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NCERT Solutions for Class 5th Maths Chapter 7 Counting faces, vertices and edges of two beautiful solids — Icosahedron and Dodecahedron

Book page 103 Updated on2026-09-19

Q1.
What do these names mean? Once you count their faces, you will know.
Answer

The names simply say how many faces the solid has.

NameBroken into partsWhat it meansNumber of faces
Icosahedronicosa + hedrontwenty + faces20
Dodecahedrondodeca + hedrontwelve + faces12

How to count the faces on your model without going wrong

  1. Hold the model with one face flat on the table. Count the faces you can see on the top and around the sides.
  2. Mark each face as you count it with a small pencil dot, so you never count the same one twice.
  3. Turn the model over and count the rest.
  4. Add. For the icosahedron you should reach 20; for the dodecahedron, 12.
Where the names come from: They are Greek words. Hedron means a face or a seat. You already know other words built the same way: a pentagon has 5 sides because penta means five, and a hexagon has 6 because hexa means six.
Two more of the family: A tetrahedron has 4 faces (tetra = four) and an octahedron has 8 (octa = eight). Along with the cube, these five solids are the only ones whose faces are all the same regular shape.
Q2.
What shapes do you see in an icosahedron and a dodecahedron? Icosahedron: ........ Dodecahedron: ........
Answer

Icosahedron: equilateral triangles.  Dodecahedron: regular pentagons.

one face of a dodecahedron a regularpentagon — 12 ofthem one face of an icosahedron an equilateral triangle — 20 ofthem
Every face of an icosahedron is the same triangle; every face of a dodecahedron is the same pentagon.
Icosahedron = 20 equilateral triangles
Dodecahedron = 12 regular pentagons
Why these two shapes: For a solid to close up neatly, the faces meeting at each corner must bend round without lying flat. Triangles and pentagons are small enough at the corner to let that happen. This is why you cannot build a solid out of regular hexagons — three hexagons at a corner lie flat, exactly as you saw in the tessellation activity.
Where you have seen them: A football is made of pentagons and hexagons stitched together. A twenty-sided dice used in board games is an icosahedron.
Q3.
Do all the faces look the same? Icosahedron: ........ Dodecahedron: ........
Answer

Icosahedron: yes.  Dodecahedron: yes. In both solids every face is exactly the same shape and the same size.

How to be sure

  1. Cut out one spare face from the leftover paper of the net — one triangle, or one pentagon.
  2. Hold it against each face of the model in turn.
  3. It covers every face exactly, with nothing sticking out and no gap left.
  4. So all the faces are identical.
Why they had to be the same: Look at the net at the end of your book. It is made of 20 identical triangles (or 12 identical pentagons) printed side by side. Folding does not change a shape's size, so every face of the finished model is the same as every other.
Not every solid is like this. A brick-shaped box has 6 rectangular faces but they are not all the same size. A solid in which all faces are identical and the same number meet at every corner is called a regular solid. There are only five of them in the whole world of mathematics.
Q4.
How many faces meet at a vertex (point)? Icosahedron: ........ Dodecahedron: ........ Do the same number of faces meet at each vertex? Icosahedron: ........ Dodecahedron: ........
Answer

Icosahedron: 5 faces meet at each vertex. Dodecahedron: 3 faces meet at each vertex. And yes — in both solids the number is the same at every single vertex.

SolidShape of each faceFaces at one vertexSame at every vertex?Number of vertices
IcosahedronEquilateral triangle5Yes12
DodecahedronRegular pentagon3Yes20

How to count at a vertex

  1. Pick one corner point of the model and put your fingertip on it.
  2. Turn the model slowly and count the faces that touch your fingertip.
  3. Do it again at two or three other corners to check you get the same number every time.
Icosahedron: 20 faces × 3 corners each = 60 corner-pieces
Each vertex uses 5 of them, so vertices = 60 ÷ 5 = 12

Dodecahedron: 12 faces × 5 corners each = 60 corner-pieces
Each vertex uses 3 of them, so vertices = 60 ÷ 3 = 20
Why the number is the same everywhere: These are regular solids. Regular means the same rule holds all over — identical faces, and the same number of them at every corner. If one corner had 4 faces and another had 5, the solid would be lumpy, not regular.
A neat swap: The icosahedron has 20 faces and 12 corners. The dodecahedron has 12 faces and 20 corners. The two numbers have simply changed places. These two solids are partners.
Q5.
How many edges do you see? Icosahedron: ........ Dodecahedron: ........ How did you count them such that you do not miss out any edge or count an edge twice?
Answer

Both solids have 30 edges.

The counting trick — count face by face, then halve

  1. Count the edges of one face. A triangle has 3; a pentagon has 5.
  2. Multiply by the number of faces. Icosahedron: 20 × 3 = 60. Dodecahedron: 12 × 5 = 60.
  3. Notice the double counting. Every edge of the solid is shared by exactly 2 faces, so each one has been counted twice.
  4. Divide by 2. 60 ÷ 2 = 30 for each solid.
Icosahedron: (20 × 3) ÷ 2 = 60 ÷ 2 = 30 edges
Dodecahedron: (12 × 5) ÷ 2 = 60 ÷ 2 = 30 edges

A second way, if you would rather count on the model:

  1. Put a small pencil mark on each edge as you count it.
  2. Work in rings — first all the edges round the top vertex, then the ring below, and so on.
  3. An edge with a mark on it has already been counted, so skip it.
Why dividing by 2 is right: An edge is the line where two faces meet, like the crease between two walls of a room. When you go round face by face, both of those faces claim the same crease. Halving removes the double claim.
Check with Euler's rule: For any such solid, faces + vertices − edges = 2. Icosahedron: 20 + 12 − 30 = 2 ✓ Dodecahedron: 12 + 20 − 30 = 2 ✓ Try it on a cube too: 6 + 8 − 12 = 2 ✓
Q6.
Can you think of any other solid shapes that have faces that look the same? Do the same number of faces meet at each common vertex? ………..
Answer

Yes — the cube, the tetrahedron and the octahedron. In all of them the same number of faces meets at every vertex.

SolidFacesShape of each faceFaces at each vertexVerticesEdges
Tetrahedron4Equilateral triangle346
Cube6Square3812
Octahedron8Equilateral triangle4612
Dodecahedron12Regular pentagon32030
Icosahedron20Equilateral triangle51230
tetrahedron 4 triangles cube 6 squares octahedron 8 triangles
Three more solids whose faces are all the same, with the same number meeting at every corner.
Why only five in all: At a corner you need at least 3 faces, and their corner angles must add up to less than 360 — otherwise the faces lie flat and never fold up into a solid. Triangles (60) allow 3, 4 or 5 at a corner. Squares (90) allow only 3. Pentagons (108) allow only 3. Hexagons (120) allow none, because 3 of them make exactly 360 and stay flat. That gives exactly five solids and no more.
Do the same number of faces meet at each common vertex? Yes — in every one of these five solids. That is exactly what makes them regular.
Q7.
You can also build some 3-D shapes using straws or ice-cream sticks and clay or play dough. Which shapes did you make?
Answer

How to build them

  1. Roll small balls of clay, about the size of a marble. These will be the corners (vertices).
  2. Cut the straws to the same length — they will be the edges.
  3. Push a straw into a clay ball to join two corners.
  4. Build the flat base first, then add the straws that go upwards and pinch them together at the top.
  5. Count as you go: corners, straws and faces. Write the three numbers down.

Sample answer — the three models in the book

ModelWhat it isClay balls (vertices)Straws (edges)Faces
The blue oneOctahedron6128 triangles
The green onePyramid584 triangles + 1 square base
The red onePentagonal prism10152 pentagons + 5 rectangles
Check each one with Euler's rule (faces + vertices − edges = 2):
Octahedron: 8 + 6 − 12 = 2 ✓
Square pyramid: 5 + 5 − 8 = 2 ✓
Pentagonal prism: 7 + 10 − 15 = 2 ✓
Why the straw model has no faces to touch: The straws only show the edges. The faces are the empty spaces between them. Looking at a shape as just its edges makes counting much easier — nothing is hidden behind a wall.
Try this: Build a cube out of 12 straws and 8 clay balls, then press gently on one corner. It wobbles. Now add 2 more straws across two opposite faces, making triangles — the wobble stops. Triangles are stiff, and that is why bridges and towers are full of them.
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