The figure is made of identical triangles.
- Complete the table for layers 8 and 17.
- If each small triangle has a height of 3 cm and a perpendicular base of 2 cm, find the area of all the triangles at the 31st layer.
Layer number |
Number of triangles |
1 |
1 |
2 |
3 |
3 |
5 |
4 |
7 |
... |
... |
8 |
(i) |
... |
... |
17 |
(ii) |
(a)
1st layer: 0 + 1 = 1 triangle
2nd layer: 1 + 2 = 3 triangles
3rd layer: 2 + 3 = 5 triangles
4th layer: 3 + 4 = 7 triangles
Formula:
Number of triangles in a layer = Pattern number + (Pattern number - 1)
Number of triangles for layer 8
= 8 + (8 - 1)
= 8 + 7
= 15
Number of triangles for layer 17
= 17 + (17 - 1)
= 17 + 16
= 33
(b)
Number of triangles in the 31st layer
= 31 + (31 - 1)
= 31 + 30
= 61
Area of one small triangle
=
12 x 3 x 2
= 3 cm
2Area of all the triangles at the 31st layer
= 3 x 61
= 183 cm
2 Answer(s): (a) 15, 33; (b) 183 cm
2