NCERT Solutions Ganita Prakash Chapter 6 The tangram pieces — Figure it Out
Book page 139 Updated on2026-09-05
Q1.
Explore and figure out how many pieces have the same area.
Answer
Place the pieces on top of one another. Taking the smallest triangle C as one unit of area, the seven pieces measure:
The tangram square: A and B are the two big triangles, F the middle-sized triangle, C and E the two small triangles, D the square and G the parallelogram.
Piece
Shape
Area (taking C = 1)
A
Large triangle
4
B
Large triangle
4
C
Small triangle
1
D
Square
2
E
Small triangle
1
F
Middle-sized triangle
2
G
Parallelogram
2
Answer — there are three groups of equal area:
A and B have the same area (4 units each).
C and E have the same area (1 unit each).
D, F and G all have the same area (2 units each), even though one is a square, one a triangle and one a parallelogram.
Check the total: 4 + 4 + 1 + 2 + 1 + 2 + 2 = 16 units — and the seven pieces do make one big square.
Q2.
How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?
Answer
Place C and E on top of the square D — together they cover it exactly.
Area of D = area of C + area of E
and area of C = area of E, so
Area of D = 2 × area of C
Answer: Shape D is twice as big as Shape C.
The relationship: C and E are equal small triangles, and the two of them join along their longest sides to form the square D. So
D = C + E and C = E = half of D
Q3.
Which shape has more area: Shape D or F? Give reasons for your answer.
Answer
Neither — D and F have exactly the same area.
Area of D = C + E = 1 + 1 = 2 units
Area of F = C + E = 1 + 1 = 2 units
Reason: the square D can be covered exactly by the two small triangles C and E. The middle-sized triangle F can also be covered exactly by the same two triangles — just place them side by side along their short sides instead. Two shapes that are made from the same pieces must have the same area.
The big idea: area does not depend on the shape. Cut a figure up and rearrange the parts — the area stays the same, even though the figure now looks completely different.
Q4.
Which shape has more area: Shape F or G? Give reasons for your answer.
Answer
Again, neither — F and G have equal areas (2 units each).
F (middle triangle) = C + E = 2 units
G (parallelogram) = C + E = 2 units
Reason: both the triangle F and the parallelogram G can be exactly covered by the two small triangles C and E. Cut the parallelogram G along its short diagonal and you get two small triangles; put those two triangles together the other way and you get F.
Check it with the square: D, F and G are all 2 units, so together they make 6 units — and 6 + 4 + 4 + 1 + 1 = 16, the whole square. ✔
Q5.
What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big?
Answer
Area of A = 4 units Area of G = 2 units
4 ÷ 2 = 2
Answer: Shape A is exactly twice as big as Shape G — not four times.
Shape A is four times as big as Shape C (the small triangle), because 4 ÷ 1 = 4. That is why it is worth fixing one piece (C) as the unit and measuring everything against it.
Comparison
Working
Answer
A compared with G
4 ÷ 2
2 times
A compared with C
4 ÷ 1
4 times
A compared with B
4 ÷ 4
equal
Q6.
Can you now figure out the area of the big square formed with all seven pieces in terms of the area of Shape C?
Answer
Just add up all seven pieces, measuring each in units of C.
A + B + C + D + E + F + G
= 4 + 4 + 1 + 2 + 1 + 2 + 2
= 16 × area of Shape C
Answer: the big square has an area equal to 16 small triangles (16 C).
Nice check: 16 is a square number, 16 = 4 × 4 — and indeed four small triangles C fit along each half-diagonal of the big square.
Q7.
Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle in terms of the area of Shape C now? Give reasons for your answer.
Answer
The rectangle also has an area of 16 × area of Shape C.
Reason: the rectangle is made from exactly the same seven pieces. Nothing was added and nothing was thrown away, so the total region covered cannot change. Rearranging pieces changes the shape, never the area.
Area of the square = 16 C
Area of the rectangle = 16 C
So they are equal
Try This: the seven pieces can also be arranged into a triangle, a trapezium and dozens of animal and human figures. Every one of them has an area of 16 C.
Q8.
Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same? Give an explanation for your answer.
Answer
The perimeters are different — the rectangle has the longer boundary, even though the two figures have the same area.
Explanation: perimeter depends on how the pieces are placed, not only on how much region they cover. When the pieces are packed into a square, the shape is as “compact” as possible and the boundary is short. Stretching the same pieces into a long rectangle pushes some edges outwards, so the boundary gets longer.
You can see the same effect with simple rectangles of area 16 sq units:
Rectangle
Area
Perimeter
4 × 4 (a square)
16 sq units
16 units
8 × 2
16 sq units
20 units
16 × 1
16 sq units
34 units
Same area every time, but the more “stretched” the figure, the bigger the perimeter. Of all figures with a given area, the square-like one has the least perimeter.