Height of a Parallelogram Formula - Steps and Examples

Key Points

• The height of a parallelogram formula is $h = \frac{A}{b}$, rearranged from the area formula $A = b \times h$.

• The height is the perpendicular distance between the base and the side opposite it, never the length of the slanted side.

• Every parallelogram has two base-and-height pairs, and a base must always be paired with the height measured perpendicular to it.

• When the slanted side $a$ and the angle $\theta$ between the sides are known, the height is $h = a \sin \theta$.

• The height is shorter than the slanted side in any parallelogram that is not a rectangle, because it is a perpendicular rather than a diagonal path.

What Is the Height of a Parallelogram Formula?

The height of a parallelogram formula comes straight from rearranging its area:

$$A = b \times h \implies h = \frac{A}{b}$$

Here $b$ is the base and $h$ is the perpendicular height to that base. The height is a distance measured at right angles, not along an edge.

Why the Slanted Side Is Not the Height

A parallelogram can be cut along a perpendicular and rearranged into a rectangle of the same base and the same perpendicular height. That rearrangement is why the area is base times height, and it uses the perpendicular distance, not the edge.

Picture pushing the top of a rectangle sideways. The slanted sides get longer while the vertical distance between top and bottom stays the same, and so does the area.

So the slanted side can grow without limit while the height and the area are unchanged. Using the edge in place of the height therefore overstates the area, and the error grows as the parallelogram leans further over.

Variable Key

Examples of the Height of a Parallelogram Formula

Example 1

A parallelogram has area 48 square cm and base 8 cm. Find its height.

$$h = \frac{48}{8} = 6$$

Final answer: 6 cm

Example 2

A parallelogram has base 10 cm, slanted side 7 cm and area 50 square cm. Find the height by using the slanted side.

The side of 7 cm is the only other length given, so treating it as the height looks reasonable, and it would predict an area of $10 \times 7 = 70$ square cm.

But the area is stated as 50, not 70. The slanted side runs diagonally between the parallel sides, so it is longer than the perpendicular gap between them.

Dividing the actual area by the base gives the true height:

$$h = \frac{50}{10} = 5$$

Final answer: 5 cm, which is shorter than the 7 cm side, as it must be.

Example 3

A parallelogram has sides 9 cm and 6 cm with an included angle of 30 degrees. Find the height to the 9 cm base.

Use the trigonometric form, taking the side that is not the base:

$$h = 6 \sin 30^\circ = 6 \times 0.5 = 3$$

Final answer: 3 cm, giving an area of $9 \times 3 = 27$ square cm.

Example 4

A parallelogram has area 72 square m. Find the height for a base of 12 m, then for a base of 9 m.

$$h_1 = \frac{72}{12} = 6, \qquad h_2 = \frac{72}{9} = 8$$

Final answer: 6 m and 8 m

Both are correct heights of the same parallelogram, each paired with its own base. A height means nothing without saying which base it belongs to.

Example 5

A parallelogram-shaped panel has area 2.4 square m and base 1.6 m. Find the height in centimetres.

$$h = \frac{2.4}{1.6} = 1.5 \text{ m} = 150 \text{ cm}$$

Final answer: 150 cm

Dividing square metres by metres leaves metres, which is the check that the answer is a length rather than an area.

Don't Confuse the Height With These Three Lengths

The slanted side and the height coincide in exactly one case, a rectangle, where the sides already meet at right angles and $\sin 90^\circ = 1$.

Where Do Students Trip Up on the Height of a Parallelogram?

Mistake 1: Using the slanted side as the height

Where it slips in: A diagram labels two side lengths, and the second one looks like the height.

Don't do this: $A = b \times a$ with $a$ the sloping edge.

The correct way: $A = b \times h$ with $h$ perpendicular. If only sides and an angle are given, compute $h = a \sin \theta$ first.

Students who look for the right-angle marker in the diagram before choosing a value get this right consistently, because the marker is what identifies a perpendicular. When no marker is drawn, no length in the figure is the height yet.

Mistake 2: Pairing a base with the wrong height

Where it slips in: A parallelogram has two bases and two heights, and a question may supply both.

Don't do this: Multiply the 12 m base by the height belonging to the 9 m base.

The correct way: Keep each base with the height measured perpendicular to it, as in Example 4. Mixing the pairs gives an area the shape does not have.

Mistake 3: Reporting the height in square units

Where it slips in: The area is in square units and the division feels like it keeps them.

Don't do this: Report a height of 6 square cm.

The correct way: Square centimetres divided by centimetres leaves centimetres. A height is a length, and checking the units is the fastest way to confirm the division went the right way.

Bottom Line

• The height of a parallelogram formula is $h = \frac{A}{b}$, rearranged from $A = b \times h$.

• The height is always perpendicular, so the slanted side cannot stand in for it.

• From a side and an included angle, the height is $h = a \sin \theta$.

• Each base pairs with its own height, and mixing the pairs produces an area the shape does not have.

• To work through mensuration with a teacher, explore a geometry tutor or math tuition.

Your Next Move

Find the height of a parallelogram with area 90 square cm and base 15 cm. Then find the height to a 12 cm base when the other side is 8 cm and the included angle is 60 degrees.

Want a live Bhanzu trainer to walk through more mensuration problems? Book a free demo class.

Read More

• Angles of a Parallelogram

• Adjacent Angles of a Parallelogram

• Altitude of a Triangle

• Basic Area Concepts

• Area vs Perimeter