Calculate the area of a parallelogram whose diagonals are of length 8cm and 12cm and intersect at an angle of 135°

Analyze the question We need the area of a parallelogram given its two diagonals (8 cm and 12 cm) and the angle between them (135°).

Concept The diagonals of a parallelogram bisect each other. This splits the parallelogram into 4 triangles, and each pair of opposite triangles is congruent.

For a parallelogram with diagonals d₁ and d₂ meeting at angle θ:

Area = ½ × d₁ × d₂ × sin θ

Derivation

• Half-diagonals are 4 cm and 6 cm.
• Each triangle has area ½ × 4 × 6 × sin 135° = 12 sin 135°.
• There are 4 triangles (the angles are 135° and 45°, and sin 135° = sin 45°), so:

Total area = 4 × 12 × sin 135° = 48 sin 135°, which matches ½ × 8 × 12 × sin 135° = 48 sin 135°.

Solution sin 135° = sin 45° = 0.7071

Area = 48 × 0.7071 = 33.94 cm² ≈ 33.9 cm²

Answer This matches option E (33.

Tip Don't forget the sine. If you choose 48 cm² (option D), you've just done ½ × d₁ × d₂ without multiplying by sin θ. Also remember sin(180° − θ) = sin θ, so 135° gives the same sine as 45°.