RS Aggarwal Class 8 Mathematics Solutions Chapter 19 Three-Dimensional Figures

Question 1. Name the faces, edges and vertices of a cuboid, cube, triangular prism and square pyramid.
Answer:
(i) A cuboid has 6 faces: ABCD, EFGH, HDAE, GCBF, HDCG and EABF. The diagram shows a cuboid with labeled vertices where opposite faces are rectangular and parallel to each other.
(ii) A cube has 6 faces: IJKL, MNOP, PLIM, OKJN, LKOP and MNIJ. The diagram displays a cube where all faces are equal squares and all edges meet at right angles.
(iii) A triangular prism has 5 faces (3 rectangular faces and 2 triangular faces): QRUT, QTVS, RUVS, QRS and TUV. The diagram shows two triangular ends connected by three rectangular faces along the length.
(iv) A square pyramid has 5 faces (4 triangular faces and 1 square face): OWZ, OWX, OXY, OYZ and WXYZ. The diagram illustrates a square base with four triangular faces meeting at a single apex point O.
In simple words: A cuboid and cube each have 6 flat surfaces called faces. A triangular prism has 5 faces - two triangular ones and three rectangular ones. A square pyramid has 5 faces - one square base and four triangle-shaped sides.

Exam Tip: Always count faces, edges and vertices systematically - draw or visualise the shape and mark each part carefully to avoid missing any.

Question 2. Write the number of edges for each of the following solids: tetrahedron, rectangular pyramid, cube and triangular prism.
Answer:
(i) A tetrahedron has 6 edges: KL, LM, LN, MN, KN and KM. The diagram shows a triangular base with three edges meeting at the top vertex.
(ii) A rectangular pyramid has 8 edges: AB, AE, AD, AC, EB, ED, DC and CB. The diagram displays a rectangular base with four edges and four slant edges connecting to the apex.
(iii) A cube has 12 edges: PL, LK, KO, OP, MN, NJ, JI, IM, PM, LI, KJ and ON. The diagram shows a cube where four edges form the top face, four form the bottom face, and four connect the two faces vertically.
(iv) A triangular prism has 9 edges: QR, RS, QS, TU, TV, UV, QT, RU and SV. The diagram illustrates two triangular faces with three edges each, plus three edges connecting the corresponding vertices of the two triangles.
In simple words: Count the straight lines where faces meet. A tetrahedron has 6, a rectangular pyramid has 8, a cube has 12, and a triangular prism has 9 edges.

Exam Tip: To find edges systematically, count the lines on the top face, bottom face, and the vertical or slant edges connecting them.

Question 3. Write the number of vertices for each of the following solids: cuboid, square pyramid, tetrahedron and triangular prism.
Answer:
(i) A cuboid has 8 vertices: A, B, C, D, E, F, G and H. The diagram shows a rectangular box with four vertices on the top face and four on the bottom face.
(ii) A square pyramid has 5 vertices: O, W, X, Y and Z. The diagram displays four vertices forming the square base and one apex vertex at the top.
(iii) A tetrahedron has 4 vertices: K, L, M and N. The diagram illustrates a triangular base with three vertices and one vertex at the top.
(iv) A triangular prism has 6 vertices: Q, R, S, T, U and V. The diagram shows three vertices forming one triangular end and three vertices forming the other triangular end.
In simple words: Vertices are the corners or points where edges meet. Count them by identifying each corner point on the solid shape.

Exam Tip: Visualise the shape and mark each corner point to ensure you count all vertices without repetition.

Question 4. For a cube, write its vertices, edges and faces.
Answer:
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: MNIJ, POKL, PLIM, OKJN, POMN and LKJI

Additional notes: A cuboid is also recognised as a rectangular cube. A triangular pyramid is called a tetrahedron.
In simple words: A cube is a box shape where all sides are equal squares. It has 8 corners, 12 equal edges, and 6 square faces that are all the same size.

Exam Tip: Remember that a cube is a special type of cuboid where all dimensions are equal - use this to quickly recall the number 8, 12, and 6.

Question 1. State Euler's relation for three dimensional figures.
Answer: Euler's relation for a three dimensional figure can be written as:

\( F - E + V = 2 \)

Here,
\( F \) - Number of faces
\( E \) - Number of edges
\( V \) - Number of vertices

In simple words: This formula shows the connection between the number of flat surfaces, edges and corners in any solid shape - when you subtract edges from faces and add vertices, you always get 2.

Exam Tip: Memorise the formula as "F minus E plus V equals 2" and always verify it by counting carefully on diagrams.

Question 2. Find the number of edges in a cuboid and a tetrahedron using any method.
Answer:
(i) A cuboid has 12 edges, specifically: AD, DC, CB, BA, EA, FB, HD, DC, CG, GH, HE, and GF. The diagram shows these edges as the lines where faces meet on the rectangular solid.
(ii) A tetrahedron has 6 edges, namely: KL, LM, MN, NL, KM and KN. The diagram illustrates a pyramid with a triangular base and three edges meeting at the apex.
In simple words: An edge is where two flat surfaces join. Count where the faces touch - the cuboid has 12 such lines and the tetrahedron has 6.

Exam Tip: Count systematically by going around the top face, then the bottom face, then the vertical edges connecting them.

Question 3. Find the number of faces in a cube, pentagonal prism, tetrahedron and pentagonal pyramid.
Answer:
(i) A cube has 6 faces: IJKL, MNOP, PLIM, OKJN, POKL and MNIJ. All faces are equal squares on this solid shape.
(ii) A pentagonal prism has 7 faces - 2 pentagons and 5 rectangles: ABCDE, FGHIJ, ABGF, AEHF, EDIH, DCHG and BCIH. This solid has two parallel pentagonal ends joined by five rectangular sides.
(iii) A tetrahedron has 4 faces: KLM, KLN, LMN and KMN. Each face is a triangle in this pyramid-like shape.
(iv) A pentagonal pyramid has 6 faces - 1 pentagon and 5 triangles: NOPQM, SNM, SNO, SOP, SQP and SMQ. The solid has one pentagonal base and five triangular faces meeting at the apex S.
In simple words: Count the flat surfaces on each shape. A cube has 6 square faces, a pentagonal prism has 7 faces, a tetrahedron has 4 triangular faces, and a pentagonal pyramid has 6 faces.

Exam Tip: For prisms, count the two end faces first, then the rectangular sides. For pyramids, count the base and then the triangular sides.

Question 4. What do you mean by a vertex?
Answer: A vertex is the point where three faces of a three dimensional figure meet. It is also called a corner of the solid.
In simple words: A vertex is simply a corner - the sharp point where edges come together on a 3D shape.

Exam Tip: Always identify vertices by finding where three or more edges meet - each such meeting point is one vertex.

Question 5. Verify Euler's relation for a square prism, tetrahedron, triangular prism and square pyramid.
Answer: Euler's relation states that for any polyhedron,

\( F - E + V = 2 \)

Here, \( F \) = Number of faces, \( E \) = Number of edges, \( V \) = Number of vertices

(i) A square prism

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

Number of faces = \( F = 2 \) squares + 4 rectangular = 6

Number of edges = \( E = 12 \)

Number of vertices = \( V = 8 \)

\( \implies (F - E + V) = 6 - 12 + 8 = 2 \)

(ii) A tetrahedron

Number of faces = \( F = 4 \)

Number of edges = \( E = 6 \)

Number of vertices = \( V = 4 \)

\( \implies (F - E + V) = 4 - 6 + 4 = 2 \)

(iii) A triangular prism

Number of faces = \( F = 2 \) triangular + 3 rectangular = 5

Number of edges = \( E = 9 \)

Number of vertices = \( V = 6 \)

\( \implies (F - E + V) = 5 - 9 + 6 = 2 \)

(iv) A square pyramid

Number of faces = \( F = 2 \) triangular + 3 rectangular = 5

Number of edges = \( E = 8 \)

Number of vertices = \( V = 5 \)

\( \implies (F - E + V) = 5 - 8 + 5 = 2 \)

In all four cases, Euler's relation holds true, confirming the formula applies universally to polyhedra.
In simple words: No matter what shape you pick - whether it is a tetrahedron, prism or pyramid - when you subtract edges from faces and add vertices, you will always get exactly 2.

Exam Tip: Always verify by carefully counting faces, edges and vertices from a clear diagram, then substitute into the formula to confirm the answer is 2.