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Math Lecture Notes
Geometry Skills

Polygon Area Puzzles link to here link to index

Many problems make finding area puzzle-like. Sometimes we can "stick together" small pieces to find a big area. Sometimes we can "remove" a small piece from a big area to get the shape in question. And sometimes either method will work!

Example 1

Find the perimeter and area of this shape.

area

When finding the area, which plan did you use?

This "subtract pieces" plan?

area

This "glue pieces together" plan?

area

Or this other "glue pieces together" plan?

area

All of those work! Which plan seems most natural varies from person to person. Our brains are not all built the same!

Let's do another example of a puzzle-like area problem.

Example 2

Find the area of this shape.

area

Measured Sides link to here link to index

When solving geometry problems do not get confused if the diagram provides too many numbers!

Consider this problem. The square has an area of 4 × 4 = 16.

area of a square

Here is the same problem with extra numbers.

area of a square

The extra numbers do nothing! The area does not change. The problem does not magically change into a perimeter problem.

Be wary! Keep the formulas in mind. Ignore extra numbers.

Problem Solving Steps link to here link to index

Recall our six-step problem solving process.

  1. Determine what you are looking for
  2. Draw pictures
  3. Name things
  4. Make equations
  5. Solve the equations
  6. Check your answer

Let's augment this list of six steps to include the details of solving a word problem with a percent sentence.

The first two steps (determining what you are looking for and drawing pictures) are usually the most difficult, especially if there is puzzle-like thinking about gluing pieces together or cutting out holes. Fortunately, the textbook almost always supplies you with the picture.

The third step is really important. In real life most shapes do not appear pre-measured. We need to realize which sides are important to measure. Some sides—such as the slanted sides of a triangle, parallelogram, or trapezoid—are not part of the formulas. So steps three and four (naming things and making equations) are intimately related.

The final two steps (solving the equations and checking your answer) are easy for geometry problems. Just plug the measured lengths into the formulas and solve. We have already seen that in a puzzle-like problem the order of operations will keep each small shape's area a separate term until the end.

So our list of steps becomes:

  1. Determine what you are looking for
  2. Draw pictures
  3. Name things
    write a "plan" for puzzle-like problems
    consider which measurements you will need
  4. Make equations
    put formulas in your "plan"
    put measured lengths in the formulas
  5. Solve the equations
  6. Check your answer

Let's do two more examples of puzzle-like area problems.

Example 3 (§9.2 # 47, Page 536)

Find the area of this shape.

area

  1. Determine what you are looking for
  2. Draw pictures
  3. Name things
    write a "plan" for puzzle-like problems
    consider which measurements you will need
  4. Make equations
    put formulas in your "plan"
    put measured lengths in the formulas
  5. Solve the equations
  6. Check your answer

Example 4 (§9.2 # 39, page 535)

Find the area of the sidewalk, which is only on two sides of the building.

area

This problem we will only start together.

  1. Determine what you are looking for
  2. Draw pictures
  3. Name things
    write a "plan" for puzzle-like problems
    consider which measurements you will need
  4. Make equations
    put formulas in your "plan"
    put measured lengths in the formulas
  5. Solve the equations
  6. Check your answer

The picture can be confusing! Try drawing the footprint of the building instead.

The problem is easy once you draw flat rectangles.

area

Circle Puzzles link to here link to index

Example 5

Find the perimeter and area of this circle.

circle example

Example 6

Find the perimeter and area of this circle.

circle example

Example 7

Find the perimeter and area of this circle.

circle example

Example 8

Find the perimeter and area of this shape.

area

  1. Determine what you are looking for
  2. Draw pictures
  3. Name things
    write a "plan" for puzzle-like problems
    consider which measurements you will need
  4. Make equations
    put formulas in your "plan"
    put measured lengths in the formulas
  5. Solve the equations
  6. Check your answer