The height of a regular rectangular truncated pyramid is 36cm, apothem is 45cm, and the sides

| January 16, 2021 | education

The height of a regular rectangular truncated pyramid is 36cm, apothem is 45cm, and the sides of the bases are proportional to the numbers 1 and 4. Calculate the areas of the bases of the truncated pyramid.

Let’s make additional constructions. We connect the points M and K of the apothem with the points O1 and O of the intersection of the diagonals of the bases, and draw them points M perpendicular to the segment OK.
In the formed right-angled МНК triangle, according to the Pythagorean theorem, we determine the length of the НC leg. HK ^ 2 = MK ^ 2 – MH ^ 2 = 45 ^ 2 – 36 ^ 2 = 2025 – 1296 = 729.
НK = 27 cm.
According to the condition, the sides of the bases are related as 1/4, then O1M / OK = 1/4.
Then OK = 4 * O1M = 4 * OH.
Also OK = OH + NK = OH + 27.
Then 4 * OH = OH + 27.
3 * OH = 27.
OH = 27/3 = 9 cm.
Then, OK = 9 + 27 = 36 cm.
BP = 2 * OK = 72 cm.
A1D1 = 2 * OH = 18 cm.
The area of ​​the larger base is: S1 = AD ^ 2 = 72 ^ 2 = 5184 cm2.
The area of ​​the smaller base is: S2 = A1D1 ^ 2 = 18 ^ 2 = 324 cm2.
Answer: The areas of the bases are equal to 324 cm2, 5184 cm2.



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