MathsCBSE Class 8 · Exploring Some Geometric Themes

Nets, Shortest Paths and Projections of Solids for Class 8

Q: How many nets does an octahedron have?

The chapter says 11 different nets, as for the cube; a dodecahedron has 43,380.

Q: Why must the net contain both the start and the end point?

A path on the solid corresponds to a path of the same length on the net, so both points must be on the same net to join them with a straight line.

Q: What if the straight line leaves the net?

Then it does not correspond to a path on the cuboid; choose another unfolding.

Q: Do I need isometric paper in the exam?

Isometric grids are mainly used to draw solids made of cubes; the chapter provides isometric paper at the end of the book.

Updated 17 October 2026 · Fact-checked

A net is a flat shape that folds into a solid: a cube has 11 nets, a regular tetrahedron 2, an octahedron 11 and a dodecahedron 43,380. An n-sided prism has n + 2 faces, 3n edges and 2n vertices; an n-sided pyramid has n + 1 faces, 2n edges and n + 1 vertices. The shortest path on a cuboid is a straight line on a net, found with the Baudhayana theorem, and projections give front, side and top views.

Understand Nets, Shortest Paths and Projections of Solids

Faces, edges, vertices. For a solid with flat faces: faces are the flat surfaces, edges are the line segments where faces meet, vertices are the points where edges meet. A cuboid has 6 faces, 12 edges and 8 vertices.

A prism has two congruent polygons as opposite faces joined by parallelograms. A prism on an $n$ -sided polygon has $n + 2$ faces, $3 n$ edges and $2 n$ vertices; for $n = 3$ (triangular prism) these are 5, 9, 6.
A pyramid has a polygonal base and a point (apex) joined to every vertex of the base. A pyramid on an $n$ -sided base has $n + 1$ faces, $2 n$ edges and $n + 1$ vertices. A triangular pyramid is a tetrahedron.

For both, $V - E + F = 2$ (for example $6 - 9 + 5 = 2$ for the triangular prism).

Nets. A net is obtained by unfolding a solid onto a plane; it can be folded back into the solid.

A cube has 11 different nets (counting nets related by rotation or flip as the same); a regular tetrahedron has 2; an octahedron (two square pyramids joined at their bases) has 11; a dodecahedron (pentagonal faces) has 43,380.
*Cylinder:* two circles and one rectangle. The rectangle has one side equal to the height and the other equal to the circumference $2 π r$ of the circle.
*Cone:* one circle and a sector of a circle whose radius is the slant height $l$ and whose arc length is the circumference of the base circle, $2 π r$ . So the sector angle is $\frac{r}{l} \times 36 0^{\circ}$ .
A sphere has no net: you cannot wrap a ball without wrinkles, gaps or overlaps using a flat sheet.

Shortest paths on a cuboid. On a plane, the shortest path between two points is the straight line. On the surface of a cuboid, imagine an ant going to a laddu. A path on the cuboid corresponds to a path of the same length on a net, and vice versa. So draw a net that contains both points, join them with a straight line, and check that the line stays inside the net. Different unfoldings give different lengths, so compare them. For a right triangle in the net, use the Baudhayana-Pythagoras theorem $d^{2} = a^{2} + b^{2}$ .

Projections (views). A solid seen from a viewpoint shows a profile. The front view, side view and top view are projections on the vertical, side and horizontal planes. We lose information when projecting a solid on a plane. The projection of a pair of parallel lines stays parallel, which helps in drawing. The shadow of an object in sunlight perpendicular to a plane is practically its projection.

A cylinder shows a rectangle from one side and a circle from the top; a cone shows a triangle and a circle.

Isometric drawing. In an orientation in which all projected edges of a cube have equal length (balance the cube on a vertex), the projection is a regular hexagon. Tiling the plane with hexagons gives an isometric grid with three directions for length, depth and height, used in engineering drawing.

Fractals are on the previous topic page of this chapter.

Nets, Shortest Paths and Projections of Solids: formulas to remember

Prism with n-sided base: $F = n + 2, E = 3 n, V = 2 n$; Faces, edges, vertices.
Pyramid with n-sided base: $F = n + 1, E = 2 n, V = n + 1$; Faces, edges, vertices.
Euler relation: $V - E + F = 2$; Holds for the prisms and pyramids here.
Cylinder net: $rectangle 2 π r \times h, two circles of radius r$; Total area = 2 pi r h + 2 pi r squared.
Cone net sector angle: $θ = \frac{r}{l} \times 36 0^{\circ}$; l is the slant height; the sector arc equals 2 pi r.
Shortest surface path on a cuboid: $d = (a + b)^{2} + c^{2}$; For opposite corners; compare the three choices of c.

How to solve Nets, Shortest Paths and Projections of Solids questions

Use these steps for nets, surface paths and views.

1For faces, edges and vertices, identify the base polygon (n sides) and use the prism or pyramid counts; check with V - E + F = 2.
2For a net, list the faces of the solid and their dimensions, and decide how they are joined; for a curved solid note which dimensions join up.
3For a surface path, draw nets that contain both points; join the points with a straight line and check it stays inside the net.
4Compute the length with the Baudhayana-Pythagoras theorem; repeat for every possible unfolding.
5The shortest is the minimum; the straight line through the solid is not a valid surface path.
6For views, imagine looking from the front, side and top and draw the outline, showing hidden edges if asked.
7Check that the three views agree with each other (heights, widths).

Nets, Shortest Paths and Projections of Solids in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Shortest path on a cuboid

Suppose an ant sits at one corner of a cuboid-shaped box of edges 3 cm, 4 cm and 12 cm and wants to reach the opposite corner by crawling only along the surface.

Find the shortest route length, and compare with the straight line through the box.

Try it first, then show the solutionSolution

Unfold in three ways: $(3 + 4)^{2} + 1 2^{2} = 193$ , $(3 + 12)^{2} + 4^{2} = 241$ , $(4 + 12)^{2} + 3^{2} = 265$ .
The smallest is $193$ ; since $13. 8^{2} = 190.44$ and $13. 9^{2} = 193.21$ , it lies between 13.8 cm and 13.9 cm.
The straight line through the box is $9 + 16 + 144 = 13$ cm, but the ant cannot go through the solid.

Answer: Between 13.8 cm and 13.9 cm (the square root of 193).

Memory hook: On the surface the ant must go over two faces; choose the unfolding with the smaller total.

Cylinder net

Imagine a tin of radius 7 cm and height 10 cm whose label is a rectangle that wraps once round the side.

Find the dimensions of the label and the total area of the net (take pi = 22/7).

Try it first, then show the solutionSolution

Label: height 10 cm and length equal to the circumference $2 \times \frac{22}{7} \times 7 = 44$ cm, so area $440$ cm $^{2}$ .
Each circle has area $\frac{22}{7} \times 49 = 154$ cm $^{2}$ ; two circles make 308 cm $^{2}$ .
Total $= 440 + 308 = 748$ cm $^{2}$ .

Answer: Label 44 cm by 10 cm; net area 748 square cm

Memory hook: The rectangle wraps around: its length is the circumference.

Tricks to remember Nets, Shortest Paths and Projections of Solids

Short memory aids. Say each one aloud once, then use it on a question the same day.

Prism: n+2, 3n, 2n
A prism on an n-gon has two bases plus n sides, 3n edges, 2n vertices.
Use it when: Faces-edges-vertices questions.
Pyramid: n+1, 2n, n+1
A pyramid has n sides plus one base, 2n edges, and n+1 vertices including the apex.
Use it when: Same questions for pyramids.
Unfold, then straight line
Shortest surface paths are straight on a net; always test each possible net.
Use it when: Ant and laddu problems.
Cylinder wraps, cone fans
The cylinder's curved face is a rectangle; the cone's is a sector.
Use it when: Drawing nets of curved solids.

Nets, Shortest Paths and Projections of Solids: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Counting for a prism · 1 mark · MCQBoard level

The number of faces, edges and vertices of a pentagonal prism are, respectively: (a) 7, 15, 10 (b) 6, 15, 10 (c) 7, 10, 15 (d) 5, 10, 15

Try it first, then show the solutionSolution

$n = 5$ : faces $n + 2 = 7$ , edges $3 n = 15$ , vertices $2 n = 10$ .

Answer: (a) 7, 15, 10

Example 2: Counting for a pyramid · 2 marks · Short answerBoard level

A pyramid has a hexagonal base. Find its numbers of faces, edges and vertices and verify V - E + F = 2.

Try it first, then show the solutionSolution

$n = 6$ : faces $= 7$ , edges $= 12$ , vertices $= 7$ . (1 mark)
$7 - 12 + 7 = 2$ . (1 mark)

Answer: 7 faces, 12 edges, 7 vertices; V - E + F = 2

Example 3: Net of a cylinder · 3 marks · Short answerHarder board level

A cylinder has radius 7 cm and height 10 cm. Describe its net with dimensions and find the total area of the net (take pi = 22/7).

Try it first, then show the solutionSolution

The net has a rectangle of height 10 cm and length equal to the circumference $2 \times \frac{22}{7} \times 7 = 44$ cm, and two circles of radius 7 cm. (1 mark)
Rectangle area $= 44 \times 10 = 440$ cm $^{2}$ ; circle area $= \frac{22}{7} \times 49 = 154$ cm $^{2}$ each. (1 mark)
Total $= 440 + 2 \times 154 = 748$ cm $^{2}$ . (1 mark)

Answer: Rectangle 44 cm by 10 cm and two circles of radius 7 cm; area 748 square cm

Example 4: Ant on a cuboid · 3 marks · Short answerHarder board level

An ant is at a corner of a cuboid with edges 3 cm, 4 cm and 12 cm. It crawls on the surface to the opposite corner. Find the length of the shortest path to one decimal place bounds, and compare with the space diagonal.

Try it first, then show the solutionSolution

The three unfoldings give $49 + 144 = 193$ , $225 + 16 = 241$ and $256 + 9 = 265$ . (1 mark)
The smallest is $193$ ; $13. 8^{2} = 190.44 < 193 < 193.21 = 13. 9^{2}$ , so it is between 13.8 cm and 13.9 cm. (1 mark)
The space diagonal is $9 + 16 + 144 = 13$ cm, shorter, but it passes through the solid and is not a path on the surface. (1 mark)

Answer: Between 13.8 cm and 13.9 cm; the space diagonal is 13 cm but is not allowed.

Example 5: Reading views of a stack of cubes · 4 marks · Long answerHarder board level

Unit cubes are stacked on a 3 x 3 grid with these heights (rows from front to back, columns left to right): row 1: 3, 1, 2; row 2: 1, 0, 2; row 3: 2, 1, 1. Find the total number of cubes, the number of squares in the top view, and the heights shown in the front view (looking along the rows, so you see the tallest stack in each column) and in the side view (tallest stack in each row).

Try it first, then show the solutionSolution

Total $= 3 + 1 + 2 + 1 + 0 + 2 + 2 + 1 + 1 = 13$ cubes. (1 mark)
Top view: every cell with at least one cube is a filled square; only one cell has height 0, so $9 - 1 = 8$ squares. (1 mark)
Front view (tallest in each column): column 1: 3, column 2: 1, column 3: 2. (1 mark)
Side view (tallest in each row): row 1: 3, row 2: 2, row 3: 2. (1 mark)

Answer: 13 cubes; 8 squares in the top view; front view heights 3, 1, 2; side view heights 3, 2, 2.

Where students lose marks in Nets, Shortest Paths and Projections of Solids

The slips examiners see most often, and what to write instead.

Mistake: Using the space diagonal as the shortest surface path
The ant must stay on the surface.
Wrong working
"Shortest path = sqrt(3^2 + 4^2 + 12^2) = 13 cm."
Right working
Compare the unfoldings: $193 \approx 13.9$ cm is the shortest surface path.
Do this: Draw the net and check the line stays inside it.
Usually costs: The whole answer.
Mistake: Using the radius instead of the circumference for the cylinder net
The rectangle wraps once around the circle.
Wrong working
"Net of a cylinder of radius 7 cm and height 10 cm: a 7 cm by 10 cm rectangle."
Right working
Rectangle: $2 π r = 44$ cm by 10 cm.
Do this: Imagine cutting the side and unrolling: its length is the circumference.
Usually costs: 2 marks.
Mistake: Wrong counts for a pyramid
A pyramid has n + 1 faces and 2n edges, not 2n faces.
Wrong working
"Hexagonal pyramid: 12 faces and 7 edges."
Right working
7 faces, 12 edges, 7 vertices.
Do this: Check with V - E + F = 2.
Usually costs: 1 mark.
Mistake: Reading the views from the wrong direction
Front view shows the tallest stack in each column when viewed along the rows; side view shows the tallest in each row.
Wrong working
"Front view: 3, 2, 2" (the row maxima).
Right working
Front view is the column maxima 3, 1, 2; side view is the row maxima 3, 2, 2.
Do this: State the viewing direction first.
Usually costs: 1 to 2 marks.

Exam tips

Always draw the net or the solid; a clear labelled figure earns marks.
For surface paths, show every unfolding you tried with its length, then state the minimum.
Quote the counts for prisms and pyramids using n and check with V - E + F = 2.
Mention the viewing direction when you give front, side and top views.
State pi as 22/7 or 3.14 when you use it.

Practice questions on Nets, Shortest Paths and Projections of Solids

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): A regular tetrahedron has only 2 different nets.
Reason (R): A cube has 11 different nets.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Do the counts of nets of one solid explain another?
Answer
(b). Both statements are true (the chapter gives 2 nets for a regular tetrahedron and 11 for a cube), but the number of nets of a cube does not explain the number for a tetrahedron.
Q2 · 2 marks · Very short answerBoard level
Find the number of faces, edges and vertices of a triangular prism, and verify V - E + F = 2.
Hint
n = 3 for a prism.
Answer
Faces $= 3 + 2 = 5$ , edges $= 3 \times 3 = 9$ , vertices $= 2 \times 3 = 6$ . $V - E + F = 6 - 9 + 5 = 2$ .
Q3 · 3 marks · Short answerHarder board level
A cone has base radius 6 cm and slant height 10 cm. Describe its net, and find the angle of the sector that forms the curved surface.
Hint
The arc of the sector equals the circumference of the base.
Answer
The net is a circle of radius 6 cm (the base) and a sector of radius 10 cm (the slant height). The arc length of the sector equals the base circumference $2 π \times 6 = 12 π$ cm, which is $\frac{θ}{360}$ of the full circumference $2 π \times 10 = 20 π$ cm. So $θ = \frac{12 π}{20 π} \times 36 0^{\circ} = 21 6^{\circ}$ .
Q4 · 4 marks · Long answerHOTS / competency
An ant crawls along the surface of a cuboid with edges 2 cm, 3 cm and 6 cm from one corner to the opposite one. Find the lengths for the three unfoldings, give bounds (one decimal digit) for the shortest, and say why the straight line of length 7 cm through the box is not a valid answer.
Hint
Try (2+3) by 6, (2+6) by 3 and (3+6) by 2.
Answer
$(2 + 3)^{2} + 6^{2} = 61$ , $(2 + 6)^{2} + 3^{2} = 73$ and $(3 + 6)^{2} + 2^{2} = 85$ . The shortest is $61$ with $7. 8^{2} = 60.84 < 61 < 62.41 = 7. 9^{2}$ , so between 7.8 cm and 7.9 cm. The space diagonal is $4 + 9 + 36 = 7$ cm, but the ant must stay on the surface, and a straight line through the solid does not lie on any net.
Q5 · 4 marks · Long answerHOTS / competency
For each requirement, name a solid and the viewpoints: (a) a rectangle from one viewpoint and a circle from another; (b) a circle from one viewpoint and a triangle from another; (c) a rectangle from one viewpoint and a triangle from another; (d) a pentagon from one viewpoint and a rectangle from another. Explain each choice.
Hint
Think of a cylinder, a cone, a triangular prism and a pentagonal prism.
Answer
(a) A cylinder: seen from the side it is a rectangle, from the top (along its axis) a circle. (b) A cone: from the top it is a circle, from the side a triangle. (c) A triangular prism lying on a rectangular face: from the end it shows a triangle, from above a rectangle. (d) A pentagonal prism: from one end it shows a pentagon, from the side a rectangle. Other solids can also work (for example a square pyramid also gives a triangle and a square), so the answers are not unique.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many nets does a cube have?Show answer
11.
2.How many nets does a regular tetrahedron have?Show answer
2.
3.What is the net of a cylinder?Show answer
A rectangle (height x circumference) and two circles.
4.How many faces, edges and vertices does a cuboid have?Show answer
6, 12 and 8.
5.What is the isometric projection of a cube?Show answer
A regular hexagon.
6.Why can a sphere not have a net?Show answer
A flat sheet cannot wrap a ball without wrinkles, gaps or overlaps.

Nets, Shortest Paths and Projections of Solids: frequently asked questions

How many nets does an octahedron have?

The chapter says 11 different nets, as for the cube; a dodecahedron has 43,380.

Why must the net contain both the start and the end point?

A path on the solid corresponds to a path of the same length on the net, so both points must be on the same net to join them with a straight line.

What if the straight line leaves the net?

Then it does not correspond to a path on the cuboid; choose another unfolding.

Do I need isometric paper in the exam?

Isometric grids are mainly used to draw solids made of cubes; the chapter provides isometric paper at the end of the book.