Answer:
The lateral surface area is 1449.58 cm².
The total surface area is 2114.69 cm².
The volume is 5948.94 cm³.
Step-by-step explanation:
The question is:
Calculate the lateral area, total and volume of a hexagonal pyramid of 16 cm of basic edge and 28 cm of lateral edge. Answer question 2.
Solution:
Consider the diagram of the hexagonal pyramid.
It is provided that:
Base = <em>a</em> = 16 cm
Lateral height = <em>b</em> = 28 cm
Compute the length of Ap = <em>h</em> using the Pythagoras theorem as follows:
![b^{2}=(\frac{a}{2})^{2}+h^{2}\\\\h^{2}=b^{2}-(\frac{a}{2})^{2}\\\\h=\sqrt{b^{2}-(\frac{a}{2})^{2}}\\\\=\sqrt{28^{2}-8^{2}}\\\\=\sqrt{720}](https://tex.z-dn.net/?f=b%5E%7B2%7D%3D%28%5Cfrac%7Ba%7D%7B2%7D%29%5E%7B2%7D%2Bh%5E%7B2%7D%5C%5C%5C%5Ch%5E%7B2%7D%3Db%5E%7B2%7D-%28%5Cfrac%7Ba%7D%7B2%7D%29%5E%7B2%7D%5C%5C%5C%5Ch%3D%5Csqrt%7Bb%5E%7B2%7D-%28%5Cfrac%7Ba%7D%7B2%7D%29%5E%7B2%7D%7D%5C%5C%5C%5C%3D%5Csqrt%7B28%5E%7B2%7D-8%5E%7B2%7D%7D%5C%5C%5C%5C%3D%5Csqrt%7B720%7D)
Compute the lateral surface area as follows:
![LSA=3a\sqrt{h^{2}+\frac{3a^{2}}{4}}=(3\cdot 16)\sqrt{720+\frac{3\cdot 16^{2}}{4}}=1449.58](https://tex.z-dn.net/?f=LSA%3D3a%5Csqrt%7Bh%5E%7B2%7D%2B%5Cfrac%7B3a%5E%7B2%7D%7D%7B4%7D%7D%3D%283%5Ccdot%2016%29%5Csqrt%7B720%2B%5Cfrac%7B3%5Ccdot%2016%5E%7B2%7D%7D%7B4%7D%7D%3D1449.58)
Thus, the lateral surface area is 1449.58 cm².
Compute the total surface area as follows:
![TSA=\frac{3\sqrt{3}}{2}a^{2}+LSA=\frac{3\sqrt{3}}{2}\cdot16^{2}+1449.58=2114.69](https://tex.z-dn.net/?f=TSA%3D%5Cfrac%7B3%5Csqrt%7B3%7D%7D%7B2%7Da%5E%7B2%7D%2BLSA%3D%5Cfrac%7B3%5Csqrt%7B3%7D%7D%7B2%7D%5Ccdot16%5E%7B2%7D%2B1449.58%3D2114.69)
Thus, the total surface area is 2114.69 cm².
Compute the volume as follows:
![\text{Volume}=\frac{\sqrt{3}}{2}\cdot a^{2}h=\frac{\sqrt{3}}{2}\times 16^{2}\times \sqrt{720}=5948.94](https://tex.z-dn.net/?f=%5Ctext%7BVolume%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Ccdot%20a%5E%7B2%7Dh%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5Ctimes%2016%5E%7B2%7D%5Ctimes%20%5Csqrt%7B720%7D%3D5948.94)
Thus, the volume is 5948.94 cm³.