has 9 edges, namely* QR, RS, QS, TU, TV, UV, QT, RU*, and *SV. *

#### Page No 215:

#### Question 3:

Write down the number of vertices of each of the following figures:

(i) Cuboid

(ii) Square pyramid

(iii) Tetrahedron

(iv) Triangular prism

#### Answer:

(i) A cuboid has 8 vertices, namely* A, B, C, D, E, F, G *and *H. *

(ii) A square pyramid has 5 vertices, namely* O, W, X, Y* and *Z*.

(iii)A tertrahedron has 4 vertices, namely* K, L, M *and *N. *

(iv) A triangular prism has 6 vertices, namely* Q, R, S, T, U* and *V*.

#### Page No 215:

#### Question 4:

Fill in the blanks:

(i) A cube has ....... vertices, ....... edges and ....... faces.

(ii) The point at which three faces of a figure meet is known as its .......

(iii) A cuboid is also known as a rectangular .......

(iv) A triangular pyramid is called a .......

#### Answer:

(i) A cube has __8__ vertices, __12__ edges and __6__ faces.

Vertices: *I, J, K, L, M, N, O* and *P *

Edges : *IJ, JN, NM, MI, PL, LK, KO, OP, PM, LI, KJ*, and *ON *

Faces :* MNJI, POKL, PLIM, OKJN, PONM *and *LKJI*

(ii) The point at which the three faces of a figure meet is known as its __vertex__.

(iii) A cuboid is also known as a rectangular __cube__.

(iv) A triangular pyramid is called a __tetrahedraon__.

#### Page No 217:

#### Question 1:

Define Euler's relation between the number of faces, number of edges and number of vertices for various 3-dimensional figures.

#### Answer:

The Euler's relation for a three dimensional figure can be expressed in the following manner:

$F-E+V=2\phantom{\rule{0ex}{0ex}}H\mathrm{ere},\phantom{\rule{0ex}{0ex}}F-\mathrm{Number}\mathrm{of}\mathrm{faces}\phantom{\rule{0ex}{0ex}}E-\hspace{0.17em}\mathrm{Number}\mathrm{of}\mathrm{edges}\phantom{\rule{0ex}{0ex}}V-\hspace{0.17em}\mathrm{Number}\mathrm{of}\mathrm{vertices}$

#### Page No 217:

#### Question 2:

How many edges are there in a

(i) cuboid

(ii) tetrahedron

(iii) triangular prism

(iv) square pyramid?

#### Answer:

(i) A cuboid has 12 edges, namely *AD, DC, CB, BA, EA, FB, HD, DC, CG, GH, HE*, and *GF. *

(ii) A tetrahedron has 6 edges, namely *KL, LM, MN, NL , KM *and* KN.*

(iii) A triangular prism has 9 edges, namely *QR, RS, SQ, TU, UV, VT, RU, SV *and *QT. *

(iv) A square pyramid has 8 edges, namely* OW, OX, OY , OZ , WX, XY, YZ* and *ZW.*

#### Page No 217:

#### Question 3:

How many faces are there in a

(i) cube

(ii) pentagonal

(iii) tetrahedron

(iv) pentagonal pyramid?

#### Answer:

(i) A cube has 6 faces, namely *IJKL, MNOP, PLIM , OKJN, POKL *and *MNJI*.

(ii) A pentagonal prism has 7 faces, i.e. 2 pentagons and 5 rectangles, namely *ABCDE, FGHIJ, ABGF, AEJF , EDIJ, DCHI *and* CBGH.*

(iii) A tetrahedron has 4 faces, namely* KLM, KLN, LMN *and* KMN*.

(iv) A pentagonal pyramid has 6 faces, i.e. 1 pentagon and 5 triangles, namely *NOPQM, SNM, SOP, SNO, SMQ* and * SQP.*

#### Page No 217:

#### Question 4:

How many vertices are there in a

(i) cuboid

(ii) tetrahedron

(iii) pentagonal prism

(iv) square pyramid?

#### Answer:

(i) A cuboid has 8 vertices, namely *A, B, C, D, E, F, G *and* H*.

(ii) A tetrahedron has 4 vertices, namely *K, L, M* and *N. *

(iii) A pentagonal prism has 10 vertices, namely *A, B, C, D, E, F, G, H, I* and *J*.

(iv) A square pyramid has 5 vertices, namely* O, W, X, Y *and *Z. *

#### Page No 217:

#### Question 5:

Verify Euler's relation for each of the following:

(i) A square

(ii) A tetrahedron

(iii) A triangular prism

(iv) A square pyramid

#### Answer:

Euler's relation is:

$F-E+V=2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Here:\phantom{\rule{0ex}{0ex}}F-\mathrm{Number}\mathrm{of}\mathrm{faces}\phantom{\rule{0ex}{0ex}}E-\hspace{0.17em}\mathrm{Number}\mathrm{of}\mathrm{edges}\phantom{\rule{0ex}{0ex}}V-\hspace{0.17em}\mathrm{Number}\mathrm{of}\mathrm{vertices}$

(i) A square prism

(There is an error in this question. It should have been a square prism rather than square.)

$\mathrm{Number}\mathrm{of}\mathrm{faces}=F=2\mathrm{squares}+4\mathrm{rectangular}=6\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{edges}=E=12\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{vertices}=V=8\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(F-E+V\right)=6-12+8=2$

(ii) A tetrahedron

$\mathrm{Number}\mathrm{of}\mathrm{faces}=F=4\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{edges}=E=6\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{vertices}=V=4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(F-E+V\right)=4-6+4=2$

(iii) A triangular prism

$\mathrm{Number}\mathrm{of}\mathrm{faces}=F=2\mathrm{triangular}+3\mathrm{rectangular}=5\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{edges}=E=9\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{vertices}=V=6\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(F-E+V\right)=5-9+6=2$

(iv) A square pyramid

$\mathrm{Number}\mathrm{of}\mathrm{faces}=F=2\mathrm{triangular}+3\mathrm{rectangular}=5\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{edges}=E=8\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{vertices}=V=5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(F-E+V\right)=5-8+5=2$

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