# The figure below is the net of a prism made up of identical triangles, what is the total area of the prism if the side of the square is 6 cm.

[a] 75 sq-cm

[b] 84 sq-cm

[c] 95 sq-cm

[d] 56 sq-cm

Answer

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Hint: Find the height of the triangles and use the fact that the area of a triangle $=\dfrac{1}{2}bh$ where b is the base and h is the height and the area of a square $={{a}^{2}}$. There are four triangles and one square. So, the net area will be the sum of 4 times the area of the triangle and area of the square.

Complete step-by-step answer:

Since the length of a side of square = 6cm.

We have AD = 6cm.

Also since the triangles are identical CG = HF

Since the total height of the diagram = 14.

We have CG+AD+HF = 14

i.e. CG+HF+6 = 14

i.e. 2CG+6 = 14

Subtracting 6 from both sides we get

2CG+6-6 = 14-6

i.e. 2CG = 8

Dividing both sides by 2 we get

CG = 4cm

Since CG = HF, we have

HF = 4 cm.

Now we know that the area of a triangle $=\dfrac{1}{2}bh$

Using we get the area of triangle ABC $=\dfrac{1}{2}\times AB\times CG=\dfrac{1}{2}\times 6\times 4=\dfrac{24}{2}=12$

Also, we know that the area of a square $={{a}^{2}}$

Using, we get the area of square ABED $=A{{D}^{2}}={{6}^{2}}=36$.

Hence the total area of the prism = 4 times the area of the triangle ABC + area of square ABED

$=4\times 12+36=48+36=84$

Hence the total area of faces of prism = 84 sq-cm.

Hence option [b] is correct.

Note: Although the figure above is referred to be of a prism, it is a pyramid. A pyramid has triangular faces, whereas a prism has rectangular faces. The diagram above is of a pyramid with a square base.

Further pyramids have only one base, whereas prisms have two.

A cone is an example of a pyramid with a circular base.

Complete step-by-step answer:

Since the length of a side of square = 6cm.

We have AD = 6cm.

Also since the triangles are identical CG = HF

Since the total height of the diagram = 14.

We have CG+AD+HF = 14

i.e. CG+HF+6 = 14

i.e. 2CG+6 = 14

Subtracting 6 from both sides we get

2CG+6-6 = 14-6

i.e. 2CG = 8

Dividing both sides by 2 we get

CG = 4cm

Since CG = HF, we have

HF = 4 cm.

Now we know that the area of a triangle $=\dfrac{1}{2}bh$

Using we get the area of triangle ABC $=\dfrac{1}{2}\times AB\times CG=\dfrac{1}{2}\times 6\times 4=\dfrac{24}{2}=12$

Also, we know that the area of a square $={{a}^{2}}$

Using, we get the area of square ABED $=A{{D}^{2}}={{6}^{2}}=36$.

Hence the total area of the prism = 4 times the area of the triangle ABC + area of square ABED

$=4\times 12+36=48+36=84$

Hence the total area of faces of prism = 84 sq-cm.

Hence option [b] is correct.

Note: Although the figure above is referred to be of a prism, it is a pyramid. A pyramid has triangular faces, whereas a prism has rectangular faces. The diagram above is of a pyramid with a square base.

Further pyramids have only one base, whereas prisms have two.

A cone is an example of a pyramid with a circular base.

Last updated date: 24th Sep 2023

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