Areas of Similar Cross Sections
Areas of Similar Cross Sections is a Grade 7 math skill in Reveal Math Accelerated, Unit 12: Area, Surface Area, and Volume, where students find the area of a cross-section of a solid that is similar to the base by using the square of the scale factor to relate the areas of similar figures. This combines knowledge of similar figures, scale factors, and area formulas.
Key Concepts
When a three dimensional figure (such as a cone or pyramid) is sliced parallel to its base, the resulting cross section is similar to the base. If the scale factor between the corresponding linear dimensions of the two similar cross sections is $k$, then the ratio of their areas is $k^2$.
$$\frac{\text{Area of Cross Section 1}}{\text{Area of Cross Section 2}} = k^2 = \left(\frac{\text{Linear Dimension 1}}{\text{Linear Dimension 2}}\right)^2$$.
Common Questions
How do you find the area of a similar cross section?
Find the scale factor between the cross section and the reference figure, then square the scale factor and multiply by the area of the reference figure. If the scale factor is k, the area of the similar figure is k^2 times the original area.
Why is the scale factor squared when comparing areas of similar figures?
Area is a two-dimensional measurement. When both length and width are scaled by k, the area is multiplied by k x k = k^2. This is different from perimeter, which is multiplied by only k.
What is the relationship between the height of a cut and the area of the cross section in a pyramid?
For a pyramid, if the cross section is made at height h above the base where the total height is H, the scale factor of the cross section to the base is (H - h) / H, and the area of the cross section is this ratio squared times the base area.
What is Reveal Math Accelerated Unit 12 about?
Unit 12 covers Area, Surface Area, and Volume, including circle area, cross sections, similarity ratios for areas, and volume calculations for 3D shapes.