# Thread: Finding Internal Volumes and Total Surface Areas of a Cylinder

1. ## Finding Internal Volumes and Total Surface Areas of a Cylinder

A Pencil case in the shape of a cylinder with a hemisphere at each end has a diameter of 8cm and a total length of 22cm.

(i) Find the internal volume of the pencil case, correct to one decimal place.

(ii) Find the Total surface area of the pencil case, correct to the nearest cm².

What I did for this, I don't think Im right so its more where did I go wrong (if at all).

(i) (pi)(r²)(h) + 2 times the hemisphere 2/3(pi)(r³)
My final answer using this method gave me 971.3cm³

(ii) 2(pi)(r)(h) + 2(pi)(r²) = 452.16 (This was for the TSA of the cylinder)
Now for the hemisphere
I did 3(pi)(r)² X 2 which gave me 301.44.
I said 301.44 + 452.16 = 753.6cm²

A Pencil case in the shape of a cylinder with a hemisphere at each end has a diameter of 8cm and a total length of 22cm.

(i) Find the internal volume of the pencil case, correct to one decimal place.

(ii) Find the Total surface area of the pencil case, correct to the nearest cm².

What I did for this, I don't think Im right so its more where did I go wrong (if at all).

(i) (pi)(r²)(h) + 2 times the hemisphere 2/3(pi)(r³)
My final answer using this method gave me 971.3cm³

(ii) 2(pi)(r)(h) + 2(pi)(r²) = 452.16 (This was for the TSA of the cylinder)
Now for the hemisphere
I did 3(pi)(r)² X 2 which gave me 301.44.
I said 301.44 + 452.16 = 753.6cm²

It appears you have a complete sphere of radius 4 cm when you consider the hemispheres at each end of the cylinder.
So, we're dealing with the volume of a sphere plus the volume of a cylinder.

Find the volume of the sphere: $V=\frac{4}{3}\pi r^3$, using r = 4.

Now, the cylinder between the two hemispheres has a height that is 8 centimeters less than the height (length) of the container which was 22 cm.

You have to subtract the radii of the two hemispheres from 22 cm. to get a cylinder height of 14 cm.

Now find the volume of the cylinder using $V=\pi r^2 h$ using r = 4 and h = 14.

Finally, add the volume of the cylinder to the volume of the sphere.

[ii] Use a similar scheme with Surface area formulas of a sphere and cylinder to complete the second part.