Area of a Trapezium and a Polygon

192. Square Secrets · Perimeter, Area, and Diagonal formulas

The perimeter is 4a, area is a², and diagonal is a√2 for any square.

a = 150a = 150d = 212.1d = 212.1Perimeter = 4a = 4(150)Perimeter = 4a = 4(150)Area = a² = 22500Area = a² = 22500A
Square formulas: Perimeter = 4a, Area = a², Diagonal = a√2. The diagonal follows from Pythagoras applied to the half-square right triangle: d² = a² + a² = 2a², so d = a√2.

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Selina ICSE: Area of a Trapezium and a Polygon

What this lesson covers

Try to break it

Drag S outward or inward to change the side a. All three readouts update at once — perimeter = 4a, area = a², diagonal = a√2. Try to find a side length where one of them disagrees with its formula — impossible.

How you build it

Construct a square with its diagonal.

  • Mark point A — one end of the diagonal. Place it in the upper-left area of the canvas.
  • Mark point C — the opposite end of the diagonal. Place it in the lower-right area. AC will be the full diagonal d = a√2.
  • Draw segment AC — this is the diagonal of the square. Every square has two equal diagonals; we start by drawing one.
  • Construct the perpendicular bisector of AC — it passes through midpoint M at 90°. The other two vertices of the square must lie on this line.
  • Mark M — click where the perpendicular bisector crosses AC. This is the midpoint: MA = MC = d/2. In a square both diagonals bisect each other here.
  • With centre M and radius MA, swing an arc. It crosses the perpendicular bisector at two points — those are B and D. MB = MD = MA = MC = half the diagonal.
  • Mark B — click where the arc meets the perpendicular bisector on one side of AC. This is the third vertex of the square.
  • Mark D — click where the arc meets the bisector on the opposite side of AC. You now have all four vertices A, B, C, D of the square.
  • Draw side AB.
  • Draw side BC.
  • Draw side CD.
  • Draw side DA to complete the square. You constructed it entirely from the diagonal — every side a = d / √2, and d = a√2!

The proof, step by step

Prove that a square has perimeter 4a, area a², and diagonal a√2.

  • Let the side of the square be 'a'.
  • The diagonal divides the square into two right-angled triangles.
  • By Pythagoras theorem, d² = a² + a².
  • Therefore, d = √(2a²) = a√2.

Worked example

A square garden has a side length of 12 meters. What is the length of the diagonal path across the garden?

The diagonal of a square is side × √2. So, 12 × √2 = 12√2 m.

  • 12√2 m — correct
  • 24 m
  • 144 m
  • 6√2 m
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