Area and Perimeter of Plane Figures

268. Area Unlocked · base × height and diagonals × half

Area formulas hold true as you drag handles to reshape figures.

ABCDbWLOd₁d₂b = 300b = 300h = 200h = 200d₁ = 200d₁ = 200d₂ = 200d₂ = 200▱ area = b × h = 300 × 200 = 60003▱ area = b × h = 300 × 200 = 60003◇ area = ½ × d₁ × d₂ = ½ × 200 × 200 = 20000◇ area = ½ × d₁ × d₂ = ½ × 200 × 200 = 20000hshearVR
The area of a parallelogram is base × height; the area of a rhombus is ½ × (product of its diagonals). Each formula depends only on the figure's key measurements, not on how slanted it looks.

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Selina ICSE: Area and Perimeter of Plane Figures

What this lesson covers

Try to break it

Drag h to change the parallelogram's perpendicular height — the area scales with h. Drag the shear handle to slide the top side without changing h; the area doesn't budge. On the rhombus side, drag V and R to change the two diagonals — the rhombus area always equals ½·d₁·d₂. Two shapes, two formulas, both unbreakable.

How you build it

Two figures, two area rules.

  • Point tool: drop corner A near the lower-left.
  • Point tool: drop corner B to the right of A — AB is the base.
  • Segment tool: join A to B — the base of the parallelogram.
  • Point tool: drop D up and to the side of A — this fixes the height and the slant.
  • Segment tool: join A to D — the slant side.
  • Parallel tool: click B, then click side AD — a line through B parallel to AD.
  • Parallel tool: click D, then click base AB — a line through D parallel to AB. It meets the previous line at C.
  • Point tool: mark C where the two parallels cross — parallelogram ABCD is done (Area = base × height). BC and DC lie on the parallels you drew.
  • Point tool: drop P on the right half — one end of the rhombus's first diagonal.
  • Point tool: drop R across from P — the other end of the first diagonal.
  • Segment tool: join P to R — the first diagonal.
  • Midpoint tool: click P then R — M drops at the exact centre, where the diagonals will cross.
  • Bisector tool: click P, then R — the perpendicular bisector through M is the line of the second diagonal.
  • Arc tool: centre on M, open to any radius — the arc cuts the bisector at two points equally far from M.
  • Point tool: mark Q where the arc meets the bisector on one side.
  • Point tool: mark S where the arc meets the bisector on the other side — MQ = MS, so the diagonals bisect each other.
  • Segment tool: join P to Q.
  • Segment tool: join Q to R.
  • Segment tool: join R to S.
  • Segment tool: join S to P. PQRS is a rhombus; its area = ½ × diagonal PR × diagonal QS.

The proof, step by step

Prove that the area of a parallelogram is base × height and of a rhombus is half the product of its diagonals.

  • Cut a right-angled triangle from one end of the parallelogram along the height line. Translate it to the opposite side. The parallelogram becomes a rectangle with the same base and height, proving Area = base × height.
  • The diagonals of a rhombus bisect each other at right angles. This splits the rhombus into four congruent right triangles. Summing their areas gives 4 × (½ × d₁/2 × d₂/2) = ½ × d₁ × d₂.

Worked example

In a parallelogram, the base measures 10 cm and the corresponding height is 8 cm. A rhombus has diagonals measuring 12 cm and 20 cm. What is the ratio of the area of the parallelogram to the area of the rhombus?

Parallelogram area = 10 × 8 = 80 cm². Rhombus area = ½ × 12 × 20 = 120 cm². Ratio = 80 : 120 = 2 : 3.

  • 1 : 1
  • 2 : 3 — correct
  • 3 : 2
  • 4 : 5
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