## heather040200 2 years ago HELP! PIC BELOS I GIVE MEDALS:) the lateral area of the right regular triangular prism is 18pi cm^2. find the total surface area. give your answer in decimal form rounded of to 2 decimal places

1. heather040200

2. mathstudent55

The lateral area of the prism is the sum of the areas of the three rectangles whose dimensions are 3 and x.

3. mathstudent55

Each rectangular face has an area of 3x, so all three faces have a combined area of 3 * 3x = 9x. The lateral area = 9x = 18pi

4. mathstudent55

9x = 18pi x = 2pi

5. mathstudent55

Now we need the area of the bases.

6. mathstudent55

Each base is a regular triangle with side = 2pi

7. mathstudent55

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8. mathstudent55

Each base of the prism is a regular (equilateral) triangle with side length of 2pi. If you drop an altitude, as the picture above shows, you create two 30-60-90 triangles. The ratio of the lengths of the legs of a 30-60-90 triangle is 1 : sqrt(3) Since the short leg measures pi, the long leg measures pi * sqrt(3) The long leg is the altitude of the large triangle, whose base measure 2pi

9. mathstudent55

The area of a triangle is A = (1/2)bh Here we have A = (1/2)(2pi)(pi * sqrt(3) ) = pi^2*sqrt(3) Since there are two bases, the area of the 2 bases is 2*pi^2*sqrt(3)

10. mathstudent55

Since the lateral area is 18pi, the total surface area is SA = (18pi + 2*pi^2*sqrt(3) ) cm^2 Now you can calculate an approximate value with your calculator.

11. heather040200

72.21

12. mathstudent55

I get 90.67