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? 3.75 square centimeters 6 square centimeters 7.5 square centimeters 12 square centimeters

Explanation:

Step1: Find the side - length of the base

The perimeter of the square base is $P = 4s$, where $s$ is the side - length. Given $P=6$ cm, then $s=\frac{6}{4}=1.5$ cm.

Step2: Find the slant height of the pyramid

The distance between the tallest points of the two pyramids is $h_{total}=2$ cm, so the height of one pyramid $h = 1$ cm. Using the Pythagorean theorem for the right - triangle formed by half of the side of the base and the height of the pyramid to find the slant height $l$. Half of the side of the base $a=\frac{s}{2}=\frac{1.5}{2}=0.75$ cm. Then $l=\sqrt{h^{2}+a^{2}}=\sqrt{1^{2}+0.75^{2}}=\sqrt{1 + 0.5625}=\sqrt{1.5625}=1.25$ cm.

Step3: Calculate the area of one triangular face

The area of a triangular face of the pyramid is $A_{triangle}=\frac{1}{2}sl$, where $s$ is the base of the triangle (side - length of the square base) and $l$ is the slant height. Substituting $s = 1.5$ cm and $l=1.25$ cm, we get $A_{triangle}=\frac{1}{2}\times1.5\times1.25 = 0.9375$ square cm.

Step4: Calculate the surface area of the 8 - sided figure

The 8 - sided figure is composed of 8 congruent triangular faces. So the surface area $A = 8\times A_{triangle}=8\times0.9375 = 7.5$ square cm.

Answer:

7.5 square centimeters