Same sides, different area
Sides alone do not fix a 4-gon's area
Four sides, one area?
The three sides of a triangle fix its area (Heron's formula). Now take a 4-gon (a quadrilateral). Can we find its area from the lengths of its four sides?
A 4-gon has four sides 3, 3, 3, 3. Is its area always the same?
What this lesson covers
The idea
The area of a quadrilateral cannot be found from the lengths of its four sides alone; more information is needed, such as an angle, a diagonal, or a property like being cyclic.
Four sides, one area?
The three sides of a triangle fix its area (Heron's formula). Now take a 4-gon (a quadrilateral). Can we find its area from the lengths of its four sides?
A 4-gon has four sides 3, 3, 3, 3. Is its area always the same?
- Yes, always 9
- Yes, always 12
- No, it can change
Tilt the shape
This shape has hinges at the corners, so the angle can change but the sides cannot. Tilt it. Make three clearly different areas.
More information is needed
The area of a quadrilateral cannot be found from the lengths of its four sides alone. More information is needed, such as an angle, a diagonal, or a property like being cyclic.
The same four sides 3, 3, 3, 3 make a square of area 9. Tilt it, and the area gets smaller, though the perimeter stays 12. The same happens for a parallelogram: its sides do not fix its area. One extra piece of information, like an angle or a diagonal, does.
Notes
The area of a 4-gon cannot be found from its four sides alone. We need more information, such as an angle, a diagonal, or a property like being cyclic.
Check yourself
A 4-gon has sides 3, 3, 3, 3. What can we say about its area?
A rhombus with sides 3 is tilted. What is its perimeter now?
Answer: 12
The perimeter is 4 × 3 = 12, whatever the angle.
A tilted rhombus with sides 3 has height 2. What is its area?
Answer: 6
Area = 3 × 2 = 6, smaller than the square's 9.
A 4-gon's four sides are known. Which extra information would help fix its area?
- It is always 9. 9 is the area of the square. A tilted rhombus with the same sides has a smaller area.
- It is always 12. 12 is the perimeter, not the area. The area changes when the shape is tilted.
- It is not fixed: tilting the shape changes it — correct. Yes. The sides stay the same, but the area changes.
- The length of one diagonal — correct. Yes. The diagonal splits the 4-gon into two triangles, and each triangle's sides are then known.
- The colour of the shape. Colour tells us nothing about the size of the area.
- The same four sides once again. That is what we already know, and it is not enough.