Area of Combined Shapes
You already know how to find the area of simple shapes like rectangles and squares. But what about shapes that are not simple rectangles? Many real-life shapes are combinations — an L-shaped room, a T-shaped garden, or a cross-shaped tile.
To find the area of such shapes, you break them into simpler shapes (rectangles and squares), find the area of each part, and add them up. This method is called the "break into parts" or "split and add" method.
This skill is very useful in real life — calculating the area of a room with a balcony, a garden with a path, or an oddly shaped plot of land.
What is Area of Combined Shapes?
Definition: A combined shape (or composite shape) is a shape made by joining two or more simple shapes like rectangles, squares, or triangles.
Method to find area of combined shapes:
- Look at the shape and identify the simple shapes it is made of.
- Draw lines to split the combined shape into rectangles, squares, or triangles.
- Find the dimensions (length and width) of each part.
- Calculate the area of each part separately.
- Add the areas of all parts.
Alternative method (subtraction):
- Sometimes it is easier to think of the shape as a large rectangle with a piece removed.
- Area = Area of large rectangle − Area of removed piece.
Types and Properties
1. L-shaped Figures
An L-shape can be split into two rectangles. Find the area of each and add.
2. T-shaped Figures
A T-shape can be split into two rectangles — the top bar and the vertical stem.
3. U-shaped Figures
A U-shape is like a rectangle with a rectangular notch cut from the top. Use the subtraction method: area of full rectangle − area of notch.
4. Cross-shaped (+) Figures
A plus-sign shape can be split into a vertical rectangle and a horizontal rectangle, then subtract the overlapping square in the middle.
5. Irregular Step-like Figures
Step-like or staircase shapes can be split into several small rectangles. Find each area and add.
Solved Examples
Example 1: Example 1: L-shaped figure (addition)
Problem: An L-shaped room has dimensions as shown: the full length is 10 m, full width is 8 m. The cutout is 4 m × 5 m. Find the area.
Solution:
- Method 1 (subtraction): Full rectangle = 10 × 8 = 80 m². Cutout = 4 × 5 = 20 m².
- Area = 80 − 20 = 60 m².
Answer: 60 m²
Example 2: Example 2: L-shape split into two rectangles
Problem: An L-shape has: bottom rectangle 12 cm × 4 cm, right rectangle 4 cm × 6 cm. Find the area.
Solution:
- Area of bottom = 12 × 4 = 48 cm².
- Area of right = 4 × 6 = 24 cm².
- Total = 48 + 24 = 72 cm².
Answer: 72 cm²
Example 3: Example 3: T-shaped figure
Problem: A T-shape has a top bar 10 cm × 3 cm and a stem 3 cm × 7 cm. Find the area.
Solution:
- Top bar = 10 × 3 = 30 cm².
- Stem = 3 × 7 = 21 cm².
- Total = 30 + 21 = 51 cm².
Answer: 51 cm²
Example 4: Example 4: U-shaped figure
Problem: A U-shaped garden: outer rectangle 14 m × 10 m, inner cutout 8 m × 4 m. Find the area.
Solution:
- Outer = 14 × 10 = 140 m².
- Cutout = 8 × 4 = 32 m².
- Area = 140 − 32 = 108 m².
Answer: 108 m²
Example 5: Example 5: Cross-shaped figure
Problem: A plus sign: vertical bar 3 cm × 9 cm, horizontal bar 9 cm × 3 cm. Find the area.
Solution:
- Vertical = 3 × 9 = 27 cm².
- Horizontal = 9 × 3 = 27 cm².
- Overlap (centre square) = 3 × 3 = 9 cm².
- Area = 27 + 27 − 9 = 45 cm².
Answer: 45 cm²
Example 6: Example 6: Path around a rectangle
Problem: A garden is 20 m × 15 m. A path 2 m wide runs around it. Find the area of the path.
Solution:
- Outer rectangle (with path) = (20 + 4) × (15 + 4) = 24 × 19 = 456 m².
- Inner rectangle (garden) = 20 × 15 = 300 m².
- Path area = 456 − 300 = 156 m².
Answer: 156 m²
Example 7: Example 7: Two rectangles joined
Problem: Two rectangles of 6 cm × 4 cm and 5 cm × 4 cm are joined along their 4 cm sides. Total area?
Solution:
- Area 1 = 6 × 4 = 24 cm².
- Area 2 = 5 × 4 = 20 cm².
- Total = 24 + 20 = 44 cm².
Answer: 44 cm²
Example 8: Example 8: Rectangle with square cut out
Problem: A rectangular card is 15 cm × 10 cm. A square of side 3 cm is cut from one corner. Find the remaining area.
Solution:
- Rectangle = 15 × 10 = 150 cm².
- Square = 3 × 3 = 9 cm².
- Remaining = 150 − 9 = 141 cm².
Answer: 141 cm²
Example 9: Example 9: Staircase shape
Problem: A staircase shape has 3 steps. Each step is 2 cm wide and 2 cm tall. The base is 6 cm wide. Find the area.
Solution:
- Bottom step = 6 × 2 = 12 cm².
- Middle step = 4 × 2 = 8 cm².
- Top step = 2 × 2 = 4 cm².
- Total = 12 + 8 + 4 = 24 cm².
Answer: 24 cm²
Example 10: Example 10: Room with a balcony
Problem: A room is 5 m × 4 m. It has a balcony 2 m × 1.5 m attached on one side. Find the total area.
Solution:
- Room = 5 × 4 = 20 m².
- Balcony = 2 × 1.5 = 3 m².
- Total = 20 + 3 = 23 m².
Answer: 23 m²
Real-World Applications
Home Design: Rooms are rarely perfect rectangles. L-shaped living rooms, kitchens with islands, and balconies all require combined area calculation for flooring and painting.
Garden Planning: Gardens with paths, flower beds, and ponds have combined shapes. You need the area to buy the right amount of soil, grass seed, or tiles.
Construction: Builders calculate the area of walls (rectangular with windows and doors cut out) to estimate paint or plaster needed.
Art and Design: Creating patterns, tiles, and craft projects often involves combining simple shapes into complex designs.
Key Points to Remember
- A combined shape is made of two or more simple shapes joined together.
- Addition method: Split into simple shapes, find each area, add them.
- Subtraction method: Start with a large rectangle, subtract the area of the removed piece.
- Always identify the dimensions of each part carefully.
- Be careful not to double-count overlapping regions (like in cross shapes).
- Units of area are always squared: cm², m².
- Draw dotted lines to show how you split the shape.
- Check: the combined area should make sense — not more than the bounding rectangle.
Practice Problems
- An L-shape: outer dimensions 12 m × 8 m, cutout 5 m × 3 m. Find the area.
- A T-shape: top bar 8 cm × 2 cm, stem 2 cm × 6 cm. Find the area.
- A rectangular garden 18 m × 12 m has a rectangular pond 6 m × 4 m in the centre. Find the area of the garden excluding the pond.
- Two squares of side 5 cm are placed side by side. What is the total area?
- A plus-sign shape: each arm is 2 cm wide and 4 cm long (from centre). Find the area.
- A room 6 m × 5 m has a cupboard area 2 m × 1 m built into one wall. What is the usable floor area?
- A frame is made from a rectangle 20 cm × 15 cm with a rectangle 16 cm × 11 cm cut from the centre. Find the area of the frame.
- A staircase has 4 steps, each 3 cm wide and 3 cm tall. Find the total area.
Frequently Asked Questions
Q1. What is a combined shape?
A shape made by joining two or more simple shapes (rectangles, squares, triangles). Also called a composite shape.
Q2. How do you find the area of a combined shape?
Split it into simple shapes, find the area of each part, and add them up. Or, use the subtraction method: full rectangle area minus removed piece area.
Q3. When should I use subtraction instead of addition?
Use subtraction when the shape looks like a rectangle with a piece cut out (U-shape, rectangle with hole). Use addition when the shape is made by joining pieces together (L-shape, T-shape).
Q4. How do I know where to split the shape?
Look for straight horizontal or vertical lines that divide the shape into rectangles. Draw dotted lines to mark the split.
Q5. What about overlapping areas?
If two parts overlap (like in a cross shape), add both areas and subtract the overlapping region to avoid counting it twice.
Q6. Can combined shapes include triangles?
Yes. At Class 6 level, most problems use rectangles and squares. In higher classes, you will combine triangles, circles, and other shapes too.
Q7. What unit do I use for area?
Area is always in square units: cm², m², km². If the dimensions are in cm, the area is in cm².
Q8. Is this topic important for exams?
Yes. Questions on area of combined shapes appear regularly in Class 6 and 7 NCERT exams.
