Parallelogram Calculator
Area, perimeter, heights, diagonals and angles from any three measurements
The area of a parallelogram is its base times its perpendicular height, A = b × h, so a parallelogram with a 12 ft base and a 7 ft height covers 84 square feet. With two sides and the angle between them, use A = a × b × sin θ; with the diagonals, A = ½ × d₁ × d₂ × sin φ. This calculator handles all three cases and also returns the perimeter, both heights, both diagonals and the interior angles.
Last updated October 2026
Method: Exact Euclidean geometry. The area comes from b × h, a·b·sin θ or ½·d₁·d₂·sin φ; diagonals and sides come from the law of cosines; angles come from the arcsine and arccosine. Values are computed in double precision and rounded only for display.
Included: Area, perimeter, the height onto each side, the short and long diagonal, both interior angles, the angle where the diagonals cross, a rectangle, rhombus and square check, and a labeled sketch of the shape.
Not included: Unit conversion (enter every length in the same unit), irregular quadrilaterals whose opposite sides are not parallel, and solid shapes such as parallelepipeds.
▱ What do you know about the parallelogram?
Measured at a right angle to the base, not along the slanted side.
Add it to get the perimeter, both angles and both diagonals.
The label is cosmetic. Enter every length in the same unit; areas come out in square units.
Area of the parallelogram
Shape: Parallelogram
📐 Sketch with labeled sides
🧮 Full breakdown
📋 Step by step
- 1. Area. A = b × h = 12 × 7 = 84 ft².
- 2. Angle. θ = arcsin(h ÷ a) = arcsin(7 ÷ 9) = 51.0576°, so the other angle is 180° − θ = 128.9424°.
- 3. Perimeter. P = 2 × (a + b) = 2 × (9 + 12) = 42 ft.
- 4. Diagonals. d = √(a² + b² ∓ 2ab·cos θ) gives 9.4465 ft (short) and 18.9938 ft (long).
- 5. Second height. A ÷ a = 84 ÷ 9 = 9.3333 ft, the distance between the two slanted sides.
Parallelogram formulas, worked examples and properties
A parallelogram is a four-sided figure whose opposite sides are parallel and equal in length. It is a rectangle pushed sideways, so its area depends on how far apart the parallel sides are, not on how far it leans. The three worked examples below match the calculator's three modes and its Try buttons.
The parallelogram formulas
Call the base b, the slanted side a, the perpendicular height h, the acute interior angle θ, the two diagonals d₁ and d₂, and the angle where the diagonals cross φ. Then:
A = b × h A = a × b × sin θ A = ½ × d₁ × d₂ × sin φ P = 2 × (a + b) h = a × sin θ d₁ = √(a² + b² − 2ab × cos θ) d₂ = √(a² + b² + 2ab × cos θ) d₁² + d₂² = 2 × (a² + b²) The first line is the one most people need, and it has a simple proof: cut a right triangle off one end along the height, slide it to the other end, and the pieces form a rectangle with the same base and height. This rearranging argument is how the Common Core grade 6 standards introduce the area of special quadrilaterals, and it explains why sliding the top edge sideways changes the sides, angles and diagonals but never the area. The second line follows because the height is the slanted side times the sine of the angle. The diagonal formulas are the law of cosines, and the last line, the parallelogram law, ties the two diagonals to the four sides.
Worked example 1: a patio laid out as a parallelogram
A paver patio runs 12 ft along the house, its far edge is also 12 ft, and its two angled ends are 9 ft long. Measured straight out from the house, the patio is 7 ft deep. The area needs only the base and that perpendicular depth:
A = b × h = 12 × 7 = 84 ft² The 9 ft ends do not enter the area, but they unlock everything else. The lean angle is θ = arcsin(7 ÷ 9) = 51.06°, so the corners measure 51.06° and 128.94°. The perimeter, the length of edge restraint to buy, is 2 × (12 + 9) = 42 ft. The diagonals, which a crew can measure to check the layout, are 9.45 ft and 18.99 ft. The distance between the two 9 ft ends is 84 ÷ 9 = 9.33 ft, the second height of the same shape.
Worked example 2: two sides and the angle between them
A drafting exercise gives sides of 15 cm and 8 cm that meet at 60°. The height on the 15 cm base is 8 × sin 60° = 6.93 cm, so the area is:
A = 15 × 8 × sin 60° = 103.92 cm² The perimeter is 2 × (15 + 8) = 46 cm. This example has a tidy diagonal: 15² + 8² − 2 × 15 × 8 × cos 60° = 225 + 64 − 120 = 169, so the short diagonal is exactly 13 cm. The long diagonal is √(225 + 64 + 120) = √409 = 20.22 cm. Entering 120° instead of 60° returns the same parallelogram, because the two interior angles always add up to 180° and an angle and its supplement have the same sine.
Worked example 3: area from the diagonals
A rhombus-shaped tile has diagonals of 12 in and 16 in that cross at right angles. The diagonal formula gives:
A = ½ × 12 × 16 × sin 90° = 96 in² The diagonals of every parallelogram cut each other in half, so each side of this tile is the hypotenuse of a right triangle with legs of 6 in and 8 in. That makes every side exactly 10 in, the perimeter 40 in and the corner angles 73.74° and 106.26°. Diagonals of 14 and 20 crossing at 60° would instead give a general parallelogram of 121.24 square units with sides of 14.80 and 8.89.
How the angle changes the area: computed table
This table keeps the sides fixed at b = 10 and a = 6 and changes only the angle, using the calculator's formulas. The perimeter is 32 in every row, while the area climbs from about a quarter of the rectangle's area to all of it.
| Angle θ | sin θ | Area | Height on b | Short diagonal | Long diagonal | Diagonals cross at |
|---|---|---|---|---|---|---|
| 15° | 0.2588 | 15.53 | 1.553 | 4.482 | 15.872 | 25.89° |
| 30° | 0.5000 | 30.00 | 3.000 | 5.664 | 15.489 | 43.15° |
| 45° | 0.7071 | 42.43 | 4.243 | 7.152 | 14.861 | 52.97° |
| 60° | 0.8660 | 51.96 | 5.196 | 8.718 | 14.000 | 58.37° |
| 75° | 0.9659 | 57.96 | 5.796 | 10.244 | 12.925 | 61.09° |
| 90° | 1.0000 | 60.00 | 6.000 | 11.662 | 11.662 | 61.93° |
Three patterns stand out. At 30° the area is exactly half of the rectangle's 60 square units, because sin 30° = 0.5. The diagonals approach each other as the angle opens and become equal only at 90°. Even then, the diagonals of this 10 by 6 rectangle cross at 61.93°, not 90°: perpendicular diagonals are the mark of a rhombus, not a rectangle.
How to use the parallelogram calculator
Choose the mode that matches what you measured. Base × height is the quickest route to the area; add the slanted side to also get the angles, perimeter, diagonals and second height. Sides + angle takes two side lengths and either interior angle. Diagonals takes both diagonals and the angle where they cross. The unit label only names the unit, it converts nothing. The sketch follows your numbers, so a typo that produces a very flat or very steep shape stands out at once.
Who uses a parallelogram calculator
Students meet the parallelogram in middle school area units and again in geometry proofs, and the step-by-step card shows the intermediate values. In landscaping and construction, angled patios, parking stalls striped at an angle and offset deck boards all form parallelograms. Quilters cut parallelogram patches for chevron blocks. Physics students use the same geometry for vectors: forces of 15 N and 8 N acting at 60° to each other add up to the long diagonal, √409 ≈ 20.22 N, pointing 20.03° away from the larger force, and the short diagonal, 13 N, is the size of their difference.
Key terms
- Base (b): the side the height is measured from. Any side can serve as the base.
- Height (h): the perpendicular distance between the base and the opposite side, also called the altitude.
- Slanted side (a): the side next to the base. It is never shorter than the height on b.
- Interior angles: the acute angle θ and the obtuse angle 180° − θ. Opposite angles are equal.
- Diagonals: the two segments that join opposite corners. They always cut each other in half.
- Rectangle, rhombus, square: parallelograms with right angles, with four equal sides, or with both.
- Parallelogram law: d₁² + d₂² = 2(a² + b²), a link between the diagonals and the sides.
Rectangle, rhombus and square compared
The special parallelograms inherit every formula above and add properties of their own. Here "No" means "not in general".
| Property | Parallelogram | Rectangle | Rhombus | Square |
|---|---|---|---|---|
| Four right angles | No | Yes | No | Yes |
| Four equal sides | No | No | Yes | Yes |
| Equal diagonals | No | Yes | No | Yes |
| Perpendicular diagonals | No | No | Yes | Yes |
| Diagonals bisect each other | Yes | Yes | Yes | Yes |
| Shortest area formula | b × h | length × width | ½ × d₁ × d₂ | side² |
What changes the result
- The angle. With the sides fixed, the area follows sin θ, which changes fastest at small angles.
- Which height you use. Each height belongs to its own base. On the patio, 7 ft pairs with the 12 ft base and 9.33 ft with the 9 ft side; 12 × 9.33 would report 112 ft² instead of 84 ft².
- Height versus slanted side. Using the slanted side as the height overstates the area unless the corners are square.
- Units. Mixing inches and feet in one calculation gives answers off by a factor of 12 or 144.
Tips for measuring a real parallelogram
- Measure the height with a framing square or a tape held at a right angle to the base, never along the angled edge.
- Measure both diagonals as a check: d₁² + d₂² should equal 2(a² + b²). For the patio, 9.45² + 18.99² ≈ 450 = 2 × (12² + 9²).
- Equal diagonals mean square corners, so the shape is a rectangle.
- Round only the final answer, not the intermediate steps.
Limitations
The calculator assumes a true parallelogram, with both pairs of opposite sides parallel and equal. A four-sided plot whose sides all differ is a general quadrilateral; split it along a diagonal and solve the two triangles with the Triangle Calculator. Angles must lie strictly between 0° and 180°, and a slanted side shorter than the height is rejected because no such shape exists. Results are shown to four decimals, far finer than any tape measure.
Related calculators
- For a four-sided shape with only one pair of parallel sides, use the Trapezoid Calculator.
- For rectangles, circles, ellipses and other common shapes side by side, use the Area Calculator.
- Each diagonal splits a parallelogram into two congruent triangles, which the Triangle Calculator can solve on their own.
- For the distance around many other shapes, use the Perimeter Calculator.
- The diagonal formulas are a direct application of the Law of Cosines Calculator.
- For flooring, pavers or sod estimates in square feet, continue with the Square Footage Calculator.
Sources
Parallelogram formulas are exact geometry rather than published statistics, so every figure here was computed from the formulas, which are documented in these primary references:
- NIST Digital Library of Mathematical Functions, §4.42 Solution of Triangles: the law of cosines (equation 4.42.5) and the triangle area formula ½·b·c·sin A (equation 4.42.7). A diagonal splits a parallelogram into two congruent triangles, which gives the diagonals and the area a·b·sin θ.
- Common Core State Standards for Mathematics, Grade 6 Geometry, standard 6.G.A.1: finding the area of special quadrilaterals by composing them into rectangles or decomposing them into triangles, the basis of the cut-and-slide argument above.
- Worked examples and tables: computed with the same formulas the calculator uses and rounded only at the last step.
⚠️ Common mistakes & edge cases
Multiplying the base by the slanted side
For the patio, 12 × 9 = 108 ft², but the true area is 12 × 7 = 84 ft². The slanted side and the height are the same length only when the corners are 90°, so base times side is the rectangle formula, not the parallelogram formula.
Pairing a height with the wrong base
A parallelogram has two heights. The 9.33 ft height of the patio is measured between the two 9 ft ends, so it multiplies 9, not 12. Both pairings give the same 84 ft²; a mixed pairing does not.
Expecting the diagonals to be equal
Equal diagonals happen only in rectangles. The patio's diagonals are 9.45 ft and 18.99 ft, and the gap widens the further the shape leans. Comparing two diagonals is a fast field test for square corners.
Assuming a diagonal bisects the corner
Only in a rhombus does a diagonal split a corner in half. With sides of 15 and 8 at 60°, the long diagonal divides the 60° corner into 20.03° next to the 15 cm side and 39.97° next to the 8 cm side.
Treating 60° and 120° as different shapes
A parallelogram always has one acute and one obtuse angle that add up to 180°, and sin 60° = sin 120°. Either angle describes the same shape, so the calculator accepts both and reports the acute one first.
Angles of 0° or 180°
At these angles the shape collapses into a line with zero area, so the calculator asks for an angle strictly between the two instead of returning a misleading zero.
❓ Frequently asked questions
How do you find the area of a parallelogram?
Multiply the base by the perpendicular height: A = b × h. The height is measured at a right angle to the base, not along the slanted side. A parallelogram with a 12 ft base and a 7 ft height covers 12 × 7 = 84 square feet, whatever the length of its slanted sides.
Is the area of a parallelogram base times the slanted side?
No, unless the shape is a rectangle. The slanted side is at least as long as the height, so multiplying by it overstates the area. With a 12 ft base, 9 ft slanted sides and a 7 ft height, the area is 84 square feet; 12 × 9 = 108 would be about 29% too much.
How do you find the perimeter of a parallelogram?
Add the base and the slanted side, then double the sum: P = 2(a + b), because opposite sides are equal. For a 12 ft base and 9 ft slanted sides, P = 2 × (12 + 9) = 42 ft. The perimeter needs the actual side length, not the height.
How do you find the height of a parallelogram?
If you know the area, divide it by the base: h = A ÷ b. If you know the slanted side and the angle between the sides, use h = a × sin θ. With a slanted side of 8 cm at 60°, h = 8 × sin 60° = 6.93 cm. Each pair of parallel sides has its own height, and the calculator reports both.
How do you find the diagonals of a parallelogram?
Use the law of cosines. The short diagonal is √(a² + b² − 2ab·cos θ) and the long one is √(a² + b² + 2ab·cos θ), with θ the acute angle. For sides of 15 cm and 8 cm at 60°, the diagonals are exactly 13 cm and √409 ≈ 20.22 cm. They are equal only when the parallelogram is a rectangle.
Can you find the area of a parallelogram from its diagonals?
Yes, if you also know the angle where they cross: A = ½ × d₁ × d₂ × sin φ. Diagonals of 12 in and 16 in crossing at 90° give ½ × 12 × 16 × 1 = 96 square inches. The two lengths alone are not enough, because the same diagonals can cross at many angles.
Can I calculate the area from the two side lengths alone?
No. Two sides do not fix the shape, because the parallelogram can lean. Sides of 10 and 6 enclose 60 square units when the corners are square, 51.96 at 60°, 30 at 30° and 15.53 at 15°. You need one more measurement: the height, an interior angle or a diagonal.
What are the main parallelogram formulas?
Area A = b × h = a × b × sin θ = ½ × d₁ × d₂ × sin φ. Perimeter P = 2(a + b). Height h = a × sin θ. Diagonals d = √(a² + b² ∓ 2ab·cos θ), and together they satisfy d₁² + d₂² = 2(a² + b²). Neighboring angles add up to 180° and opposite angles are equal.
Are rectangles, rhombuses and squares parallelograms?
Yes. A parallelogram is any quadrilateral with two pairs of parallel sides. A rectangle has four 90° angles, a rhombus has four equal sides, and a square has both. Every formula here applies; for a rectangle sin 90° = 1, so a × b × sin θ becomes length times width.
What is the parallelogram law?
The squares of the two diagonals add up to the squares of all four sides: d₁² + d₂² = 2(a² + b²). For sides of 15 and 8 with diagonals 13 and √409, the left side is 169 + 409 = 578 and the right side is 2 × (225 + 64) = 578. It is a quick check on field measurements.
💡 Good to know
Leaning does not change the area
A 12 ft by 7 ft rectangle and the patio parallelogram share a 12 ft base and a 7 ft height, so both cover 84 ft². The parallelogram simply needs more edging: 42 ft around instead of 38 ft.
Every parallelogram is two triangles
A diagonal cuts it into two identical triangles, each with area ½ × a × b × sin θ. Doubling that gives a × b × sin θ, which is why triangle tools and parallelogram tools always agree.
Square corners give the most area
For fixed sides a and b, the area a × b × sin θ is largest at θ = 90°. Any lean shrinks it: sides of 10 and 6 enclose 60 square units as a rectangle but only 30 at 30°.