Secondary Mathematics · Worked Repair Guide 40
Plane mensuration is the mathematics of length, perimeter and area in two dimensions. The formulas are simple only when the correct base, perpendicular height and exposed boundary have been identified. Composite figures become reliable when they are decomposed into familiar regions and then recomposed without double-counting.
This guide develops one central habit: separate area decomposition from perimeter tracing. Internal edges may matter when splitting a figure into pieces but disappear from the external perimeter. A sloping side may belong to the boundary but not be the perpendicular height used in an area formula.
All dimensions and figures below are original teaching constructions. Use Units, Scale and Measurement for unit conversion and Surface Area, Volume and Composite Solids for the separate three-dimensional route.
1. Perimeter measures boundary length
A rectangle 8 cm by5 cm has perimeter2(8+5)=26 cm.
Perimeter uses linear units because it measures a path around the boundary.
Entry check: area40 cm² and perimeter26 cm describe different properties of the same rectangle.
When the figure is composite, trace only the outside boundary unless the question explicitly includes an internal edge.
2. Rectangle area is length times perpendicular width
Area of a rectangle=length×width.
For8 cm by5 cm, area=40 cm².
Area uses square units because two independent length directions are multiplied.
Do not report cm when the calculation is measuring a two-dimensional region.
3. Triangle area uses a perpendicular height
Area of a triangle=1/2×base×perpendicular height.
If base=12 cm and perpendicular height=7 cm, area=42 cm².
A sloping side of length7 cm cannot automatically replace the perpendicular height.
Extend the base line if necessary; the altitude may fall outside an obtuse triangle.
4. A parallelogram uses base times perpendicular height
Area of a parallelogram=base×perpendicular height.
For base10 cm and height6 cm, area=60 cm².
A sloping side length does not affect the area unless it helps determine the perpendicular height.
Geometrically, a triangular piece can be shifted from one side of a parallelogram to the other to form a rectangle with the same base and height.
5. A trapezium averages the parallel sides
For parallel sides a and b separated by perpendicular height h:
Area=1/2(a+b)h.
If a=8 cm,b=14 cm,h=5 cm, area=1/2(22)(5)=55 cm².
The height is the perpendicular distance between the parallel lines, not one of the sloping sides.
6. The trapezium formula includes triangles and parallelograms as limiting cases
If one parallel side shrinks to0, the trapezium formula becomes1/2bh: the triangle formula.
If the parallel sides are equal, a=b, the area becomes1/2(2a)h=ah: the parallelogram formula.
This connection reduces isolated memorisation.
Different named shapes can share one structural area relationship.
7. A kite or rhombus can use its perpendicular diagonals
Where the diagonals are perpendicular, area=1/2d₁d₂.
If a kite has perpendicular diagonals12 cm and9 cm, area=54 cm².
The formula works because the diagonals split the shape into triangles whose combined bases and heights produce the same product.
Do not assume arbitrary quadrilateral diagonals are perpendicular.
8. Composite area usually means add simple regions
An L-shape can be split into two non-overlapping rectangles.
Suppose one rectangle is8×3 and another5×4, joined without overlap. Total area=24+20=44 square units.
Any valid decomposition should produce the same total area.
Choose a split that minimises missing lengths and arithmetic.
9. Composite area can also mean subtract a cut-out
A10×8 rectangle has a3×2 rectangular corner removed.
Remaining area=80−6=74 square units.
Subtraction is often simpler than partitioning the remaining L-shape into several rectangles.
Area decomposition is flexible; use the representation that is easiest to verify.
10. Missing lengths can be recovered from alignment
In an orthogonal composite figure, a total horizontal width may be split into known parts.
If total width is12 and one horizontal section is5, the aligned remainder is7.
This is a consequence of collinear segments partitioning the same total distance.
Do not invent equal lengths merely because the drawing looks symmetric.
11. Composite perimeter is not the sum of the component perimeters
If two rectangles share an internal edge, adding their separate perimeters counts the shared edge twice even though it is not exposed.
For joined shapes, perimeter should usually be found by tracing the final outer boundary.
If two component perimeters are used, subtract twice the shared internal edge length.
Boundary logic and area logic are different operations.
12. Work an L-shape perimeter carefully
Start with a10×8 rectangle and remove a3×2 rectangle from one corner.
The removed corner deletes boundary lengths3 and2 but creates new exposed inner-step lengths3 and2.
Therefore the perimeter remains2(10+8)=36 units.
This surprising invariance occurs only for a rectangular corner notch whose new step edges match the removed outer lengths.
13. Semicircles and sectors can join polygonal figures
A rectangle width10 has a semicircle of diameter10 attached along one side.
The semicircle radius is5.
Added area=1/2π(5²)=25π/2.
For perimeter, the shared diameter is internal and is replaced by the semicircular arc length5π.
This connects plane mensuration to circle measure.
14. Shaded regions are often differences of familiar areas
If a circle radius5 is inscribed in a square side10, shaded corner area outside the circle is square area−circle area.
Area=100−25π.
The word “shaded” is not a formula. Identify which complete regions combine to produce it.
Draw a set-style inclusion picture mentally: outer region minus inner region.
15. Scale factors change perimeter and area differently
If every length of a similar plane figure is multiplied by k, perimeter is multiplied by k and area by k².
For scale factor3, perimeter factor=3 and area factor=9.
If areas are in ratio25:4, the positive length scale factor is5:2.
Area ratios should not be used directly as length ratios.
16. Reverse mensuration problems solve for a missing dimension
A trapezium has area84 cm², parallel sides10 cm and18 cm, and height h.
84=1/2(28)h=14h, so h=6 cm.
A parallelogram of area96 cm² and base12 cm has perpendicular height8 cm.
Keep the meaning of the unknown dimension attached to the algebra.
17. Trigonometry may supply the height needed for area
A parallelogram has adjacent sides10 cm and7 cm with included angle40°.
The perpendicular height relative to base10 is7 sin40°.
Area=10×7 sin40°=70 sin40°≈45.0 cm².
This is the same structural idea as triangle area1/2ab sinC, without the one-half factor because the full parallelogram contains two congruent triangles.
18. Capstone: composite trapezium and semicircle
An invented figure consists of a trapezium with parallel sides14 cm and8 cm, height6 cm, with a semicircle of diameter8 cm attached along the shorter parallel side.
Trapezium area=1/2(14+8)6=66 cm².
Semicircle radius4, area=1/2π(16)=8π cm².
Total area=66+8π cm².
For perimeter, the shared8 cm diameter is internal. The semicircular arc contributes4π cm, while the other three trapezium sides must be read or calculated from the specific diagram.
19. Independent practice
- Find the perimeter of an8×5 rectangle.
- Find its area.
- Find triangle area for base12,height7.
- Find parallelogram area for base10,height6.
- Find trapezium area for parallel sides8 and14,height5.
- A kite has perpendicular diagonals12 and9. Find area.
- An L-shape is made from non-overlapping rectangles8×3 and5×4. Find total area.
- A10×8 rectangle has a3×2 corner removed. Find remaining area.
- Find the perimeter of the notched figure in Question8.
- Total horizontal width12 includes a section5. Find the aligned remainder.
- A rectangle has a semicircle of diameter10 attached. Find the semicircle area.
- Find the semicircular arc length in Question11.
- A circle radius5 is inscribed in a square side10. Find area outside the circle.
- Similar figures have length factor3. Find perimeter factor.
- Find area factor in Question14.
- Similar figures have area ratio25:4. Find the positive length ratio.
- A trapezium has area84, parallel sides10 and18. Find height.
- A parallelogram has area96 and base12. Find perpendicular height.
- A parallelogram has adjacent sides10 and7 with included angle40°. Find area.
- A trapezium has parallel sides14 and8,height6, with a semicircle diameter8 attached. Find total area.
20. Worked answers
1. 26.
2. 40.
3. 42.
4. 60.
5. 55.
6. 54.
7. 44.
8. 74.
9. 36. The corner notch replaces equal removed horizontal and vertical boundary lengths.
10. 7.
11. 25π/2.
12. 5π.
13. 100−25π.
14. 3.
15. 9.
16. 5:2.
17. 6.
18. 8.
19. 70sin40°≈45.0.
20. 66+8π.
21. Diagnose mensuration errors by separating inside from boundary
Common failures include using a sloping side as a perpendicular height, adding component perimeters and double-counting internal edges, confusing square and linear units, using an area scale factor as a length factor, or forgetting that a shared diameter becomes internal when a semicircle is attached.
A useful repair note says “area needs perpendicular height”, “trace external boundary”, “shared edge is internal”, or “length factor k gives area factor k²”.
Then solve the same figure with a different valid decomposition. Agreement between two independent area routes is a strong check.
22. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Units, Scale and Measurement, Angles, Similarity and Geometric Reasoning, and Circle Geometry for neighbouring foundations.
Within Batch10, continue to Radian Measure, Arc Length, Sector and Segment Area, Graphical Solutions: Equations, Intersections and Roots, or Three-Dimensional Trigonometry and Spatial Diagrams.
23. Sources and scope
The plane figures, dimensions and practice questions are original teaching material. Area and perimeter formulas are used in their standard Euclidean sense.
For current Singapore Secondary curriculum and assessment scope, consult the relevant MOE/SEAB syllabus for the learner’s subject level and year. Match kite/rhombus and trigonometric-area extensions to the actual course being studied.
Complete Secondary Mathematics repair route: browse all 48 worked repair guides or return to the Secondary Mathematics Learning Hub.