Area between two shapes
Key Notes :
| Introduction |
Sometimes we need to find the area covered between two shapes.
This usually happens when one shape is inside another (like a circle inside a square, or a rectangle inside another rectangle).
The area between two shapes = Area of the larger shape – Area of the smaller shape.
| Formula |
Area between two shapes=Area of Outer Shape−Area of Inner Shape
| Steps to Solve |
- Identify the outer shape and the inner shape.
- Find the area of the outer shape using the correct formula.
- Find the area of the inner shape using the correct formula.
- Subtract:
- Required Area=Outer Area−Inner Area
| Examples |
Example 1: Rectangle with a smaller rectangle cut out
- Outer rectangle: length = 12 cm, breadth = 8 cm
- Inner rectangle: length = 6 cm, breadth = 4 cm
Outer Area=12×8=96 cm2
Inner Area=6×4=24 cm2
Area between two rectangles=96−24=72 cm2
Example 2: Square with a circle inside
- Square side = 14 cm
- Circle radius = 7 cm
Area of square=14×14=196 cm2
Area of circle=πr2=22/7×7×7=154 cm2
Area between square and circle=196−154=42 cm2
| Common Shapes and Their Areas |
- Rectangle → l×b
- Square → a2
- Triangle → 1/2×base×height
- Circle → πr2
- Parallelogram → base×height
| Key Points |
- Always subtract smaller shape area from larger shape area.
- Be careful with units (cm², m², etc.).
- Sometimes inner shapes may be circles, rectangles, or triangles.
- This concept is often used in real life, e.g., finding garden paths, borders, or rings.
Learn with an example
▶️ What is the area of the shaded region?
Write your answer as a whole number or a decimal rounded to the nearest hundredth.
_______ square metres
To find the area of the shaded region, subtract the area of the inner shape from the area of the outer shape. Start by finding the area of the inner shape.
Find the base and height of the inner triangle.
base: 8 m
height: 12 m
Use these numbers in the formula.
inner area = 1/2 . base . height
= 1/2 . 8 . 12
= 48
Now find the units. The lengths are measured in metres, so the area is measured in square metres.
The area of the inner triangle is 48 square metres.
Next, find the area of the outer shape.
Find the base and height of the outer triangle.
base: 13 m
height: 17 m
Use these numbers in the formula.
outer area = 1/2 base . height
= 1/2 . 13 . 17
= 110.5
As with the inner triangle, the lengths of the outer triangle are measured in metres. So, the area is measured in square metres.
The area of the outer triangle is 110.5 square metres.
Finally, subtract the inner area from the outer area to find the area of the shaded region.
shaded area = 110.5 – 48
= 62.5
The area of the shaded region is 62.5 square metres.
▶️ What is the area of the shaded region?
_____ square centimetres
To find the area of the shaded region, subtract the area of the inner shape from the area of the outer shape. Start by finding the area of the inner shape.
Find the side length of the inner square.
side: 10 cm
Use this number in the formula.
inner area=side
side = 10 . 10
=100
Now find the units. The lengths are measured in centimetres, so the area is measured in square centimetres.
The area of the inner square is 100 square centimetres.
Next, find the area of the outer shape.
Find the side length of the outer square.
side: 19 cm
Use this number in the formula.
outer area=side . side
=19 . 19
=361
As with the inner square, the lengths of the outer square are measured in centimetres. So, the area is measured in square centimetres.
The area of the outer square is 361 square centimetres.
Finally, subtract the inner area from the outer area to find the area of the shaded region.
shaded area = 361 – 100
= 261
The area of the shaded region is 261 square centimetres.
▶️ What is the area of the shaded region?
______ square metres
To find the area of the shaded region, subtract the area of the inner shape from the area of the outer shape. Start by finding the area of the inner shape.
Find the side length of the inner square.
side: 6 m
Use this number in the formula.
inner area = side . side
= 6 . 6
= 36
Now find the units. The lengths are measured in metres, so the area is measured in square metres.
The area of the inner square is 36 square metres.
Next, find the area of the outer shape.
Find the side length of the outer square.
side: 17 m
Use this number in the formula.
outer area=side . side
= 17 . 17
= 289
As with the inner square, the lengths of the outer square are measured in metres. So, the area is measured in square metres.
The area of the outer square is 289 square metres.
Finally, subtract the inner area from the outer area to find the area of the shaded region.
shaded area = 289 – 36
= 253
The area of the shaded region is 253 square metres.
Let’s practice!🖊️
