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 – 1. AREAS
OF PLANE SHAPES; CONNECT: Areas, Perimeters – 2. PERIMETERS OF
PLANE SHAPES and CONNECT: Volume, Surface Area – 1. VOLUMES OF
SOLIDS.
You may also need to review Pythagoras’ Theorem – if so, please refer to
CONNECT: Pythagoras’ Theorem
The surface area of a 3D shape is “the total area of the outside” 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:
If we draw in the
edges that are
blocked from
sight by the
other faces, we
can see the
shapes of all the
faces:
The faces are all rectangles and so each of their areas is A = l x w units2.
For each of the top and bottom faces, the area is 3cm x 4cm = 12cm2.
For each of the left and right faces, the area is 4cm x 12cm = 48cm2.
For each of the front and back faces, the area is 3cm x 12cm = 36cm2.
So the total surface area is 12 x 2 + 48 x 2 + 36 x 2 cm2
that is, 192cm2.
The units are square units because we are measuring area.
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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.)
h
h
l
l
So the area of the curved surface of the cylinder is
𝐴 = 2𝜋𝑟 × ℎ units2.
(2𝜋𝑟 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:
𝐴 = 2𝜋𝑟 × ℎ units2 ← open at both ends
𝐴 = 2𝜋𝑟 × ℎ + 𝜋𝑟 2 units2 ← open at one end
OR
𝐴 = 2𝜋𝑟 × ℎ + 2 × 𝜋𝑟 2 units2 ← closed at both ends.
OR
Example: Find the surface area of the closed cylinder over the page.
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r=6cm
h=10.3cm
We need to use the formula 𝐴 = 2𝜋𝑟 × ℎ + 2 × 𝜋𝑟 2 units2 because the
cylinder is closed.
So, A = 2 x 𝜋 x 6 x 10.3 + 2 x 𝜋 x 62 cm 2
Using a calculator we obtain 614.495 523 cm2, that is, approximately
614.496cm2.
Surface Area of a Cone
The curved surface area of a cone is a little more involved to calculate. We
need to know more than just our usual measurements – the radius of the
circular base and the perpendicular height of the cone. We also need to know
the slant height (s) of the cone.
Diagram retrieved January 31, 2013, from
http://www.mathsteacher.com.au/year10/ch14_measurement/18_cone/20cone.htm
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If we know the slant height of the cone, we can use the formula
𝐴 = 𝜋𝑟𝑠 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:
𝐴 = 𝜋𝑟𝑠 units 2 ← open cone
𝐴 = 𝜋𝑟𝑠 + 𝜋𝑟 2 units 2 ← closed cone
Example: Find the surface area of the following closed cone:
s = 8.1cm
r = 5.1cm
Curved surface area = 𝜋𝑟𝑠 cm2
= 𝜋 x 5.1 x 8.1 cm2
= 129.779 192 5… cm2
Area of base = 𝜋𝑟 2 cm2
= 𝜋 x 5.12 cm2
= 81.712 824 92… cm2
So the total area = 129.779 192 5… + 81.712 824 92… cm2
= 211.492 017 4… cm2
Round this to approximately 211.5cm2
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But what if we have a cone and do not know s?
For example, find the total surface area of this closed cone:
s
h = 4.1 mm
r = 3.2mm
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 Pythagoras’ Theorem, 𝑠 2 = 𝑟 2 + ℎ2
𝑠 2 = 3.22 + 4.12
= 27.05
To find s, take the square root of 27.05, so √27.05 = 5.200 961 45…
So, s is 5.200 961 45… mm.
possible, don’t round yet.)
(To make the final answer as accurate as
Now, the curved surface area = 𝜋𝑟𝑠 mm2
= 𝜋 x 3.2 x 5.200 961 45… mm2
= 52.285 767 3…mm2
and the area of the base = 𝜋𝑟 2 mm2
= 𝜋 × 3.22 mm2
= 32.169 908 77…mm2
So the total surface area is 52.285 767 3… + 32.169 908 77…mm2
= 84.455 676 07…mm2
Only now do we round and so the area is approximately 84.5mm2
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If you need help with any of the Maths covered in this resource (or any other Maths
topics), you can make an appointment with Learning Development through
Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your
campus.
REFERENCES
De Klerk, J. (2007). Illustrated Maths Dictionary. 4th ed. Pearson. Melbourne.
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