Length and Area in 2-D Figures

Every area formula is a rectangle in disguise. Know where each one comes from and you will never mix them up.

You need carpet for an L-shaped room. There is no formula for an L.

But there is a formula for a rectangle, and an L is two rectangles that happen to be joined.

carpet for an L-shaped room piece A piece B …or cut here? 5 m 2 m 4 m no single formula fits this shape — but two rectangles do

Cut along the dashed line. The top piece is 5 × 2, the bottom is 7 × 2 — add them and you have the floor.

Cut the other way instead and you get two different rectangles, and the same total. Both cuts are right.

You could also take the big enclosing rectangle and subtract the missing corner. Also right.

That is the whole method for awkward shapes: cut into pieces you have formulas for, or subtract what is not there. The shape is not the problem; finding a single formula for it was.

A parallelogram rearranged into a rectangle of equal area parallelogram: b × h same rectangle, same area h cut the triangle off one end, slide it to the other — nothing is lost

Where the formulas come from

Rectangle: A = bh. Rows of unit squares — h rows of b squares each.

Parallelogram: A = bh too. Cut a right triangle off one end and slide it to the other; you have made a rectangle. Nothing was added or lost.

Triangle: A = 12bh. Any triangle is exactly half of a parallelogram — copy it, rotate the copy 180°, and the two fit together.

Trapezoid: A = 12(b1 + b2)h. Two copies make a parallelogram with base b1 + b2.

Height means perpendicular height

The h in every formula is the straight-up distance between the two bases, not the slanted side. On a tilted parallelogram, the slanted side is longer and will give you an answer that is too big.

Circles

C = 2π r = π d and A = π r2.

Keep them apart by their units. Circumference is a length, so it uses r to the first power. Area is a square measure, so it uses r2.

Why A = π r2: slice the circle into thin wedges and alternate them point-up and point-down. They interlock into something very close to a rectangle of height r and width C2 = π r. So A = π r · r.

Composite figures: add or subtract

Break the shape into rectangles, triangles, and parts of circles. Find each area, then combine.

Often subtraction is faster. A circle cut out of a square: find the square, find the circle, subtract. A shaded border: find the outer shape and take away the inner one.

Three composite areas: a rectangle plus a half circle, a square minus a circle, and a border found as outer minus inner rectangle addhalf circle on top subtracta circle cut out subtracta border r = 4 84 8×4 + ½π×4² = 32 + 8π ≈ 57.1 r = 3 10 10² − π×3² = 100 − 9π ≈ 71.7 8 × 6 108 10×8 − 8×6 = 80 − 48 = 32 often subtraction is faster: the whole shape, take away the hole

Draw the cut lines

Physically draw lines on the figure to split it up, and label every piece with its own dimensions before computing anything. Doing all the drawing first, then all the arithmetic, is much more reliable than alternating.

Perimeter of composite shapes

Perimeter follows the outside edge only. When you cut a shape into pieces, the cut lines are not part of the perimeter.

For a shape with a semicircular top, the curved part contributes 12 · 2π r = π r, and the straight edge it replaced is no longer on the boundary.

Same perimeter, different area

A 1 × 11 rectangle and a 6 × 6 square both have perimeter 24, but their areas are 11 and 36.

For a fixed perimeter, the closer to a square, the larger the area. Perimeter does not determine area.

A 1 by 11 rectangle and a 6 by 6 square both have perimeter 24, but their areas are 11 and 36; areas of all six whole-number rectangles with perimeter 24 every rectangle here has perimeter 24 1 × 11: P = 24, A = 11 6 × 6: P = 24, A = 36 area of each, perimeter 24 111×11202×10273×9324×8355×7366×6 the closer to a square, the bigger the area — perimeter does not decide area

Worked examples

Composite by subtraction

A square of side 10 cm has a circle of diameter 10 cm cut out of it. Find the remaining area. Use π ≈ 3.14.

  1. Square: 10 × 10 = 100 cm2 The whole shape before cutting.
  2. Circle radius = 5 cm Radius is half the diameter — do not use 10 here.
  3. Circle: π (5)2 = 25π ≈ 78.5 cm2 Square the radius, then multiply by π.
  4. 100 - 78.5 = 21.5 cm2 Subtract what was removed.

About 21.5 cm2.

A trapezoid

A trapezoid has parallel sides 7 m and 13 m, and height 6 m. Find its area.

  1. Add the parallel sides: 7 + 13 = 20 The two bases go together first.
  2. Halve it: 20 ÷ 2 = 10 This is the average base — the width of the equivalent rectangle.
  3. 10 × 6 = 60 Average base times height.

60 m2.

Working backwards

A triangle has area 54 cm2 and base 12 cm. Find its height.

  1. Write the formula: 54 = 12(12)h Put in what you know.
  2. Simplify: 54 = 6h Half of 12 is 6.
  3. h = 9 Divide both sides by 6.

9 cm.

Words to know

Perimeter
Distance all the way around the outside.
Area
Space inside, measured in square units.
Circumference
The perimeter of a circle: C = 2π r.
Composite figure
A shape built from simpler ones.
π
The circumference of any circle divided by its diameter — about 3.14159.

Mistakes to watch for

Using the slanted side as the height. In a parallelogram or triangle the height must be perpendicular to the base — the straight-up distance, not the sloping edge. The slant is always longer, so the area comes out too big.

Using the diameter where the formula asks for the radius. Since r is squared, using 10 instead of 5 makes the area four times too large. Halve it and write it down before substituting.

Including an internal cut line in a perimeter. The dashed line you drew to split the shape is not a wall — nobody walks along it. Perimeter follows only the outside edge.

Adding areas of pieces that overlap. If your cuts double-count a corner, the total is too big. Make the pieces meet exactly, edge to edge, with nothing shared.

The word area is Latin for a level piece of open ground — a threshing floor. It meant a real patch of earth long before it meant a number.

Egyptian surveyors were called harpedonaptai, "rope-stretchers," and re-measured farmland after every Nile flood so taxes could be set fairly.

Geometry itself is Greek for "earth measurement." The subject is named after that job.

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