Question

An 8-sided die is formed when two square pyramids are attached by their bases. The perimeter of the base shape is 6 centimeters. The distance between the tallest points of the two pyramids is 2 centimeters. What is the surface area of the 8-sided figure?

Answer 1

The surface area of the 8-sided figure is 7.5 cm²

Explanation:

The perimeter of the base of each square pyramid is 6 centimeters.

So, each side length of the base
`=(6)/(4)= 1.5\ cm.`

Two pyramids are attached by their bases and the distance between the tallest points of the two pyramids is 2 centimeters.

So, the height of each pyramid = 1 centimeter.

Now, lateral surface area of each pyramid

`=2a\sqrt{(a^2)/(4)+h^2}`

`=2(1.5)\sqrt{((1.5)^2)/(4)+(1)^2}\\ \\ =3\sqrt{(2.25)/(4)+1}\\ \\ =3√(1.5625)\\ \\ =3(1.25)=3.75\ cm^2`

So, the total lateral surface area of two pyramids
`=2(3.75)=7.5\ cm^2`

Hence, the surface area of the 8-sided figure is 7.5 cm²

Answer 2

Consider one pyramid
Side length of base = 1.5cm and its height is 1 cm
Slant height of one of the lateral faces = sqrt(1^2 + 0.75^2) = 1.25 cm
Area of one of the triangular faces = 0.5 * 1.5* 1.25 = 0.9375 cm^2

There are 8 of these so the required surface area = 8 * 0.9375
= 7.5 cm^2 Answer