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Parallelogram calculator

Two sides with angle, base + side + height, or base + side + area; get height, perimeter, area and both diagonals

Formula

S=ab\sin A=ah,\quad h=b\sin A

S = ab sin(theta), P = 2(a + b)

Enter adjacent sides a, b and angle theta; S = ab sin(theta).

ab∠A / °

Presets

Tap an edge, angle or region to see how it is computed

Sides and height

Angles

Perimeter, area, diagonals

Relationh3.064

h=b\sin A=4\times0.766=3.064

Area S18.385

Perimeter P20

Height h3.064

Diagonals4.598

a6b4h3.064∠A50∠B130P20S18.385BD4.598AC9.102

Steps

7 / 7

1Given
$a=6,\ b=4,\ \angle A=50^\circ$
Collect the given quantities first.
2Find height
$h=b\sin A=4\times\sin50^\circ=3.064$
Height = 2 x area / base, or from a trig ratio.
3Area
$S=ah=6\times3.064=18.385=ab\sin A$
Apply the area formula; the unit is squared.
4Perimeter
$P=2(a+b)=2\times(6+4)=20$
Add all sides to get the perimeter.
5Cosine rule
$BD=\sqrt{a^2+b^2-2ab\cos A}=4.598$
c^2 = a^2 + b^2 - 2ab cosC.
6Angle sum
$\angle B=180^\circ-50^\circ=130^\circ$
Triangle angles sum to 180 degrees; an n-gon sums to (n-2) x 180.
7Result
$a=6,\ b=4,\ h=3.064,\ P=20,\ S=18.385$
Collect all computed quantities.

Concept

A parallelogram has two pairs of parallel sides: opposite sides and angles are equal and the diagonals bisect each other.

Area = base x height, where the height is perpendicular to the chosen base, not the adjacent side.

Going further

Diagonal identity: The diagonals satisfy d1^2 + d2^2 = 2(a^2 + b^2), the parallelogram law.
When it becomes a rectangle: At 90 degrees the parallelogram is a rectangle with equal diagonals; at 0 degrees it collapses into a segment.
Vector view: If the sides are vectors u and v, the area is ∣u x v∣, which explains S = ab sin(theta).

Common mistakes

The height never exceeds the adjacent side b.
Use base x height, not the product of two adjacent sides.
When solving for the height, pick the base perpendicular to it.

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