Length and Area in 2-D Figures — Print
Every area formula is a rectangle in disguise. Know where each one comes from and you will never mix them up.
What the practice questions cover
These questions use 13 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Circumference — The perimeter of a circle: C = 2π r.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Base — The number being multiplied. In 53 the base is 5.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Height — The perpendicular distance from the base to the opposite corner.
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.
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.
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.
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.
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.
- Square: 10 × 10 = 100 cm2 The whole shape before cutting.
- Circle radius = 5 cm Radius is half the diameter — do not use 10 here.
- Circle: π (5)2 = 25π ≈ 78.5 cm2 Square the radius, then multiply by π.
- 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.
- Add the parallel sides: 7 + 13 = 20 The two bases go together first.
- Halve it: 20 ÷ 2 = 10 This is the average base — the width of the equivalent rectangle.
- 10 × 6 = 60 Average base times height.
60 m2.
Working backwards
A triangle has area 54 cm2 and base 12 cm. Find its height.
- Write the formula: 54 = 12(12)h Put in what you know.
- Simplify: 54 = 6h Half of 12 is 6.
- 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.
Problem 1
Find the area of a rectangle 7 cm by 9 cm.
Worked answer
63
7 × 9 = 63 cm2
Problem 2
Find the area of a triangle with base 10 and height 6.
Worked answer
30
12(10)(6) = 30
Problem 3
Find the area of a parallelogram with base 12 and height 5.
Worked answer
60
12 × 5 = 60
Problem 4
Find the circumference of a circle with radius 3. Use π ≈ 3.14.
Worked answer
18.84
2 × 3.14 × 3 = 18.84
Problem 5
Find the area of a circle with radius 5. Use π ≈ 3.14.
Worked answer
78.5
52 = 25
3.14 × 25 = 78.5
Problem 6
A trapezoid has parallel sides 7 m and 13 m, and height 6 m. Find its area.
Worked answer
60
7 + 132 = 10
10 × 6 = 60 m2
Problem 7
A triangle has area 54 cm2 and base 12 cm. Find its height.
Worked answer
9
54 = 12(12)h = 6h
h = 9 cm
Problem 8
Find the area of a circle with diameter 14. Use π ≈ 227.
Worked answer
154
r = 7
227 × 49 = 22 × 7 = 154
Problem 9
A square of side 10 cm has a circle of diameter 10 cm cut out. Find the remaining area. Use π ≈ 3.14.
Worked answer
21.5
Square: 100 cm2
Circle: 3.14 × 25 = 78.5 cm2
100 - 78.5 = 21.5 cm2
Problem 10
A rectangle has perimeter 30 and length 9. Find its width.
Worked answer
6
18 + 2w = 30
2w = 12
w = 6
Problem 11
A 10 × 6 rectangle has a 3 × 4 rectangle cut from one corner. Find the remaining area.
Worked answer
48
10 × 6 = 60
3 × 4 = 12
60 - 12 = 48
Problem 12
A circle has circumference 31.4. Find its radius. Use π ≈ 3.14.
Worked answer
5
31.4 = 6.28r
r = 31.4 ÷ 6.28 = 5
Problem 13
A shape is a 10 × 6 rectangle with a semicircle of diameter 6 on one end. Find its total area. Use π ≈ 3.14.
Worked answer
74.13
Rectangle: 10 × 6 = 60
Semicircle: 12 × 3.14 × 32 = 14.13
60 + 14.13 = 74.13
Problem 14
Among rectangles with whole-number sides and perimeter 24, which has the greatest area?
- 1 × 11
- 3 × 9
- 4 × 8
- 6 × 6
Worked answer
6 × 6
Areas: 11, 27, 32, 36
The square wins. For a fixed perimeter, the closer to square, the larger the area.
Problem 15
A circular path has outer radius 10 and inner radius 7. Find the area of the ring. Use π ≈ 3.14.
Worked answer
160.14
Outer: 3.14 × 100 = 314
Inner: 3.14 × 49 = 153.86
314 - 153.86 = 160.14
Note π(102 - 72) = 51π, not π(3)2.
Problem 16
A triangle has its base along the full 12-unit side of a 12 × 8 rectangle, with its third vertex on the opposite side. Find its area.
Worked answer
48
Base = 12, height = 8
12(12)(8) = 48
Exactly half the rectangle — and it does not matter where along the side the apex sits.