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CONNECT: Volume, Surface Area
2. SURFACE AREAS OF SOLIDS
If you need to know more about plane shapes, areas, perimeters, solids or volumes of solids, please refer to CONNECT: Areas, Perimeters โ 1. AREAS OF PLANE SHAPES; CONNECT: Areas, Perimeters โ 2. PERIMETERS OF PLANE SHAPES and CONNECT: Volume, Surface Area โ 1. VOLUMES OF SOLIDS.
You may also need to review Pythagorasโ Theorem โ if so, please refer to CONNECT: Pythagorasโ Theorem
The surface area of a 3D shape is โthe total area of the outsideโ of the shape (De Klerk, 2007, p. 129). So, to work out the surface area of a prism, you need to work out the area of each face and add these areas together. Before we calculate the surface area of an example, have a think about the units we will use to measure surface area.
Example: Calculate the surface area of this rectangular prism:
The faces are all rectangles and so each of their areas is A = l x w units^2.
For each of the top and bottom faces, the area is 3cm x 4cm = 12cm 2.
For each of the left and right faces, the area is 4cm x 12cm = 48cm 2.
For each of the front and back faces, the area is 3cm x 12cm = 36cm 2.
So the total surface area is 12 x 2 + 48 x 2 + 36 x 2 cm 2
that is, 192cm 2.
The units are square units because we are measuring area.
If we draw in the edges that are blocked from sight by the other faces, we can see the shapes of all the faces:
Surface area of a cylinder
Take an ordinary A4 sheet of paper. Now roll it into a cylinder so that the edges just meet. Do you realise that you have used a rectangle to make the curved surface of a cylinder? Place your cylinder so that it stands on its circular base. The length of the rectangle is now the circumference of the circle (base) and the width of the rectangle is now the height of the cylinder. (Or vice versa, depending on which way you rolled the cylinder, but that will not make any difference to our formula.)
So the area of the curved surface of the cylinder is
๐ด = 2๐๐ ร โ units^2.
( 2 ๐๐ is the length of the circumference, which is the same as l , the length of
the rectangle.)
If the cylinder is open at both ends, the total surface area of the cylinder is just the area of the curved surface. If it is closed at one end, we need to add the area of that circle and if it is closed at both ends, we need to add the area of both circles. So, the total surface area of a cylinder could be any of the following:
๐ด = 2๐๐ ร โ units^2 โ open at both ends
OR ๐ด = 2๐๐ ร โ + ๐๐ 2 units^2 โ open at one end
OR ๐ด = 2๐๐ ร โ + 2 ร ๐๐ 2 units^2 โ closed at both ends.
Example: Find the surface area of the closed cylinder over the page.
h
l
h
l
If we know the slant height of the cone, we can use the formula
๐ด = ๐๐๐ units 2
to find the area of the curved surface. So for an open cone, this formula gives the total surface area of the cone.
But if we have a closed cone, we need to add the area of the circle as well.
So, the total surface area of a cone will be:
๐ด = ๐๐๐ units 2 โ open cone
๐ด = ๐๐๐ + ๐๐ 2 units 2 โ closed cone
Example: Find the surface area of the following closed cone:
Curved surface area = ๐๐๐ cm 2
= ๐ x 5.1 x 8.1 cm 2
= 129.779 192 5โฆ cm^2
Area of base = ๐๐ 2 cm 2
= ๐ x 5.1 2 cm 2
= 81.712 824 92โฆ cm 2
So the total area = 129.779 192 5โฆ + 81.712 824 92โฆ cm 2
= 211.492 017 4โฆ cm 2
Round this to approximately 211.5cm 2
r = 5.1cm
s = 8.1cm
But what if we have a cone and do not know s?
For example, find the total surface area of this closed cone:
We know the radius and the perpendicular height of the cone, but we need to know the slant height. You can see that the radius, vertical height and slant height form the sides of a right-angled triangle, where the slant height ( s ) is the hypotenuse.
By Pythagorasโ Theorem, ๐ 2 = ๐ 2 + โ 2
๐ 2 = 3.2^2 + 4.1^2
To find s , take the square root of 27.05, so (^) โ27.05 = 5.200 961 45โฆ
So, s is 5.200 961 45โฆ mm. (To make the final answer as accurate as possible, donโt round yet.)
Now, the curved surface area = ๐๐๐ mm 2
= ๐ x 3.2 x 5.200 961 45โฆ mm 2
= 52.285 767 3โฆmm 2
and the area of the base = ๐๐ 2 mm 2
= ๐ ร 3.2^2 mm 2
= 32.169 908 77โฆmm 2
So the total surface area is 52.285 767 3โฆ + 32.169 908 77โฆmm 2
= 84.455 676 07โฆmm 2
Only now do we round and so the area is approximately 84.5mm 2
r = 3.2mm
h = 4.1 mm s
