Geometry

A scale drawing multiplies every length by the same scale factor $k\text{.}$ Maps and blueprints work this way. Lengths scale by $k\text{;}$ areas scale by $k^2\text{.}$

How to do it:

1. Read the scale as "drawing unit to real unit."

2. Multiply a drawing length by the real-unit factor.

3. For area, square the scale factor; do not reuse $k$ for area.

Worked example: Scale $1$ cm $:$ $5$ m, and a wall is drawn $7.4$ cm. Real length is $7.4 \times 5 = 37$ m.

Check: $7.4 + 5 = 12.4$ mixes units; that is not a scale move.

Complementary angles sum to $90°\text{.}$ Supplementary angles sum to $180°\text{.}$ Vertical angles across an intersection are equal. A transversal across parallel lines makes corresponding and alternate interior angles equal, and co-interior angles supplementary.

How to do it:

1. Name the relationship (complement, supplement, vertical, or parallel-line pair).

2. Write one equation from that relationship.

3. Solve, then find the partner angle.

Worked example: Complementary to $34°$ is $90° - 34° = 56°\text{.}$ If two lines meet and one angle is $112°\text{,}$ the vertical angle is also $112°\text{.}$

Check: the adjacent angle to $112°$ is $68°\text{,}$ not another $112°\text{.}$ Adjacent angles on a line are supplementary.

For radius $r$ and diameter $d = 2r\text{:}$

How to do it:

1. Confirm you have radius, not diameter, before using $2\pi r$ or $\pi r^2\text{.}$

2. Circumference is $2\pi r\text{.}$

3. Area is $\pi r^2\text{.}$ Doubling $r$ doubles $C$ but quadruples $A\text{.}$

Worked example: $r = 6$ cm gives $C = 12\pi$ cm. $r = 10$ in gives $A = 100\pi$ in$^2\text{.}$

Check: $r = 6$ does not make $C = 6\pi\text{.}$ Replay both formulas: Circumference and Area of a Circle.

Volume is cubic space. Surface area is the outer skin in square units. A right rectangular prism has $V = \ell w h$ and $SA = 2(\ell w + \ell h + w h)\text{.}$ A right cylinder has $V = \pi r^2 h$ and lateral $SA = 2\pi r h$ (unwrap the side into a rectangle). Total cylinder $SA$ adds the two bases: $2\pi r^2 + 2\pi r h\text{.}$

How to do it:

1. Name the solid and whether you need volume, lateral area, or total area.

2. For a box, multiply the three edges for volume, or add the three pair-faces for $SA\text{.}$

3. For a cylinder, volume is $\pi r^2 h\text{;}$ lateral area is $2\pi r h\text{.}$

Worked example: A $5$ by $4$ by $3$ prism has volume $60\text{.}$ A cylinder with $r = 3$ in and $h = 8$ in has lateral $SA = 48\pi$ in$^2\text{.}$

Check: $2\pi r h$ is not volume; volume already used $\pi r^2 h\text{.}$ Replay a cylinder in Volume of a Cylinder.

A cross-section is the 2-D shape a plane cuts from a 3-D figure. The shape depends on the cut's angle and place.

How to do it:

1. Picture the solid.

2. Picture the plane: parallel to a face, through an axis, or tilted.

3. Name the 2-D shape, not the solid.

Worked example: A cut parallel to a face of a rectangular prism is a rectangle congruent to that face. A horizontal cut of a square pyramid, parallel to the base, is a smaller square.

Check: a vertical cut through a pyramid's apex is a triangle, not another square.