Transcript 11-7 Areas & Volumes of Similar Solids

Areas & Volumes of Similar
Solids
Objective:
1) To find relationships between the
ratios of the areas & volumes of similar
solids.
Similar Solids
Similar Solids – Have the same shape, &
all their corresponding dimensions are
proportional.
– Proportional – Equal Ratios
6in
Heights must be
proportional!
Radii must be
proportional!
4in
10in
15in
Ex.1: Are the following pairs of
solids proportional??
No!
Yes
– Not the same shape.
– 2x as big or ½ as large.
8ft
2ft
4cm
2ft
4cm
1ft
1ft
4cm
4cm
4cm
1 = 1 = 4
2 2 8
4ft
Th (10-12)
If side (similarity) ratio is a:b, then
1) Ratio of their corresponding areas a2:b2.
2) Ratio of their volumes is a3:b3.
Ex.2: Surface area ratio
Find the side (similarity) ratio of two similar
cylinders with surface areas of 98ft2 & 2ft2.
– Write areas as a ratio.
– Reduce
–√
2
98ft = 49ft2 = 7ft
2ft2
1ft2
1ft
** The height of the large cylinder is 7x bigger than the
smaller cylinder.
** The radius of the large cylinder is 7x bigger than the
smaller cylinder.
Ex.3: Volume Ratio
Two similar square pyramids have volumes of
48cm3 & 162cm3. The surface area of the
larger pyramid is 135cm2. Find the surface of
the smaller pyramid.
– First find the side ratio.
Write the volumes as a ratio.
Reduce
48cm3 = 8cm3 = 3√8ft3
3√
162cm3 27cm3 3√27ft3
2cm
=
3cm
– Set up a surface area ratio
22 = x
32 135
4 = x
9
135
x = 60cm2
What have we learned??
In order for two solids to be similar they
must be
– The same shape
– Corresponding parts have to be proportional
If the side ratio is a:b, then
– Area ratio is a2:b2
– Volume ratio is a3:b3