Find the side surface of a regular triangular prism with a height of 5 cm if the straight line passing

| June 20, 2021 | education

Find the side surface of a regular triangular prism with a height of 5 cm if the straight line passing through the center of the upper base and the middle of the side of the lower base is inclined to the base plane at an angle of 60 degrees.

Let MRKM1R1K1 be a regular triangular prism. О and О1 are the centers of the bases. C – middle of the РK.

Consider a triangle OO1C: the angle O is 90 ° (OO1 is the height of the prism), OO1 = 5 cm (this is the height). Let us express the tangent of the angle О1СО: tg = ОO1 / OC. tg60 ° = √3.

√3 = 5 / OС; OС = 5 / √3.

OS is the radius of the inscribed circle of the regular triangle MRK.

r = a√3 / 6 (the radius of the inscribed circle of a regular triangle).

From here we find the side of the triangle:

5 / √3 = a√3 / 6.

a = 5 * 6: (√3 * √3) = 10 (cm).

This means that the side of the base is 10 cm.

The side surface is equal to the area of ​​three rectangles with sides of 5 cm and 10 cm:

Side = 3 * (5 * 10) = 3 * 50 = 150 (cm²).



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