What does it mean?
Area counts the space covered inside a boundary. A composite shape can be split into familiar pieces whose interiors do not overlap. Adding their areas then counts every part once. Another method surrounds the shape with a simple rectangle and subtracts the missing region.
The area of a trapezium is ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height between them. You can see this by averaging the two parallels, then multiplying by height, or by splitting into a rectangle and a triangle. A composite L-shape can be a large rectangle minus a cut-out rectangle, or two rectangles added. Use the marked lengths. Units of area are square units, such as cm². The diagram is not to scale.
Learning goal: Calculate area of trapezia and composite polygons by decomposing shapes. The grade is a suggested starting point; school sequences vary. Use the first worked example to check readiness, and Area and perimeter: the difference, examples and practice if an earlier idea needs revisiting.
See the idea
Read each diagram with its labels, then follow the worked examples. Practice questions include their own measurements and information.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Choose a method and explain why it works
Choose a decomposition that uses given lengths. Find missing lengths by subtracting matching totals before calculating areas. For a trapezium, identify the parallel sides and their perpendicular separation: area = (a + b)h/2. For each part, write the formula, substitution and square unit separately before combining results.
Notation: Trapezium: ½(a + b)h. L-shape: add rectangles or subtract a rectangle.
Key rule and when to use it
Add areas of non-overlapping parts, or subtract a missing region from a larger shape. Rectangle area = lw; trapezium area = ½(a + b)h with perpendicular h.
Worked examples, step by step
Example 1: A trapezium has parallel sides 10 cm and 6 cm and perpendicular height 4 cm. Find the area.
- Identify a = 10, b = 6, h = 4. The slant sides are not used for area.
- ½(10 + 6) = 8, then 8 × 4 = 32.
- Area = 32 cm².
Answer: 32 cm²
Check: The area sits between the rectangle 10 × 4 = 40 and the rectangle 6 × 4 = 24. 32 is in between.
Example 2: An L-shape is a 12 cm by 8 cm rectangle with a 5 cm by 3 cm rectangle cut from one corner. Find the area.
- This is not a trapezium. Decompose: large rectangle minus cut-out.
- 12 × 8 = 96. Cut-out 5 × 3 = 15.
- Area = 96 − 15 = 81 cm².
Answer: 81 cm²
Check: The L-shape is less than 96 cm² by the 15 cm² corner, so 81 cm² is consistent.
Example 3: A 9 m by 6 m rectangle has a 3 m by 2 m corner removed. Find the remaining area.
- The enclosing rectangle has area 9 × 6 = 54 m².
- The missing corner has area 3 × 2 = 6 m².
- Remaining area = 54 − 6 = 48 m². The boundary length is a separate question.
Common mistakes and important boundaries
A slanted side is not necessarily a perpendicular height. Adding all visible lengths finds neither area nor, if internal cuts are included, the correct outside perimeter. If pieces overlap, subtract the overlap so that it is not counted twice. Diagrams need not be drawn to scale.
A point to explain
A learner may multiply the three sides of a trapezium, use a slant as height, or add outer lengths instead of areas.
Compare an uncertain attempt with the closest worked example. Look at the meaning of each operation, unit and label before repeating the arithmetic.
Area of Composite Shapes: Split or Subtract — practice
Work on paper first. Open each answer afterwards and compare the reasoning. Print the questions with or without the separate answer key; no account is required.
Name: ____________________ Date: __________
In the trapezium area formula ½(a + b)h, what is h?
Answer and a way to check
The perpendicular height between the parallel sides.
h joins the two parallels at 90°. A slanted edge is not h.
Parallels 8 cm and 4 cm, height 5 cm. Which expression is the area?
Answer and a way to check
½(8 + 4) × 5
Add the parallels, take half, then multiply by the height.
Trapezium, parallel sides 12 m and 8 m, height 5 m. Find the area.
Answer and a way to check
50 m²
Add 12 and 8, take half, then multiply by 5.
Trapezium, parallel sides 9 cm and 5 cm, height 6 cm. Find the area.
Answer and a way to check
42 cm²
½(9 + 5) × 6. Do not add 9 + 5 + 6.
L-shape: large rectangle 10 cm by 7 cm minus a 4 cm by 2 cm cut-out. Find the area.
Answer and a way to check
62 cm²
Compute 10 × 7, then 4 × 2, then subtract the areas.
Why can you not use a 7 cm slanted edge as h in a trapezium with parallels 10 cm and 6 cm?
Answer and a way to check
h must be perpendicular to the parallel sides. A slanted edge is not the height.
Find the 90° join between the parallels. That length is h.
L-shape 12 cm by 8 cm minus 5 cm by 3 cm. A student computes 12 + 8 + 5 + 3 = 28 and calls it the area. What is the mistake?
Answer and a way to check
The student added lengths instead of subtracting areas of rectangles.
Area uses products of pairs: large rectangle minus small rectangle.
A patio is a trapezium with parallels 9 m and 5 m and height 4 m, joined to a 3 m by 2 m rectangle. Find the total area.
Answer and a way to check
34 m²
First ½(9 + 5) × 4. Then add 3 × 2. Do not mix metres with square metres.
Area of Composite Shapes: Split or Subtract — answer key
The perpendicular height between the parallel sides. h joins the two parallels at 90°. A slanted edge is not h.
½(8 + 4) × 5 Add the parallels, take half, then multiply by the height.
50 m² Add 12 and 8, take half, then multiply by 5.
42 cm² ½(9 + 5) × 6. Do not add 9 + 5 + 6.
62 cm² Compute 10 × 7, then 4 × 2, then subtract the areas.
h must be perpendicular to the parallel sides. A slanted edge is not the height. Find the 90° join between the parallels. That length is h.
The student added lengths instead of subtracting areas of rectangles. Area uses products of pairs: large rectangle minus small rectangle.
34 m² First ½(9 + 5) × 4. Then add 3 × 2. Do not mix metres with square metres.
Questions students and parents ask
Can adding the component perimeters give the outside perimeter?
Not directly when parts share an edge. Shared boundaries lie inside the combined shape and would be counted twice. Trace the outside boundary instead.
Can two different splits both be right?
Yes. Correct non-overlapping decompositions cover the same whole region, so their total areas should agree. Comparing two methods is a useful check.
Is this a complete set of textbook exercise solutions?
No. This guide adapts BananaLearn’s own teaching examples and practice. It is not official NCERT material or a promise of the exact chapter sequence in a particular board, country or textbook edition.
What if the examples are still difficult?
Revisit Area and perimeter: the difference, examples and practice. Try the first example with its model, then attempt one fresh question independently. A correct copied answer is not evidence that the method can be used without support.
Help at home
Ask the learner to trace the outside region and shade each proposed part in a different colour. Check for gaps or double counting before arithmetic.
Keep one example of the learner’s own working. A repeated difficulty across fresh questions can guide a discussion with their teacher; one short practice sheet does not establish a diagnosis or a full mastery profile.
Connect this idea to the next skill
About this guide
Based on BananaLearn’s Grade 8 course lesson “Split, Then Find the Area”. The explanations have been adapted for independent reading; course diagrams are reused where they fit the idea, and missing classroom givens are made explicit in questions.
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