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Name _____________________
Worksheet 7.4
Similar Relationships with Areas & Volumes
1. Two similar rectangles have lengths 18 in. and 12 in. What is the ratio of their widths? What
is the ratio of their areas? Explain the validity of your answer (i.e. why does your answer
make sense?).
Let’s say that I build a creature out of blocks, and then build a similar version (exactly the
same shape, but a different size). If the new version is three times as large (in each dimension)
as the starting version, how many more times is it greater in volume? Explain the validity of
your answer (i.e. why does your answer make sense?).
How does the length ratio of similitude,
1
, for two similar objects compare to:
2
i. the ratio of similitude of areas,
A1
?
A2
ii. to the ratio of similitude of volumes,
V1
?
V2
2. A barn was painted by your company (yes, you own a barn painting company!) and it used 100
gallons of paint. You must estimate how much paint is needed to paint a similarly shaped barn
twice as big in all dimensions. How many gallons are required?
3. Given two similarly shaped containers, the smaller is filled and emptied into the larger until the
larger one is filled. It takes 64 of the smaller to fill the larger. If the larger container is 20 inches
tall how tall is the smaller one?
4. Forward-thinking city planners envision placing a large dome over their town to protect them
from germs, space invaders, and bears (spring-fall threat only). A large scale model of the town is
constructed on a football field to show the dome exactly as it will look. The actual town has
100,000 times the area of the model. In the model you notice that Main Street is 8 feet long.
How long is the real Main Street?
5. The edges of a rectangular solid (i.e. prism) are in the ratio of 5:4:3. Find their lengths if the
volume of the solid is 202.5 cubic inches. Sketch a diagram.
6. Two similar containers have volumes of 27,000 cubic cm and 1,000 cubic cm. It takes 200 square
cm to paint the surface area of the smaller container. How much paint does it take to paint the
surface area of the larger container?
Practice Problems for Test
7.
ABCD is a trapezoid with BC parallel to AD and AD =2 BC. Diagonals AC and BD intersect
at X.
(a) Find the ratio of the areas of triangles BCX and AXD
(b) Prove that triangles AXB and CXD have equal area.
8.
ABCDEFGH is a frustum of a square based pyramid. Base ABCD has sides of length 3 and
the upper base EFGH has sides of length 1. The slant edges AE, BF,CG and DH are all of
length 2.
(a) Find the volume of the frustum.
(b) Find the total surface area of the frustum.
9.
ABCD is a parallelogram and AP, BQ, CR and DS are all perpendicular to the line PSQR.
AP=12 in, DS =16 in, CR= 10 in, PS =5 in, SQ = 2 in. Find the area of ABCD.
Note this problem is 2-dimensional.
10. ABCD is a parallelogram. If E is the midpoint of AB and the area of ABCD is 60, find the areas
of regions I, II, III, and IV.
C
D
III
IV
II
I
A
E
B
11.
A frustum could be constructed from the following schematic. Find the height, volume, and
total surface area of the frustum.
120
3cm
2cm
12.
A regular Octahedron has all edges equal to 10 inches
(a) Find the surface area of the octahedron
(b) Find the volume of the octahedron
(c) The octahedron can be inscribed in a cube such that its vertices are at the midpoint of each
face of the cube. Find the volume contained in the cube which is outside the octahedron.