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FM Surface Area

1.4 Scaling Factor and its Applications

Similar Figures and Shapes

Two shapes are similar if they share the same shape but not the size.

  • Their corresponding angles are equal;
  • Their corresponding sides are in the same ratio.

Calculating the Scaling Factor

  • The scaling factor is a numerical value representing the scale of one shape/object to a similar shape (with a different scale).
  • The scaling factor can be calculated as the proportional difference between the lengths of the two shapes/objects, the square root of the proportional difference between their areas, or the cubic root of the proportional difference between their volumes:

k=\frac{L_{A}}{L_{B}}=\sqrt{\frac{A_{A}}{A_{B}}}=\sqrt[3]{\frac{V_{A}}{V_{B}}}

Note: this calculates the scaling factor for shape/object A with respect to shape/object B. For example, a scaling factor of 2 means A has twice the length of B.

  • The scaling factor is dimensionless (i.e. has no units).
  • k>0The scaling factor is larger than 0.
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1.3 Surface Area and Volume of Composite Shapes

Note: if you cannot remember the formulas for the surface area and volume of common shapes, revise notes for 1.1 Surface Area of Common Shapes and 1.2 Volume of Common Shapes.

Guide to Finding the Surface Area of a Composite Shape

  • Composite shapes are shapes that are created by merging multiple simple shapes.
  • Due to the large amount of combinations possible, it is not possible to derive formulas for each.
  • To find the surface area of a composite shape, you need to first identify the simple shapes it is made up of. The surface area of the composite shape is equal to the sum of each shape, minus the surface area which has been removed to merge it into the shape:
Read More »1.3 Surface Area and Volume of Composite Shapes

1.1 Surface Area of Common Shapes

Note: remember that area is measured in units squared (m^2, cm^2, etc.).

Surface Area of Spheres

  • The surface area of a sphere is found using:

A=4 \pi r^{2}

Where r is the radius of the sphere.

Example

Picture 2

We wish to find the surface area of the above sphere. We do this by first identifying the values we need, in this case we have a radius of 10mm. Now, we substitute this value into our formula to find the surface area:

A=4 \pi r^{2}=4 \pi 10^{2}=1256.64 \mathrm{~mm}^{2}

Surface Area of Cylinders

Read More »1.1 Surface Area of Common Shapes