Perimeter and Area
Perimeter measures the length around a boundary; area measures the surface enclosed by it. This lesson develops the standard area formulae from geometric ideas, then applies them to missing lengths, composite shapes, coordinates and reverse problems.
Foundation & Higher. The geometry on this page belongs to both tiers. Algebraic or bounds-based extensions are difficult because of the method used, not because the underlying area geometry changes.
Perimeter and area answer different questions
Perimeter is the complete outside boundary, measured in units such as cm or m. Area is the amount of surface covered, measured in square units such as cm² or m².
A 7 cm by 4 cm rectangle has perimeter 7+4+7+4=22 cm, but area 7×4=28 cm². The units help show that these are different quantities.
Rectangle, triangle and parallelogram
Rectangle: A=bh. A diagonal divides a rectangle into two equal triangles, giving triangle area A=½bh. A parallelogram can be cut and rearranged into a rectangle with the same perpendicular height, giving A=bh.
The height is always measured perpendicular to the chosen base. A sloping side is not automatically the height.
Worked example: a triangle has base 12 cm and perpendicular height 7 cm. Area = ½×12×7=42 cm².
Trapezia, kites and rhombi
For a trapezium with parallel sides a and b and perpendicular height h, A=½(a+b)h. Think of this as average parallel-side length × height.
Worked example: parallel sides 8 cm and 14 cm, height 5 cm: A=½(8+14)×5=11×5=55 cm².
Kites and rhombi can be split into triangles using their diagonals. When the diagonals are perpendicular, the familiar result A=½d₁d₂ follows from adding the triangle areas. Understanding the split is safer than memorising an isolated formula.
Composite areas — add or subtract
Break an irregular shape into familiar pieces, or place it inside a larger simple shape and subtract the missing part. Mark every length you know before calculating.
Worked example: an 11 m by 8 m rectangle has a 3 m by 2 m rectangular corner removed. Area = 11×8−3×2=88−6=82 m².
For the perimeter of the cut-out shape, trace the new outside boundary. Removing area can create new exposed edges, so you cannot simply subtract the cut-out perimeter.
Reverse problems and missing lengths
Area formulae can be used backwards. If a triangle has area 54 cm² and base 12 cm, then 54=½×12×h=6h, so h=9 cm.
Likewise, if a parallelogram has area 72 cm² and perpendicular height 8 cm, its base is 72÷8=9 cm. Keep the geometric meaning visible while rearranging.
Coordinates and area
Coordinates can supply lengths without introducing a new area formula. For an axis-aligned rectangle with vertices (2,1), (8,1), (8,5), (2,5), the width is 8−2=6 and height is 5−1=4, so area=24 square units.
For more complicated coordinate polygons, split into rectangles and triangles whose bases and perpendicular heights can be read from the grid.
Make units compatible first
Do not multiply measurements expressed in different units. If a rectangle is 1.2 m by 80 cm, convert first: 80 cm=0.8 m, so area=1.2×0.8=0.96 m².
Remember that area units are squared. The full unit-conversion method belongs in Units, Scale Drawings and Maps; here the conversion is applied to geometry.
Think Like a Mathematician
Most area formulae are not isolated facts. Triangles are half of suitable parallelograms or rectangles; parallelograms rearrange to rectangles; trapezia can be paired or decomposed. Seeing those relationships makes unfamiliar composite problems easier.
Common Mistakes
Using a sloping side as the height: area needs the perpendicular height.
Trapezium formula: a and b are the parallel sides, not the sloping pair.
Assuming perimeter and area scale together: doubling every side doubles perimeter but multiplies area by four.
Cut-outs: subtracting a cut-out area can create extra exposed perimeter.
Practice — perimeter and area
1. Find the perimeter and area of a 9 cm by 4 cm rectangle.
Perimeter=26 cm; area=36 cm².
2. Find the area of a triangle with base 15 cm and perpendicular height 8 cm.
½×15×8=60 cm².
3. A parallelogram has base 11 m and perpendicular height 6 m. Find its area.
66 m².
4. A trapezium has parallel sides 7 cm and 13 cm and height 4 cm. Find its area.
½(7+13)×4=40 cm².
5. A 12 m by 9 m rectangle has a 4 m by 3 m corner removed. Find the remaining area.
108−12=96 m².
6. A triangle has area 45 cm² and base 10 cm. Find its perpendicular height.
45=5h, so h=9 cm.
7. A rectangle measures 1.5 m by 60 cm. Find its area in m².
60 cm=0.6 m; area=0.9 m².
8. A rectangle has vertices (−2,1), (5,1), (5,6), (−2,6). Find its area.
Width=7, height=5, area=35 square units.
9. Every side of a square is doubled. What happens to its perimeter and area?
Perimeter doubles; area becomes four times as large.
10. Explain why a 10 cm sloping side of a parallelogram cannot automatically be used as h in A=bh.
h must be the perpendicular distance between the parallel bases.