Similar Shapes - Areas and Volumes

Consider how both dimensions change.

1. Area comes from multiplying two lengths, and each length is multiplied by 4.
2. So every area is multiplied by 4\times4=4^2.
3. The area scale factor is 16.

Find the length scale factor before scaling the area.

1. The bases 5\,\text{cm} and 10\,\text{cm} correspond, so the length scale factor is 10\div5=2.
2. Areas are multiplied by the square of the length scale factor: 2^2=4.
3. The larger area is 18\times4=72\,\text{cm}^2.

Volume scales with the cube of the length scale factor.

1. The heights correspond, so the length scale factor from the smaller to the larger ornament is \frac74.
2. Volumes are multiplied by the cube of the length scale factor: \left(\frac74\right)^3=\frac{343}{64}.
3. The larger volume is 192\times\frac{343}{64}. Since 192\div64=3, this is 3\times343=1029\,\text{cm}^3.

Recover the length ratio from the area ratio.

1. Areas scale by the square of the length scale factor, so lengths scale by the square root of the area ratio.
2. \sqrt9:\sqrt{16}=3:4, so each length on the larger tile is \frac43 of the matching length on the smaller tile.
3. The corresponding side is 9\times\frac43=12\,\text{cm}.

Use a cube root to change from a volume scale factor to a length scale factor.

1. The volume scale factor is 432\div54=8.
2. Volumes scale by the cube of the length scale factor, so the length scale factor is \sqrt[3]{8}=2.
3. The larger height is 3\times2=6\,\text{cm}.

Go from surface areas to lengths, then from lengths to volumes.

1. Surface areas scale by the square of the length scale factor, so take square roots: the length ratio is \sqrt{16}:\sqrt{25}=4:5.
2. Volumes scale by the cube: the volume ratio is 4^3:5^3=64:125.
3. The larger volume is 1280\times\frac{125}{64}. Since 1280\div64=20, this is 20\times125=2500\,\text{cm}^3.

Take cube roots before working out the area scale factor.

1. Volumes scale by the cube of the length scale factor, so take cube roots: the length ratio is \sqrt[3]{27}:\sqrt[3]{64}=3:4.
2. Surface areas scale by the square: the surface-area ratio is 3^2:4^2=9:16.
3. The larger surface area is 81\times\frac{16}{9}=9\times16=144\,\text{cm}^2.

Use the square of the scale and keep the area units consistent.

1. A scale of 1:40 means every real length is 40 times the length on the drawing.
2. So every real area is 40^2=1600 times the area on the drawing.
3. Change the real area to \text{cm}^2: 1\,\text{m}^2=100\times100=10\,000\,\text{cm}^2, so 12.8\,\text{m}^2=128\,000\,\text{cm}^2.
4. The area on the drawing is 128\,000\div1600=80\,\text{cm}^2.

Turn each ratio into a ratio of lengths, then link the two ratios through B.

1. Volumes scale by the cube of the length scale factor, so height of A : height of B =\sqrt[3]8:\sqrt[3]{27}=2:3.
2. Surface areas scale by the square, so height of B : height of C =\sqrt9:\sqrt{25}=3:5.
3. B is 3 parts in both ratios, so height of A : height of B : height of C =2:3:5.
4. C is 25\,\text{cm}, so one part is 25\div5=5\,\text{cm} and the height of A is 2\times5=10\,\text{cm}.

Square the length multiplier, then interpret the new area as a percentage.

1. The length multiplier is 1.20.
2. The area multiplier is 1.20^2=1.44.
3. The area is 144\% of the original, an increase of 44\%.

Convert the percentage increase to a multiplier before taking a square root.

1. An increase of 120\% means the new area is 100\%+120\%=220\% of the original, so the area scale factor is 2.2.
2. Areas scale by the square of the length scale factor, so the length scale factor is \sqrt{2.2}=1.4832\ldots
3. The enlarged height is 15\times1.4832\ldots=22.248\ldots\,\text{cm}, which is 22.2\,\text{cm} to 3 significant figures.

Check both pairs of corresponding lengths.

1. Two shapes are similar only if every pair of corresponding lengths has the same scale factor.
2. Lengths: 9\div6=1.5. Widths: 8\div4=2. These are different, so the rectangles are not similar and Mia's method is incorrect.
3. The area scale factor is really \dfrac{9\times8}{6\times4}=\dfrac{72}{24}=3, not 1.5^2=2.25.
4. Rectangles are not automatically similar (their widths and lengths can grow by different amounts), and 1.5^3 would be a volume scale factor.

Reverse the operation used to scale volume.

1. If the length scale factor is k, the volume scale factor is k^3.
2. So k^3=1.5 and the length scale factor is \sqrt[3]{1.5}.

Find the length ratio, then form a linear equation.

1. Simplify the area ratio: 98:200=49:100 (divide both by 2).
2. Lengths scale by the square root of the area ratio, so the length ratio is \sqrt{49}:\sqrt{100}=7:10.
3. So \dfrac{x+1}{2x-1}=\dfrac{7}{10}. Cross-multiply: 10(x+1)=7(2x-1), so 10x+10=14x-7.
4. So 4x=17 and x=\dfrac{17}{4}. Check: the sides are 5.25\,\text{cm} and 7.5\,\text{cm}, in the ratio 7:10.

Use the common density to relate mass to volume.

1. Both ornaments are made of the same material, so mass is proportional to volume: the volume ratio is 54:250=27:125.
2. Volumes scale by the cube of the length scale factor, so take cube roots: the length ratio is \sqrt[3]{27}:\sqrt[3]{125}=3:5.
3. The larger height is 12\times\frac53=20\,\text{cm}.

The difference in areas corresponds to the difference between the area-ratio parts.

1. Areas scale by the square of the length scale factor, so the area ratio is 2^2:3^2=4:9.
2. The difference of 9-4=5 ratio parts is the extra 180\,\text{cm}^2.
3. One part is 180\div5=36\,\text{cm}^2.
4. The smaller panel is 4 parts: 4\times36=144\,\text{cm}^2.

Convert the volume ratio to a surface-area ratio before finding the percentage increase.

1. Simplify the volume ratio: 256:500=64:125 (divide both by 4).
2. Take cube roots for the length ratio: \sqrt[3]{64}:\sqrt[3]{125}=4:5.
3. Square for the surface-area scale factor: \left(\frac54\right)^2=\frac{25}{16}=1.5625.
4. The surface area is multiplied by 1.5625, which is an increase of 0.5625\times100=56.25\%.

Find the volume ratio, then use density equals mass divided by volume.

1. Simplify 144:225=16:25, then take square roots: the length ratio is 4:5.
2. Cube for the volume ratio: 4^3:5^3=64:125. So the volumes can be written as 64v and 125v for some number v.
3. Density =\dfrac{\text{mass}}{\text{volume}}: density of P=\dfrac{384}{64v}=\dfrac{6}{v} and density of Q=\dfrac{500}{125v}=\dfrac{4}{v}.
4. So k=\dfrac{6}{v}\div\dfrac{4}{v}=\dfrac64=1.5.

Find the fraction of the original volume removed using the height ratio.

1. The removed cone is similar to the original cone because the cut is parallel to the base.
2. The length scale factor is \frac6{15}=\frac25, so the volume fraction removed is \left(\frac25\right)^3=\frac8{125}.
3. The removed volume is 500\times\frac8{125}=32\,\text{cm}^3.
4. The remaining volume is 500-32=468\,\text{cm}^3.

Express the shaded area as a fraction of the whole triangle.

1. Triangles ADE and ABC are similar, since DE is parallel to BC.
2. AB=6+4=10\,\text{cm}, so the length ratio is AD:AB=6:10=3:5 and the area ratio is 3^2:5^2=9:25. Triangle ADE is \frac{9}{25} of triangle ABC.
3. The shaded trapezium is the rest: 1-\frac9{25}=\frac{16}{25} of triangle ABC.
4. So \frac{16}{25} of the area is 112\,\text{cm}^2, and the whole area is 112\div\frac{16}{25}=112\times\frac{25}{16}=175\,\text{cm}^2.