how to find the area of a parallelogram

To find the area of a parallelogram, you mostly use the same idea as for a rectangle: base times height.

How to Find the Area of a Parallelogram

1. Basic formula (most common)

If you know the base bbb and the height hhh (the perpendicular distance from the base to the opposite side), then:

Area=b×h\text{Area}=b\times hArea=b×h

The height is always drawn at a right angle (90°) to the base, even if the sides are slanted.

Example: If b=8b=8b=8 cm and h=5h=5h=5 cm, then area =8×5=40=8\times 5=40=8×5=40 cm².

You can imagine “cutting off” a triangle from one side of the parallelogram and sliding it to the other side to form a rectangle; the base and height stay the same, so the area stays b×hb\times hb×h.

2. When you know two sides and the angle

Sometimes you don’t have the height but you know:

Two adjacent sides aaa and bbb
The angle θθθ between them

Then you can use:

Area=a×b×sin⁡(θ)\text{Area}=a\times b\times \sin(θ)Area=a×b×sin(θ)

This works because b×sin⁡(θ)b\times \sin(θ)b×sin(θ) is effectively the height relative to side aaa.

Example: If a=6a=6a=6 cm, b=4b=4b=4 cm, and θ=30°θ=30°θ=30°:
Area=6×4×sin⁡(30°)=24×0.5=12\text{Area}=6\times 4\times \sin(30°)=24\times 0.5=12Area=6×4×sin(30°)=24×0.5=12 cm².

If the angle is 90°, the parallelogram is actually a rectangle, and this formula reduces to a×ba\times ba×b.

3. When you know the diagonals and the angle between them

If you are given:

Diagonal lengths d1d_1d1 and d2d_2d2
The angle φφφ between the diagonals

Then:

Area=12×d1×d2×sin⁡(φ)\text{Area}=\tfrac{1}{2}\times d_1\times d_2\times \sin(φ)Area=21×d1×d2×sin(φ)

This is another trigonometric method that uses the diagonals instead of sides.

Example: If d1=10d_1=10d1=10 cm, d2=8d_2=8d2=8 cm, and φ=60°φ=60°φ=60°:
Area=12×10×8×sin⁡(60°)\text{Area}=\tfrac{1}{2}\times 10\times 8\times \sin(60°)Area=21×10×8×sin(60°).

4. Quick step-by-step guide

Identify what you know
- Base and perpendicular height → use A=b×hA=b\times hA=b×h.
- Two sides and included angle → use A=a×b×sin⁡(θ)A=a\times b\times \sin(θ)A=a×b×sin(θ).
- Two diagonals and the angle between them → use A=12d1d2sin⁡(φ)A=\tfrac{1}{2}d_1d_2\sin(φ)A=21d1d2sin(φ).

Plug values into the right formula
- Be sure angles are in the correct mode if you are using a calculator (degrees vs radians).

Attach the correct units
- If lengths are in cm, area is in cm²; if in m, area in m², etc.

5. Mini HTML table of key formulas

html

<table>
  <thead>
    <tr>
      <th>Given</th>
      <th>Formula for Area</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Base b and height h</td>
      <td>A = b × h</td>
    </tr>
    <tr>
      <td>Sides a, b and angle θ between them</td>
      <td>A = a × b × sin(θ)</td>
    </tr>
    <tr>
      <td>Diagonals d₁, d₂ and angle φ between them</td>
      <td>A = ½ × d₁ × d₂ × sin(φ)</td>
    </tr>
  </tbody>
</table>

TL;DR: For most school problems, draw the height perpendicular to the base and use A=b×hA=b\times hA=b×h; switch to the sine formulas only when you’re given angles instead of height.

Information gathered from public forums or data available on the internet and portrayed here.

how to find the area of a parallelogram