Total surface area

A right solid is a 3D shape for which the vertical sides are perpendicular to the bases. In all such shapes, the volume is area of base * height.

For example:
volume of cube: area of square base * height (also known as side * side * side)
volume of a right cylinder: area of circular base * height (pi * r^2 * h)

In this case, the bases are right triangles (3, 4, 5 is a right triangle since 3^2 + 4^2 = 5^2). The area of the solid's base is base*height/2 (formula for area of triangles). In this case, base=3 and height=4, so area of the solid's base is 3*4/2 = 6.

The volume of the solid is area of base * height, so we can write 30 = 6 * height ==> height is 5.

Now we know all the dimensions. bases are 3-4-5 triangles, and all vertical lines are 5 (heights).

Surface area is sum of all the areas. We already know the areas of the bases (6 and 6). All we need are the areas of the three vertical sides.

Note that the vertical sides are all rectangles with bases 3, 4, and 5, and with height=5. So the areas of the three vertical sides are 3*5=15, 4*5=20, and 5*5=25.

Now we have all the areas. Bases are 6 and 6. Vertical sides are 15, 20 and 25. Surface area is 6+6+15+20+25 = 72.

-Patrick

First off, because you have a 3-4-5 triangle as the base, you know that your triangle is right and has an area of 3*4/2 = 6. Since your total volume is 30, the height of the solid is therefore 30/6 = 5. Now just count up the areas of the surfaces:

Bottom base: 6 (as determined above)
Top base: 6 (same as bottom base)
Face1: 3*5 = 15 (area of rectangle = side * height)
Face2: 4*5 = 20 (area of rectangle = side * height)
Face3: 5*5 = 25 (area of rectangle = side * height)
Sum = 72

Is the answer 60?