Area & Volume of Similar Shapes Flashcards

All 8 cards in this deck

How do you find a missing length on a shape similar to a given one?

e.g. two similar shapes have corresponding lengths $4$ cm and $10$ cm; find the length on the larger shape corresponding to $3$ cm on the smaller

Divide a pair of corresponding lengths to get the length scale factor $k$, then multiply.

$k = 10 \div 4 = 2.5$, so the length is $3 \times 2.5 = 7.5$ cm.

For two similar shapes with length scale factor $k$, what is the area (or surface area) scale factor?

e.g. $k = 3$

$k^2$.

For $k = 3$ the area scale factor is $9$.

For two similar solids with length scale factor $k$, what is the volume scale factor?

e.g. $k = 3$

$k^3$.

For $k = 3$ the volume scale factor is $27$.

What are the steps to find the surface area ratio of two similar solids from their volume ratio?

e.g. volume of A : volume of B $= 27 : 8$

1. Cube root each part to get the ratio of lengths: $\sqrt[3]{27} : \sqrt[3]{8} = 3 : 2$
2. Square each part to get the ratio of surface areas: $3^2 : 2^2 = 9 : 4$

What are the steps to find the ratio of lengths of three similar solids A, B and C?

e.g. area of A : area of B $= 4 : 25$ and volume of B : volume of C $= 27 : 64$

1. Square root the area ratio: A : B $= 2 : 5$
2. Cube root the volume ratio: B : C $= 3 : 4$
3. Scale so B matches (common multiple $15$): A : B : C $= 6 : 15 : 20$

True or false? A model is built to a scale factor of $\frac{1}{10}$, so the volume of the model is $\frac{1}{10}$ of the volume of the real object.

False. Volume scales by $k^3$, so the model's volume is $\frac{1}{1000}$ of the real volume.

What are the steps to work out how many times a smaller similar container must be emptied to fill a larger one?
e.g. surface area of container A : surface area of container B $= 4 : 9$

1. Square root the area ratio for lengths: $\sqrt{4}:\sqrt{9}=2:3$
2. Cube for volumes: $2^3:3^3=8:27$
3. Divide and round up: $27\div 8=3.375$, so 4 times

What are the steps to find the volume of a frustum made by cutting a cone with a plane parallel to its base?
e.g. the small cone removed has length scale factor $\frac{1}{2}$ of the whole cone, and the whole cone has volume $80$ cm$^3$

1. The small cone is similar to the whole cone, so volume factor $=\left(\frac{1}{2}\right)^3=\frac{1}{8}$
2. Volume of small cone $=80\times\frac{1}{8}=10$ cm$^3$
3. Subtract: $80-10=70$ cm$^3$