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Altitude of a Parallelogram

Altitude of a Parallelogram
Height of a Parallelogram

The distance between opposite sides of a parallelogram. Formally, the shortest line segment between opposite sides. Altitude also refers to the length of this segment.

 

Parallelogram with a vertical dashed line labeled "altitude h" drawn perpendicular between its two horizontal parallel sides.

Key Formula

A=b×hA = b \times h
Where:
  • AA = Area of the parallelogram
  • bb = Length of the base (the side the altitude is drawn to)
  • hh = Altitude (perpendicular height) corresponding to that base

Worked Example

Problem: A parallelogram has a base of 10 cm and an altitude of 6 cm drawn to that base. Find the area of the parallelogram.
Step 1: Identify the base and the corresponding altitude. The base is the side the altitude is perpendicular to.
b=10 cm,h=6 cmb = 10 \text{ cm}, \quad h = 6 \text{ cm}
Step 2: Apply the area formula for a parallelogram.
A=b×h=10×6A = b \times h = 10 \times 6
Step 3: Calculate the result.
A=60 cm2A = 60 \text{ cm}^2
Answer: The area of the parallelogram is 60 cm².

Another Example

Problem: A parallelogram has sides of length 12 cm and 8 cm. The altitude drawn to the 12 cm side is 5 cm. Find the altitude drawn to the 8 cm side.
Step 1: Find the area using the 12 cm base and its corresponding altitude of 5 cm.
A=12×5=60 cm2A = 12 \times 5 = 60 \text{ cm}^2
Step 2: The area stays the same regardless of which side you treat as the base. Set up the equation using the 8 cm side as the base.
60=8×h260 = 8 \times h_2
Step 3: Solve for the unknown altitude.
h2=608=7.5 cmh_2 = \frac{60}{8} = 7.5 \text{ cm}
Answer: The altitude drawn to the 8 cm side is 7.5 cm.

Frequently Asked Questions

Is the altitude of a parallelogram the same as its side?
No. The altitude is the perpendicular distance between two opposite sides, not the length of a slanted side. Only in a rectangle — where the sides meet at right angles — does a side double as the altitude. In a general parallelogram, the altitude is shorter than the slanted side.
Can a parallelogram have two different altitudes?
Yes. A parallelogram has two pairs of opposite sides, so there are two distinct altitudes — one for each pair. Each altitude pairs with its corresponding base to give the same area. If the two pairs of sides have different lengths, the two altitudes will have different lengths.

Altitude of a Parallelogram vs. Side (slant height) of a Parallelogram

The altitude is measured perpendicular to a base and represents the true height. A side is measured along the boundary of the shape and may be tilted at an angle. In a non-rectangular parallelogram, the altitude is always shorter than the slanted side connecting the two parallel sides. Students sometimes confuse the two when computing area, but only the altitude — not the slant side — belongs in the area formula.

Why It Matters

The altitude is essential for computing the area of any parallelogram. Because the area formula A=b×hA = b \times h requires the perpendicular height, not a slanted side, understanding the altitude is critical in geometry problems involving area. This concept also extends to triangles, trapezoids, and other shapes where height must be measured perpendicularly.

Common Mistakes

Mistake: Using the slanted side length instead of the perpendicular altitude in the area formula.
Correction: The altitude must be perpendicular to the base. If the parallelogram is not a rectangle, the slanted side is longer than the altitude and will give an incorrect (too large) area.
Mistake: Pairing the altitude with the wrong base.
Correction: Each altitude corresponds to a specific pair of parallel sides. If you measure the altitude to the longer side, you must multiply by the length of that longer side — not the shorter side.

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