Question

Manuela is making a crazy math construction. She creates 360 small pyramids out of paper she folds, then glues together identical sides until she makes a large pyramid. One small pyramid contains a square base with side lengths a and 4 equilateral triangles with base a and height h. The height h is 1.7 cm. The length a is 2 cm.
In square centimeters, what is the minimum amount of paper Manuela uses to create 360 small pyramids? (Assume there is no wasted paper.)

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Answer 1

area ABCD =4 cm^2
area DFC = (1/2) 2*1.7 =1.7 cm^2
4 triangle:, hence the area of 4 triangles is 4*1.7 = 6.8cm^2
hence total area is 4+6.8 =10.8 cm^2
just *360 to get the ansser

Answer 2

Well, we are told to have 360 small pyramids made of paper.
We are told the base is a square (with side 'a') and that the 'faces' of the pyramids are equilateral triangles (with sides of 'a' as well)

So now, if we can find how much paper we need for one small pyramid we can tell how much we need for all the 360 pyramids.
To do so we need to calculate the 'surface' of every small pyramid. that is basically the sum of areas of the base and sides.

The base is a square so we have to calculate the area of a square:
$$
\text{base_area} = \text{side_length}^2 \\
\text{base_area} = (2_{cm})^2 = 4_{cm^2}
$$
The area of a 'face' will be the area of the triangle:
$$
\text{face_area} = \frac{\text{triangle_base} \cdot \text{triangle_height}}{2} \\
\text{face_area} = \frac{(2_{cm})\cdot(1.7_{cm})}{2} = 1.7_{cm^2}
$$
The surface is basically a sum of all those areas (the base and the 4 faces):
$$
\text{pyramid_surface} = \text{base_area} + 4\cdot \text{face_area} \\
\text{pyramid_surface} = 4_{cm^2} + 4\cdot1.7_{cm^2} = 10.8_{cm^2}
$$
So 360 pyramids:
$$
\text{used_paper} = 360 \cdot \text{pyramid_surface} = 360 \cdot 10.8_{cm^2} = 3888_{cm^2}
$$