RS Aggarwal Solutions for Class 6 Chapter 16 Triangles

Exercise 16.1

Question 1. Name any four objects from your environment, which have the form of
(i) A cuboid
(ii) A cube
Answer:
(i) A lunch box, a compass box, a book, and a duster.
(ii) A disc, a chalk box, a cubical cabin, and a tissue box.
In simple words: Cuboids are rectangular boxes like lunch containers and books. Cubes are equal-sided shapes like dice and tissue boxes.

Exam Tip: Remember that cuboids have unequal dimensions while cubes have all edges equal - this distinction helps identify real-world examples quickly.

Question 2. Draw a diagram to represent a cuboid. Label its vertices as P, Q, R, S, T, U, V and W. Now write the names of its faces and edges.
Answer: The diagram of a cuboid is shown below:


Faces:
PQRS (bottom)
TUVW (top)
TPQU (front)
WSRV (back)
TPSW (left)
UVRQ (right)

Edges:
PQ, QR, RS, SP
TU, UV, VW, WT
WS, SR, RV, VW
UV, VR, RQ, QU
In simple words: A cuboid has 6 faces (flat surfaces), 12 edges (lines where faces meet), and 8 vertices (corners). The top and bottom faces are opposite, as are the front-back and left-right pairs.

Exam Tip: Always ensure all 6 faces are named and all 12 edges are listed - missing any will result in lost marks.

Question 3. Draw a diagram to represent a cube. Label its vertices as A, B, C, D, E, F, G and H. Now write the names of its faces and edges.
Answer: The cube given below:



A cube has 6 faces and 12 edges.

Faces:
ABCD, EFGH, BVGF, ADHE, ABFE and CDHG.

Edges:
AB, BC, CD, DA, EF, FG, GH, HE, BF, CG, AE and HD.
In simple words: A cube is a special cuboid where all edges are equal in length. It has the same number of faces, edges, and vertices as a cuboid - just with all sides being identical.

Exam Tip: A cube is a special case of a cuboid - memorize that both have 6 faces, 12 edges, and 8 vertices, but only a cube has all equal edges.

Question 4. Figure represents a cuboid. The lengths of the edges AE, EF and FG are indicated as l, b and h respectively. Indicated the lengths of all other edges.
Answer:
AE = DH = BF = CG = l

EF = AB = CD = GH = b

FG = EH = AD = BC = h
In simple words: In a cuboid, opposite edges have equal lengths. The three distinct measurements (length, width, height) repeat across all parallel edges. So if one edge measures l, all edges parallel to it also measure l.

Exam Tip: Use the property that opposite edges in a cuboid are always equal - this saves time identifying multiple edge lengths.

Question 5. In figure, if the face EFGH is taken as the base, the name the lateral faces. Also, name the line segment represent the height of the cuboid. (fig. from book)
Answer: Following are the lateral faces for the base EFGH.

AEHD, AEFB, BFGC, DHGC.

AE or DH or BF or CG are the line segments representing the height of the cuboid.
In simple words: Lateral faces are the four vertical faces surrounding the base. When EFGH is the bottom, the four sides standing up are the lateral faces. The height is any vertical edge connecting the base to the top.

Exam Tip: Lateral faces are always the sides of a solid (never the top or bottom). The height is always perpendicular to the base you've chosen.

Question 6. In fig. name the four diagonals of the cuboid. (fig. from book)
Answer: The four diagonals of the cuboid CE, BH, AG and DF.
In simple words: Diagonals are line segments that go from one corner of the cuboid through the interior to another corner, not lying on any face. A cuboid has exactly four such diagonals.

Exam Tip: Diagonals pass through the interior of the solid - they are not edges or face diagonals. Each connects opposite vertices.

Question 7. In fig., name the (fig. from book)
(i) face parallel to BFGC
(ii) faces adjacent to BFGC
(iii) three edges which meet in the vertex G.
Answer:
(i) The faces parallel to BFGC is AEHD.

(ii) The faces adjacent to BFGC are BCDA, DCGH, ABFE, and EFGH.

(iii) GF, GH, CG.
In simple words: Parallel faces are opposite and never touch. Adjacent faces share an edge with the given face. Edges meeting at a vertex are the three edges that connect at that corner point.

Exam Tip: A cuboid face has exactly one parallel face, four adjacent faces, and at each vertex exactly three edges meet.

Question 8. Fill in the blanks to make the following statements true:
(i) A cuboid has _____ vertices.
(ii) A cuboid has _____ edges.
(iii) A cuboid has _____ faces.
(iv) The number of lateral faces of a cuboid is _____.
(v) A cuboid all of whose edges are equal is called a _____.
(vi) Two adjacent faces of a cuboid meet in a line segment called its _____.
(vii) Each edge of a cuboid can be obtained as a line segment called in which two _____ meet.
(viii) _____ edges of a cube (or cuboid) meet at each of its vertices.
(ix) A _____ is a cuboid in which all the six faces are squares.
(x) The three concurrent edges of a cuboid meet at a point called the _____ of the cuboid.
Answer:
(i) eight
(ii) twelve
(iii) six
(iv) four
(v) cube
(vi) edge
(vii) adjacent faces
(viii) three
(ix) cube
(x) vertex or corner.
In simple words: A cuboid has 8 corners, 12 edges forming its skeleton, and 6 flat surfaces. Its 4 side faces are lateral. A cube is the special cuboid with equal edges and square faces. Three edges always converge at each corner point.

Exam Tip: These basic properties are frequently tested - memorize all six key numbers (8 vertices, 12 edges, 6 faces, 4 lateral faces).

Question 9. In each of the following, state if the statement is true (T) or false (f):
(i) Number of faces in a cuboid and the number of faces in a cube are equal.
(ii) A cube has twelve vertices.
Answer:
(i) True
(ii) False
In simple words: Both cuboids and cubes have 6 faces, so statement (i) is true. However, both have 8 vertices, not 12, making statement (ii) false. A cube has 12 edges, which might cause confusion.

Exam Tip: Be careful not to confuse the number of vertices (8), edges (12), and faces (6) - a common mistake is mixing up edges and vertices.

Question 10. For the cuboid shown: (fig. from book)
(i) What is the base of this cuboid?
(ii) What are the lateral faces of this cuboid?
(iii) Name one pair of opposite faces. How many pairs of opposite faces are there. Name them.
(iv) Name all the faces of this cuboid which have X as a vertex. Also, name those which have VW as a side.
(v) Name the edges of this cuboid which meet at the vertex P. Also name those faces which meet at this vertex.
Answer:
(i) UVWX is the base of a cuboid.

(ii) The lateral faces for the base UVWX are UXSP, QVWR, PQVU and SXWR.

(iii) Any one pair of opposite faces among the lateral faces of the base are POPU and SXWR or UXSP and QVWR.

There are two pairs of opposite faces among the lateral faces of the base of the cuboid.

(iv) The faces, which have one of the vertex as X, are UVWX, UXSP and SXWR.

The faces, which have VW as side, are QVWR and UVWX.

(v) Edges which meet at P are UP, SP, and PQ.

Faces which meet at vertex P are PQRS, UPSX, and PQVU.
In simple words: The base is the face chosen as the bottom. The lateral faces are the four sides going upward from that base. Opposite faces don't share any edge. Corners touch three faces and three edges each. A side of a face is one of its edges.

Exam Tip: Always identify the base first, then determine which faces are lateral. Remember that opposite faces appear in three pairs for any cuboid.

Question 11. The dimensions of a cuboid with vertices A, B, C, D, E, F, G and H are as shown
(i) Which edges are of length 4 cm? Which edges are of length 5 cm?
(ii) Which faces have area equal to 20cm²?
(iii) Which faces have the largest area? What is this largest area?
(iv) Which faces have a diagonal equal to 5 cm?
(v) What is the area of the base of this cuboid?
(vi) Do all the lateral faces have the same area?
Answer:
(i) The edges of 4 cm length are AD, EH, BC, and FG.

The edges of 5 cm length are AB, EF, CD and GH.

(ii) The faces having dimensions of 5 cm × 4 cm would have an area of 20 cm². And such faces are ABCD and EFGH.

(iii) ABCD and EFGH have the largest area of 20 cm².

(There are three pairs of opposite faces of equal area. The area of opposite faces are: 3 × 4 cm², 4 × 5 cm², and 3 × 5 cm². And among these, 4 × 5 cm² is the largest.

(iv) The faces having sides of 3 cm and 4 cm respectively would have the diagonal of 5 cm. (As hypotenuse of a right- angles triangle is: 3² + 4² = 5²).

Therefore, the faces ADHE and BCGF have the diagonal of 5 cm.

(v) The base has q dimension of 4 cm × 5 cm, so area of abase is: 4 × 5 = 20 cm².

(vi) No, all lateral faces do not have the same area. The two lateral faces have an area of 3 × 5 = 15 cm² and rest of the two lateral faces have an area of 3 × 4 = 12 cm².
In simple words: A cuboid with three different dimensions has edges in three different lengths. The six faces form three pairs of equal opposite faces, with three different areas. The longest face (largest area) uses the two longest dimensions. A diagonal within a face can be calculated using the Pythagorean theorem.

Exam Tip: For a cuboid with dimensions p, q, r: there are 4 edges of each length, giving three pairs of faces with areas pq, qr, and pr. Use the Pythagorean theorem for face diagonals.

Exercise 16.2

Question 1. Give two new examples of each of the for three-dimensional shapes:
(i) Cone
(ii) Sphere
(iii) Cylinder
(iv) Cuboid
(v) Pyramid
Answer:
(i) A school bell and a funnel.
(ii) A tennis ball and a model of a globe.
(iii) Drink cans and delivering pipes for a water and gas.
(iv) A match box and brick.
(v) A paper- weight and a tower like the Eiffel tower.
In simple words: Cones look like ice cream cones with a circular base tapering to a point. Spheres are perfectly round like balls. Cylinders have two circular bases connected by a curved side like a can. Cuboids are rectangular boxes. Pyramids are like Egyptian tombs with a base and triangular sides meeting at a tip.

Exam Tip: When naming real-world examples, choose common objects that clearly show the defining features of each shape - avoid ambiguous items.

Question 2. What is the shape of:
(i) instrument box
(ii) a brick
(iii) a match box
(iv) a rod- roller
(v) a sweet laddoo
Answer:
(i) My instrument box is in the shape of a cuboid.
(ii) A brick is in the shape of the cuboid.
(iii) A match - box is in the shape of a cuboid.
(iv) A road - roller is in the shape of a cylinder.
(v) A sweet laddoo is shaped like a sphere.
In simple words: Tool boxes, bricks, and matchboxes are all rectangular containers (cuboids). Road rollers have cylindrical drums for flattening surfaces. Sweet laddoos are spherical (round) sweets.

Exam Tip: When identifying shapes of everyday objects, look at their key characteristics: cuboid = rectangular, cylinder = round with flat ends, sphere = completely round.

Objective Type Questions:

Question 1. Total number of faces of a cuboid is
(a) 4
(b) 6
(c) 8
(d) 12
Answer: (b) 6
In simple words: A cuboid always has six faces - three pairs of opposite rectangular surfaces.

Exam Tip: This is a fundamental property - memorize it as 6 faces, 12 edges, 8 vertices for a cuboid.

Question 2. Total number of edges of a cuboid is
(a) 4
(b) 6
(c) 8
(d) 12
Answer: (d) 12
In simple words: A cuboid has 12 edges - four on the top face, four on the bottom, and four connecting them vertically.

Exam Tip: Count the edges systematically: 4 around the base + 4 around the top + 4 vertical = 12 total.

Question 3. Number of faces of a cuboid is
(a) 4
(b) 6
(c) 8
(d) 12
Answer: (c) 8
In simple words: The question asks for vertices, not faces. A cuboid has 8 vertices or corner points where edges meet.

Exam Tip: Be careful - this question uses awkward wording. The answer refers to vertices (corners), not faces. A cube/cuboid has 8 vertices.

Question 4. Which one of them is example of cuboid?
(a) a dice
(b) a football
(c) a gas pipe
(d) an ice- cream cone
Answer: (a) A dice
In simple words: A dice is a cube (a special cuboid with all edges equal). A football is a sphere, a gas pipe is a cylinder, and an ice-cream cone is a cone.

Exam Tip: A cube is a special type of cuboid - all cubes are cuboids, but not all cuboids are cubes.

Question 5. A brick is an example of
(a) cube
(b) cuboid
(c) prism
(d) cylinder
Answer: (b) Cuboid
In simple words: Bricks are rectangular boxes with length greater than width and height, making them cuboids (not cubes, which would have equal dimensions).

Exam Tip: Distinguish between cubes and cuboids: if all dimensions are equal, it's a cube; if they differ, it's a cuboid.

Question 6. A gas pipe is an example of
(a) cone
(b) a cylinder
(c) cube
(d) sphere
Answer: (b) A cylinder
In simple words: Gas pipes are cylindrical tubes - long circular tubes with two flat circular ends at top and bottom.

Exam Tip: Cylinders are defined by having two circular faces and a curved side surface connecting them.

Question 7. If the base radius and height of a right circular cone are 3 cm and 4 cm in lengths, then the slant height is
(a) 5 cm
(b) 2 cm
(c) 25 cm
(d) 6 cm
Answer: (a) 5 cm

L = \( \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \) cm.

In simple words: The slant height of a cone is found using the Pythagorean theorem - it forms the hypotenuse of a right triangle with the radius and vertical height as the other two sides.

Exam Tip: Always use the formula L = √(r² + h²) for slant height, where r is the base radius and h is the vertical height.

Question 8. The number of faces of a triangular pyramid is
(a) 3
(b) 4
(c) 6
(d) 8
Answer: (b) 4

A pyramid is called a triangular pyramid if its base is a triangle.

In simple words: A triangular pyramid (tetrahedron) has 4 faces - one triangular base and three triangular sides meeting at an apex point.

Exam Tip: The number of faces of a pyramid equals the number of sides of its base plus one (the base itself). For a triangular base: 3 + 1 = 4 faces.

Question 9. The number of edges of a triangular pyramid is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 6
In simple words: A triangular pyramid has 6 edges - three forming the triangular base and three going from the base vertices up to the apex.

Exam Tip: For any pyramid: edges = 2 × (number of sides of base). For a triangular base: 2 × 3 = 6 edges.

Question 10. A tetrahedron is a pyramid whose base is a
(a) triangle
(b) square
(c) rectangle
(d) quadrilateral
Answer: (a) Triangle
In simple words: A tetrahedron has a triangular shape as its foundation. It is a three-dimensional solid with four triangular faces in total.

Exam Tip: Remember that "tetra" means four and "hedron" means face - a tetrahedron has exactly 4 triangular faces, which is why the base must be a triangle.