CMDLearn

Learn

Class 8 Maths: Exploring Some Geometric Themes, 8 practice questions with answers

CMDLearn ·

These are 8 practice questions on Chapter 4, Exploring Some Geometric Themes, of the new NCERT Class 8 Maths textbook, Ganita Prakash, Part 2. Each has a worked answer and the textbook page it comes from. They cover how far a fern's copies shrink, which fractal tree adds the same length at every step, counting the faces, edges and vertices of a cube with a corner cut off, whether a prism and a pyramid can match in faces and edges, the edges that need tape when a cube net is folded, the area of a box's net, the shade of a tilted card, and the outline of an isometric drawing of a cuboid. Try each one on paper before you open the answer.

These questions are our own, drafted with AI help. Every answer was re-worked by a test script and each question was compared with the chapter's examples, activities and exercises, but no practising teacher has reviewed them yet. If you find a mistake, please tell us.

What the chapter says you need

Facts from the NCERT textbook (Ganita Prakash, Grade 8, Part 2, Chapter 4, printed pages 70 to 102):

Fractals

Question 1. A fern frond is 27 cm long. Each leaflet looks like a copy of the whole frond one third as long, each part of a leaflet is a copy one third as long again, and so on. (a) How long is a copy three levels down? (b) A magnifying glass makes things look 3 times as long. How many such glasses, used one after another, make a copy three levels down look as long as the whole frond?

Answer: (a) 1 cm; (b) 3. (a) Each level is one third of the one above: 27 → 9 → 3 → 1 cm. (b) Each glass multiplies the look by 3: 1 → 3 → 9 → 27 cm, so 3 glasses bring it back to the size of the whole frond. This is what self-similar means: each small copy, enlarged by the right amount, looks like the whole. (Self-similar shapes and the fern, page 70.)

Question 2. Two fractal trees start from the same trunk. In tree A, every newest branch splits into 2 branches half as long; in tree B, into 3 branches half as long. For which tree does every step add exactly the same total length as the step before? For the other tree, how does the added length change from step to step?

Answer: tree A; for tree B the added length is 1.5 times as much each step. At each step the number of newest branches is multiplied by the number of splits, and each new branch is half as long. For tree A, 2 × 1/2 = 1, so each step adds the same length as the step before. For tree B, 3 × 1/2 = 1.5, so each step adds 1.5 times as much as the step before, and the tree's total length keeps growing faster and faster. (Self-similar shapes, page 70.)

Solids and their nets

Question 3. A corner is sliced off a cube by one straight cut through three points, one on each of the three edges that meet at that corner. Count the faces, edges and vertices of the solid that is left.

Answer: 7 faces, 15 edges, 10 vertices. Faces: the cube's 6 faces are all still there (three of them are trimmed), plus the new triangular face: 7. Edges: the cube's 12 edges are all still there (three are shortened), plus the 3 sides of the triangle: 15. Vertices: the cut removes 1 corner and makes 3 new ones, so 8 − 1 + 3 = 10. Check with faces + vertices − edges: 7 + 10 − 15 = 2, the same as for the cube (6 + 8 − 12 = 2). (Faces, edges and vertices, pages 78 and 79.)

Question 4. A prism whose two matching faces have n sides has n + 2 faces and 3n edges, and a pyramid whose base has m sides has m + 1 faces and 2m edges. Can a prism and a pyramid have both the same number of faces and the same number of edges? Explain.

Answer: no. The same number of faces needs n + 2 = m + 1, so m = n + 1. The same number of edges needs 3n = 2m = 2(n + 1) = 2n + 2, so n = 2. But a polygon must have at least 3 sides, so no prism and pyramid can match in both. (For example, a triangular prism has 5 faces and 9 edges, and the square pyramid with 5 faces has only 8 edges.) (Prisms and pyramids, page 79.)

Question 5. A cube is made by folding a paper net of six squares. When it is folded, how many of the cube's 12 edges are already joined by a fold, and how many need tape or a glued flap?

Answer: 5 are folds; 7 need tape or a flap. A net of six squares is joined into one piece by the sides the squares share, and six squares need exactly 5 shared sides to hold together in one piece without any loop. Each shared side becomes a fold, which is one edge of the cube. The cube has 12 edges, so the other 12 − 5 = 7 edges are where two cut sides of the net meet, and each needs tape or a flap stuck to the next face. (Nets of a cube, flaps and tape, page 80.)

Question 6. A closed box 4 cm long, 3 cm wide and 2 cm tall is opened out into a net. What is the total area of the net, and why does every net of this box have the same area?

Answer: 52 square cm; a net is made of the box's own six faces. The box has two faces 4 × 3 = 12, two faces 4 × 2 = 8 and two faces 3 × 2 = 6 square cm, so any net, which is just these six rectangles laid out flat, has area 2 × (12 + 8 + 6) = 52 square cm. Different nets arrange the same six faces differently, so the area never changes (flaps for glue would add extra). (Nets of cuboids, pages 80 and 81.)

Projections and views

Question 7. A rectangular card 10 cm by 13 cm stands on its 10 cm edge and is tilted until its top edge is 5 cm above the ground. At noon the sunlight falls straight down. What fraction of the card's own area does its shade on the ground cover?

Answer: 12/13. With the light falling straight down, the shade is the card's projection on the ground. The 10 cm edge on the ground stays 10 cm. Each 13 cm side rises 5 cm, so its shade is the third side of a right triangle with hypotenuse 13 and height 5: the square root of 169 − 25 = 144, which is 12 cm. The shade is 10 × 12 = 120 square cm and the card is 10 × 13 = 130 square cm, so the shade covers 120/130 = 12/13 of the card. (Projections, page 88; shadows when sunlight falls straight down, page 94.)

Question 8. On isometric dot paper, a cube 1 unit on each side is drawn with an outline that is a regular hexagon of side 1 unit. What shape is the outline of a cuboid 2 units long, 3 units wide and 4 units tall drawn the same way, and how long is each side of the outline?

Answer: a hexagon (not regular) with sides 2, 3, 4, 2, 3 and 4 units. In an isometric drawing every edge of the cuboid is drawn along one of three directions at its true length in units. The outline goes round the solid along six edges: two of length 2, two of length 3 and two of length 4, with matching edges opposite each other, so the outline is a hexagon whose opposite sides are equal and parallel. (Drawing solids on isometric paper, pages 97 to 99.)

Practise more, and check your marks

Where CMDLearn fits

CMDLearn is building an AI tutor for Classes 1 to 10 that explains a step and checks the child's working, but leaves the writing to the child. It is still in development, and no child uses it yet. If you would like to hear when there is something to try, you can join the waitlist.

Sources

All opened and read on 30 October 2026. Page numbers are the printed page numbers of the textbook; printed page = PDF page + 69.

The rules are from the textbook pages above. The questions, worked answers and checks are our own, and every number was re-computed by script. This guide is general information; follow your teacher and the textbook.