The area of a rectangle can be found when we know the lengths of its sides. Is the same true for a parallelogram? That is, can we find the area of a parallelogram when we know the lengths of its sides? Why or why not? (Hint: What happens to the area of a parallelogram if we decrease or increase the angle between the adjacent sides while keeping the lengths fixed?)
Answer: No. With sides fixed, changing the angle between them changes the height, and so the area. Sides 5 and 3 can give area 15 (a rectangle), 7.5 (angle 30°) or almost 0 (very flat).
Step-by-step solution
Idea: Area = base × height. The height depends on how much the slant side leans, which the side lengths alone do not tell us.
- Take a parallelogram with sides 5 cm and 3 cm, hinged at the corners (like four rods). With base 5, the height is how far the top side is above the base, and that depends on the angle between the sides.1 mark
- Angle 90°: height 3, area 15 cm2. Angle 30°: height = half of 3 = 1.5, area 7.5 cm2 (the 30°-60°-90° triangle has the short side half the hypotenuse). As the angle shrinks towards 0°, the area shrinks towards 0. Same sides, different areas, so the sides alone do not fix the area. A rectangle is the special case where the angle is fixed at 90°.1 mark
No. A parallelogram with given sides can lean by different amounts; its height and hence its area change with the angle. We need one more piece of information, such as the height or an angle.
Answer to write in the exam
Area = base × height
Sides 5, 3: angle 90° ⇒ area = 5 × 3 = 15
Angle 30° ⇒ height = 1.5 ⇒ area = 7.5
Same sides, different areas
∴ No; side lengths alone do not determine the area.
Common mistakes that cost marks
- Multiplying the two side lengths for any parallelogram. That only works for a rectangle.
- Thinking a parallelogram is rigid like a triangle. With sides fixed it can still change shape.
How this can come in the exam
Assertion–Reason (1 mark)
Assertion (A): Two parallelograms with sides 6 cm and 4 cm must have equal areas.
Reason (R): Area of a parallelogram = base × height.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
A is false: the height depends on the angle between the sides. R is true.
Try one yourself
A rhombus has sides of 4 cm. Find its area when it is a square, and when one angle is 30°.
Show answer
Square: 4 × 4 = 16 cm2. Angle 30°: height = 2 cm, area = 4 × 2 = 8 cm2.
More questions like this
- You may wonder, like earlier, is there a gap in our argument? What would we do if angle EFG is obtuse and the triangle were shaped like triangle EFG in the figure? Please work out the answer to this question.
- Do you see why the two triangles fit together to make a parallelogram? (If you study the angles in the figure (e.g., ∠B’C’A’ and ∠BCA), you will see why this is so. Keep in mind the criterion by which we check whether two lines are parallel.)
- Since ΔABD and ΔACD have equal area, you may wonder — Can we divide ΔABD using straight cuts into two or more pieces that we can then rearrange to exactly cover ΔACD? What do you think? Is it possible?
- Suppose we are given two polygons P and Q with equal area. Will it always be possible to divide one of them using straight cuts into two or more pieces and then rearrange the pieces to exactly cover the other polygon? Try this out for familiar shapes, e.g.,
- Think of various rectangles with perimeter 40 units (the sides do not have to be integers).