The area of a parallelogram is 600 sq m if its height is 15 m what is the length of its base

As the name suggests, a parallelogram is a quadrilateral formed by two pairs of parallel lines. It differs from a rectangle in terms of the measure of angles at the corners. In a parallelogram, the opposite sides are equal in length, and opposite angles are equal in measure, while in a rectangle, all angles are 90 degrees.

In this article, you will learn how to calculate the area of a parallelogram using the parallelogram area formula.

To find how its area is different from other quadrilaterals and polygons, visit the previous articles.

How to Find the Area of a Parallelogram?

The area of a parallelogram is the space enclosed by 2 pairs of parallel lines.  A rectangle and a parallelogram have similar properties, and therefore, the area of a parallelogram is equal to the area of a rectangle.

Area of a Parallelogram Formula

Consider a parallelogram ABCD shown below. The area of the parallelogram is the space bounded by the sides AD, DC, CB, and AB.

The area of parallelogram formula states;

Area of a parallelogram = base x height

A = (b * h) Sq. units

Where b = the base of a parallelogram and,

h = The altitude or the height of a parallelogram.

The height or altitude is the perpendicular line (usually dotted) from the vertex of a parallelogram to any of the bases.

Example 1

Calculate the area of a parallelogram whose base is 10 centimeters and height is 8 centimeters.

Solution

A = (b * h) Sq. units.

A = (10 * 8)

A = 80 cm2

Example 2

Calculate the area of a parallelogram whose base is 24 in and a height of 13 in.

Solution

A = (b * h) Sq. units.

= (24 * 13) square inch.

= 312 square inches.

Example 3

If the base of a parallelogram is 4 times the height and the area is 676 cm², find the parallelogram’s base and height.

Solution

Let the height of the parallelogram = x

and the base = 4x

But, the area of a parallelogram = b * h

676 cm² = (4x * x) Sq. units

676 = 4x2

Divide both sides by 4 to get,

169 = x2

By finding the square root of both sides, we get,

x = 13.

Substitute.

Base = 4 * 13 = 52 cm

Height = 13 cm.

Therefore, the base and height of the parallelogram are 52 cm and 13 cm, respectively.

Apart from the area of the parallelogram formula, there are other formulas for calculating the area of a parallelogram.

Let’s take a look.

How to find the area of a parallelogram without a height?

If the parallelogram’s height is unknown to us, we can use the trigonometry concept to find its area.

Area = ab sine (α) =ab sine (β)

Where a and b are the length of parallel sides, and either β or α is the angle between the sides of the parallelogram.

Example 4

Find the area of a parallelogram if its two parallel sides are 80 cm and 40 cm and the angle between them is 56 degrees.

Solution

Let a = 80 cm and b = 40 cm.

The angle between a and b = 56 degrees.

Area = ab sine (α)

Substitute.

A = 80 × 40 sine (56)

A = 3,200 sine 56

A = 2,652.9 sq.cm.

Example 5

Calculate the angles between the two sides of a parallelogram if its side lengths are 5 m and 9 m and the area of the parallelogram is 42.8 m2.

Solution

Area of a parallelogram = ab sine (α)

42.8 m2 = 9 * 5 sine (α)

42.8 = 45 sine (α)

Divide both sides by 45.

0.95111= sin (α)

α = sine-1 0.95111

α = 72°

But β + α = 180°

β = 180° – 72°

= 108°

Therefore, the angles between the two parallel sides of the parallelogram are; 108° and 72°.

Example 6

Calculate the height of a parallelogram whose parallel sides are 30 cm and 40 cm, and the angle between these two sides is 36 degrees. Take the base of the parallelogram to be 40 cm.

Solution

Area = ab sine (α) = bh

30 * 40 sine (36) = 40 * h

1,200 sine (36) = 40 * h.

Divide both sides by 40.

h = (1200/40) sine 36

= 30 sine 36

h = 17.63 cm

So, the height of the parallelogram is 17.63 cm.

How to find the area of a parallelogram using diagonals?

Suppose d1 and d2 are the diagonals of the parallelogram ABCD, then the area of the parallelogram is given as,

A = ½ × d1 × d2 sine (β) = ½ × d1 × d2 sine (α)

Where β or α is the angle of intersection of the diagonals d1 and d2.

Example 7

Calculate the area of a parallelogram whose diagonals are 18 cm and 15 cm, and the angle of intersection between the diagonals is 43°.

Solution

Let d1 = 18 cm and d2 = 15 cm.

β = 43°.

A = ½ × d1 × d2 sine (β)

= ½ × 18 × 15 sine (43°)

= 135sine 43°

= 92.07 cm2

Therefore, the area of the parallelogram is 92.07 cm2.

If you have any problems with the geometry of a parallelogram, check this parallelogram area calculator (and also its twin brother, parallelogram perimeter calculator).

Whether you want to calculate the area given base and height, sides and angle, or diagonals of a parallelogram and angle between them, you are in the right place. Don't ask how to find the area of a parallelogram; just give the calculator a try!

Below you can find out how the tool works – the parallelogram area formulas and neat explanation are all you need to understand the topic.

A parallelogram is a simple quadrilateral with two pairs of parallel sides. Every rectangle is a parallelogram, as well as every rhombus and square. Remember, it doesn't work the other way around!

Which formulas does the parallelogram area calculator use?

• Area given base and height

  area = base × height

  Did you notice something? The formula for the area of a parallelogram is pretty much the same as for a rectangle! Why is it so? Have a look at the picture: a parallelogram can be divided into a trapezoid and a right triangle and rearranged to the rectangle.

  Learn more about rectangle area with our area of a rectangle calculator.

• Area given sides and the angle between them

  area = a × b × sin(angle)

Does it ring a bell? This formula comes from trigonometry and is used, for example, in our triangle area calculator – the parallelogram may be seen as two congruent triangles. The adjacent angles in the parallelogram are supplementary, so you can choose whichever angle you want because sin(angle) = sin(180° - angle).

• Area given diagonals of a parallelogram and the angle between them

  area = ½ × e × f × sin(angle)

  The formula comes from trigonometry as well. Do you want to know where it comes from?

  Divide the parallelogram into two triangles, and assume that our e diagonal is the "base" for both new triangles.

  What's the height of that triangle? Use the sine function. It's (f/2) × sin(angle)!

  The area of the triangle is equal to our "base" e times height: e × (f/2) × sin(angle)

  The parallelogram consists of two such triangles, so the area equals e × f × sin(angle).

Are you still not sure our parallelogram area calculator works? We will show you step by step:

1. Have a look at your exercise. What is given, what is unknown? Choose the right calculator part for your needs. Assume that we want to calculate the area knowing the diagonals of a parallelogram and the angle between diagonals.

2. Enter the given values to the right boxes. Assume 5 in, 13 in, and 30° for the first diagonal, the second one, and the angle between them, respectively.

3. The calculator displays the area of a parallelogram value. It's 32.5 in² in our case.

Check out our area calculators for other shapes, such as rhombus area calculator, circle area calculator, and trapezoid area calculator.

To determine the area given the adjacent sides of a parallelogram, you also need to know the angle between the sides. Then you can apply the formula: area = a × b × sin(α), where a and b are the sides, and α is the angle between them.

The area of a parallelogram can be determined from its diagonals, provided that you also know the angle between the diagonals.

If e and f are the lengths of the diagonals and φ is the angle between them, then the area can be calculated as follows: area = ½ × e × f × sin(φ).

It is possible to find the area of a parallelogram without height! For instance, it suffices to know one of the following things:

1. The length of adjacent sides and the angle between them – use trigonometry.
2. The length of diagonals and the angle between them, using the formula – use trigonometry.
3. The length of diagonals and one side – use Heron's formula.

The answer is 75. We use the formula that says the area is equal to ½ times the product of the lengths of the diagonals times the sine of the angle between them. As our diagonals are perpendicular, the angle between them is 90° and sin 90° = 1. Hence, the calculation we need to perform is ½ × 10 × 15 = 75.