Perimeter and Area

1Perimeter

Perimeter is the length of the border path. Walk each outer side once and add, in one unit.

Counting tiles inside is area, not perimeter.

Figure. Perimeter is the walk around the 13-by-7 rectangle: 2(13+7) = 40. The arrows follow the border. That count is in centimetres, not tiles.

How it works

1. Walk the outsideEach outer side once.
2. Add those lengthsOne unit.
3. Call that PThe border.

2Area

Area is the cover that tiles with no gap and no overlap. Unit squares are the usual tile at this class.

The lace-count is not the cover.

Figure. Area is the cover that fills without gaps. A 4-by-3 grid of unit squares is 12 square units. Counting the walk around would answer a different question.

How it works

1. Choose a unit tileA square unit.
2. Cover without gapsAnd no overlap.
3. Count the tilesThat count is A.

3Area of a Triangle

A triangle that is half of a tiled rectangle (or parallelogram) has area half of that rectangle: A=\frac{1}{2}\times base \times height when height is perpendicular to that base.

Using a slant side as “height” is a miss.

Figure. A 6-by-4 rectangle has area 24. The diagonal cuts it into two equal triangles, so each has area 12, which is (1/2)×base×height. The triangle is half the tiling rectangle, not half the walk.

How it works

1. Name a baseOne side.
2. Drop a perpendicular height to that baseThe height.
3. Take half of base × heightThe triangle’s cover.

4Perimeter is the border path

Say this again as a job-word: P answers “how long is the fence?” not “how much grass?”

A longer fence can enclose less grass.

Figure. Perimeter is the border path. The arrows walk the rim of the same 13-by-7 rectangle and stop there. The inside tiles are a different count.

How it works

1. Ask fence or grassWhich job?
2. If fence, add outside sidesP.
3. Keep A out of that sumDifferent job.

5Area is the cover that tiles without gaps

If circles leave holes, those holes are not counted as the circle-tiles’ cover of the rectangle.

A complete tile-measure has no leftover holes of the tiling.

Figure. Area tiles the inside without gaps and without leftover slivers. A 3-by-2 pack of unit squares is 6 square units. Overlapping or leaving a crack would break the count.

How it works

1. See leftover holesNot a complete cover by those tiles.
2. Switch to tiles that fit, or count holes separatelyHonest A.
3. Do not ignore holesThey are not cover.

6A triangle is half a parallelogram that tiles

Two copies of the same triangle can make a parallelogram. That is why the half-formula exists — not because “triangles look smaller”.

An unrelated triangle is not automatically half of a random rectangle.

How it works

1. See two copies joiningA parallelogram.
2. Area of one is half of thatThe reason.
3. Then use ½bh with perpendicular hThe formula.

7Units of length and units of cover are different

cm is a length unit. cm² is a cover unit. Adding 3 cm to 4 cm² is a mash.

Write the unit that matches the job.

Figure. A 9-by-9 square has walk 2(9+9) = 36 cm and cover 9×9 = 81 square centimetres. Centimetres and square centimetres are different units; swapping them is a different quantity.

How it works

1. P gets a length unitcm, m.
2. A gets a square unitcm², m².
3. Never add themTwo kinds.

8A miss: adding sides to get area

3+4+5 is a perimeter of a triangle (if those are the sides), not an area. Area needs tiles or ½bh.

The miss is using the border-sum as cover.

Figure. Adding the sides 8 and 7 gives 15, which is not the area. The 8-by-7 cover is 8×7 = 56 square units. The sum is a fragment of the walk, not the tiles.

How it works

1. See a side-sumThat is P if they are the outer sides.
2. Ask if the question wanted coverThen do not use that sum.
3. Switch to tiles or ½bhArea.