Before we can find any of the three items mentioned, we need the height. The diameter is 10, so the radius is 5. A right triangle with hypotenuse 13 and leg 5 forms. The height is h. Use the pythaogrean theorem to solve for h
5^2+h^2 = 13^2
25+h^2 = 169
h^2 = 169-25
h^2 = 144
h = sqrt(144)
h = 12
The height is 12. We now have enough info to find the volume, the lateral area and surface area.
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Volume
V = (1/3)*pi*r^2*h
V = (1/3)*3.14*5^2*12
V = 314 cubic cm
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Lateral Area
LA = pi*r*L
LA = 3.14*5*13
LA = 204.1 square cm
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Surface Area
SA = 2*pi*r + pi*r*L .... note how we add on the lateral area to the bottom circular area
SA = 2*3.14*5 + 3.14*5*13
SA = 235.5 square cm