A shape with no formula of its own
A floor is 7 m long and 5 m wide, but a corner 3 m by 2 m is missing. The floor is an L-shape, and there is no formula for the area of an L-shape.
There are formulas for rectangles, triangles, parallelograms and trapeziums. So change the question into one about shapes you can already find the area of. There are two ways to do that: subtract the missing piece from a bigger shape, or split the shape into smaller ones.
The floor is 7 by 5 with a corner 3 by 2 marked for removal.
Subtract the missing corner
Fill in the missing corner, and the floor would be a whole rectangle 7 m by 5 m, with area 7 × 5 = 35 square meters. The corner is a rectangle 3 m by 2 m, with area 3 × 2 = 6 square meters.
The floor is the whole rectangle with the corner taken away: 35 − 6 = 29 square meters.
With the 6 squares of the corner gone, 35 − 6 = 29 squares are left. Count them to check.
Or split it into rectangles
Draw a line straight across the floor where the corner begins. Below the line is a strip the full 7 m long. Its height is what is left of the 5 m once the corner's 2 m is taken off: 5 − 2 = 3 m. Its area is 7 × 3 = 21 square meters.
Above the line is a block 2 m high. Its length is what is left of the 7 m once the corner's 3 m is taken off: 7 − 3 = 4 m. Its area is 4 × 2 = 8 square meters. Together the two rectangles make 21 + 8 = 29 square meters, the same answer as before.
The line across splits the floor into a block 4 by 2 on top and a strip 7 by 3 below: 8 + 21 = 29.
Any cut gives the same area
The cut can go the other way too. Cut the floor upright where the corner begins, and it splits into a piece 4 m wide and 5 m high, 4 × 5 = 20 square meters, and a piece 3 m wide and 3 m high, 3 × 3 = 9 square meters. 20 + 9 = 29 square meters again. However the shape is cut, the pieces cover the same floor, so their areas add up to the same total.
Pick whichever way needs the fewest sums. Working it out a second way, with a different cut or by subtracting, is a good check.
two rectangles with A₁ + A₂ = 38: the cut can go either way and the total does not change, because area is conserved under decomposition
Make the notch 3 by 2 and box it
This L is 8 by 5. Drag the notch to 3 by 2, then try each button: cut upright, 25 + 9 = 34; cut across, 24 + 10 = 34; box it, 40 − 6 = 34. Every way gives 34.
Find the missing sides first
Each piece needs both of its sides. Often one of them is not written on the drawing, and you find it by subtracting from a side you do know, as 7 − 3 = 4 and 5 − 2 = 3 did above.
Two slips are common. One is to answer 35, the area of the whole rectangle, forgetting to take the corner away. The other is to add up the sides, 7 + 5 + 7 + 5 = 24, which is the distance around the edge, the perimeter, and not the area.
Worked example: Three Squares in a Row
Question Three identical squares of side 10 cm are placed in a row with their bases on one line. Each square overlaps the next by a strip 3 cm wide. Find the total length of the row and the total area covered.
1.Length: the first square is 10 cm; each of the next two adds 10 − 3 = 7 cm.
Three squares of 10, each sharing a 3 cm strip with the next. 2.Total length = 10 + 7 + 7 = 24 cm.
Length: 10 + 7 + 7 = 24 cm. 3.Each overlap strip is 3 × 10 = 30 cm2, and there are two of them.
Two strips of 30 cm² are counted twice. 4.The first and third squares are 14 cm apart at their left edges, more than a side, so they do not overlap each other.
The first and third squares start 14 cm apart, so they never meet. 5.Area = 3 × 100 − 2 × 30 = 240 cm2; or 24 × 10 = 240 cm2.
Area: 300 − 60 = 240 cm², which is 24 × 10.
Answer: 24 cm; 240 cm2
Common mistakes
- Subtracting three overlaps for three squares; there are only two places where neighbors meet.
- Giving the length as 30 − 3 = 27 cm, taking off one strip instead of two.