Example 1
- a = 8, h = 5
40.00
What is the parallelogram area at a = 8 and h = 5? 40.00.
Parallelogram area calculator
Enter base a and perpendicular height h (a > 0, h > 0). Area is A = a·h. With a = 8 and h = 5 the result is 40.00. With 10 and 6 the result is 60.00. Height is the perpendicular distance between the bases — not the slanted side length. This calculator does not use sine or diagonals; the a·b·sin(γ) formula is on a separate page.
Result
Enter base a and height h.
The formula needs a > 0 and h > 0. The two-sides-and-angle variant lives on the parallelogram area calculator with sine.
How it works
a8
h5
A = a·h40
Parallelogram area from base and height is a classical Euclidean formula: A = a·h. Symbol a is the length of the chosen base, and h is the perpendicular height to the line containing that base. The calculator uses the numbers you enter in Base a and Height h. You do not enter an angle or a second side here — the A = a·b·sin(γ) variant lives on the neighboring page.
The same result follows from a shear: a parallelogram with base a and height h has the same area as a rectangle with sides a and h, because a shear keeps height and parallel sides. Euclid in the Elements (Book I) proves that parallelograms with the same base and the same height have equal areas. The result panel shows a, h, the product, half (triangle area ½ah), and the ratio a/h.
Domain: a > 0 and h > 0. An empty field leaves the result blank — it is not area 0. A value ≤ 0 is rejected. Worked examples: 8 and 5 give 40; 10 and 6 give 60; 12 and 4 give 48; 7 and 7 give 49; 15 and 2 give 30. A triangle with the same a and h has half the area.
A comma and a period in the fields are equivalent. The header unit switch changes the cm² or in² label and does not convert an entered 8. A slanted side does not replace h — height is perpendicular.
The result-panel diagram shows a parallelogram with base a and height h labelled. The shear on the drawing is illustration only; A = a·h does not depend on an angle, because height is already measured perpendicularly.
Mathematical model: parallelogram area base × height (Euclidean geometry; Euclid, Elements I). It is not a separate “index” or 0–100 scale — only A from a and h.
Formula
A = a · h, where a > 0 is the base length and h > 0 is the height perpendicular to that base.
Equivalently with sides a, b and angle γ: h = b · sin(γ), then A = a · h = a · b · sin(γ) — that variant is on the sine page.
Model: parallelogram area base × height (Euclidean geometry; Euclid, Elements I).
Theory, shear, and height
Definition
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Area A relative to a chosen base of length a and height h (the perpendicular segment between the lines containing the bases) is A = a·h. The calculator uses the positive lengths a and h you enter; the result has dimension length² (cm² or in² according to the label).
Height h is not the length of the neighboring slanted side. If you know side b and included angle γ between a and b, then h = b·sin(γ) and A = a·b·sin(γ) — that variant lives on the sine page. Here h is already given as a perpendicular length.
Historical and conceptual context
Euclid in the Elements (Book I) introduces parallelograms and proves equal areas for a shared base and shared height. Modern analytic geometry and linear algebra sharpen the same fact: the absolute value of the determinant built from the side vectors is the parallelogram area. A shear keeps height and area, which explains why a rectangle with sides a and h and a “tilted” parallelogram with the same a and h share the same A.
In school, A = a·h appears before trigonometry: you only need the idea of height. The vector form |a × b| needs an angle or components. This page stays with the height definition.
Formal properties
Domain: a > 0, h > 0. Homogeneity: scaling a and h by λ gives A′ = λ²A. Symmetry: swapping the labels “base” and “height” does not change the numeric product, but geometrically the height relative to side a is always the perpendicular to a.
Link to a triangle: a diagonal splits the parallelogram into two triangles of area ½ah each. Hence, a triangle with base a and height h has half the parallelogram area. A rectangle and a rhombus are special parallelograms; at a right angle h is the other side.
Full derivation
Step 1. Let the base lie on a segment of length a. Height is the distance between the two parallel lines containing the pairs of sides — call it h.
Step 2. By a shear, slide vertices along the base direction until angles become right angles, without changing h or a. You obtain a rectangle with sides a and h.
Step 3. Rectangle area is a·h. A shear preserves area, so parallelogram area is also A = a·h.
Step 4 (vector check). With u = (a, 0) and v = (b cos γ, b sin γ) we have |det(u, v)| = a·b·|sin γ| = a·h when h = b·|sin γ|.
Special cases
For a = 8, h = 5: A = 40. For a = 10, h = 6: A = 60. For a = h = 7: A = 49 (a square when angles are right and sides equal). For a = 15, h = 2: A = 30 — a low, long strip. For a = 12, h = 4 and a = 6, h = 8: both give 48 — same product, different shape.
If you enter a slanted side instead of h, the result is no longer the area of that figure. When you know only sides and an angle, use the sine page. When you know rhombus diagonals, use the rhombus page.
Numeric and computational notes
The calculator multiplies a by h and formats the result to two decimal places with a local separator. An empty field is not treated as zero. The unit switch changes the unit label and does not convert values in the fields — 8 stays 8.
Comma and period are equivalent as decimal separators. The calculator does not derive height from sides (that needs Pythagoras or sine) — it takes h as given. The result has dimension length².
The diagram shows a slanted parallelogram with height h marked — the shear angle in the drawing is visual only (about 60°) and does not enter the formula. The result A = a·h does not depend on that angle as long as h is the true height to the chosen base a. If you switch the base to another side, you must also take the height relative to that side; the product remains the same area.
The panel also shows the half ½ah as the area of a triangle with the same base and height. That is not a second input mode — only a teaching check. With a = 8 and h = 5 the area is 40 and the triangle is 20. With a = 10 and h = 6: 60 and 30. With a = 12 and h = 4: 48 and 24.
Do not confuse height with a slanted side or a diagonal. For side and angle use the sine page (A = a·b·sin γ). For rhombus diagonals use the rhombus page. Here the only computation is the product of positive a and h.
Related ideas and sources
Related models: rectangle area (a·b at a right angle), triangle area (½ah), rhombus area (½e·f or a²·sin γ), parallelogram area with sine (a·b·sin γ), trapezoid ½(a+b)·h. Euclid, Elements I; MathWorld — Parallelogram; Wikipedia — Parallelogram. The calculator is educational and does not replace a geometry course.
Sources (listed below): Euclid, Elements I; MathWorld — Parallelogram; Wikipedia — Parallelogram; school geometry texts (A = a·h).
How to use
- Enter base a (e.g. 8) — a positive length in cm or inches according to the switch.
- Enter height h perpendicular to that base (e.g. 5), not the slanted side length.
- Click Calculate or wait for the live result. 8 and 5 give 40. 10 and 6 give 60.
- Check the panel: a, h, product, half (triangle), and the diagram with height marked.
- Area from two sides and an angle lives on the neighboring sine page — here you need h.
Symbol glossary
A = a·h. For a = 8 and h = 5 the area is 40. A triangle with the same a and h has half: 20.
- A
- Parallelogram area. 8·5 = 40. 10·6 = 60. 12·4 = 48.
- a
- Base length (a > 0). Form field Base a.
- h
- Height perpendicular to a (h > 0). Not a slanted side. Field Height h.
- ½ah
- Triangle area with the same base and height — half of A. For 8 and 5: 20.
Examples
Example 2
- a = 10, h = 6
60.00
What is the area at a = 10 and h = 6? 60.00.
Example 3
- a = 12, h = 4
48.00
What is the area at a = 12 and h = 4? 48.00.
Example 4
- a = 7, h = 7
49.00
What is the area at a = h = 7? 49.00.
Example 5
- a = 15, h = 2
30.00
What is the area at a = 15 and h = 2? 30.00.
Example 6
- a = 9, h = 3
27.00
What is the area at a = 9 and h = 3? 27.00.
Example 7
- a = 6, h = 8
48.00
What is the area at a = 6 and h = 8? 48.00.
Example 8
- a = 11, h = 5
55.00
What is the area at a = 11 and h = 5? 55.00.
Related calculators
Common questions
What does this calculator compute?
Parallelogram area from A = a·h, where a is the base length and h is the height perpendicular to that base. For a = 8 and h = 5 the result is 40. It does not compute a·b·sin(γ) here — that lives on the sine page.
How is this different from the sine formula?
Here you enter base and height. On the sine page you enter two adjacent sides and the included angle. Both models give the same area when h = b·sin(γ), but the form fields differ.
Why do 8 and 5 give 40, same as an 8 by 5 rectangle?
At a right angle the height is the other side. The parallelogram still uses A = a·h. A rectangle is the special case of a parallelogram.
Is rhombus area from diagonals e·f/2 here?
No. The diagonal rhombus has its own page. You do not enter e or f here. With diagonals 6 and 8 the rhombus area is 24 — another formula.
What happens with an empty field or a ≤ 0?
Empty a or h leaves the result blank (it is not area 0). A value ≤ 0 is rejected with a message — base and height must be positive.
Is a triangle with 8 and 5 the same 40?
No. A triangle with base 8 and height 5 has area ½·8·5 = 20. The parallelogram is 40 — the whole figure, not half.
Can I mix centimeters and inches?
Not in one run. Both dimensions in cm or both in inches. The header unit switch changes the cm²/in² label and does not convert the numbers you entered.
Can I enter a slanted side as h?
No. h is the perpendicular segment to the base (or to the line containing the base). A slanted side is longer and spoils the 40 result for the 8 and 5 example.
Does trapezoid ½(a+b)·h replace this area?
No. A trapezoid has two bases. Here one base and h. For 8 and 5 the result is 40; a trapezoid still needs a second base.
What is the area for 10 and 6?
60. That is the second example on this page. For 11 and 5 you get 55. For 15 and 2 you get 30.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on a, h, and the length-unit label. The currency and unit settings in the page header do not affect this result.
Where does A = a·h come from?
From Euclidean geometry: parallelograms with the same base and the same height have equal areas (Euclid, Elements, Book I). A shear keeps height and area; you can reassemble a rectangle with the same a and h.
How do I calculate the area of a parallelogram?
Multiply the base by the perpendicular height: A = a·h. With a = 8 and h = 5 the result is 40.00. Do not treat the slanted side as height — h is the right-angle distance between the parallel sides.
What is the formula for the area of a parallelogram?
Most often A = a·h (base × height). Equivalently A = a·b·sin(γ) when you know two sides and the included angle — that variant is on the sine page. Here you enter only a and h.
What is the area of a parallelogram with base 8 cm and height 5 cm?
40 cm² (shown here as 40.00 in the chosen scale). Enter a = 8 and h = 5. That is the classic school example: 8·5 = 40.
Can I enter the slanted side as the height?
No. Height h is always perpendicular to the base. The slanted side is longer than or equal to h; equality holds only for a rectangle. Entering the slant instead of h overstates the area.
How does this differ from the sine calculator?
Here A = a·h with no angle. The sine page computes A = a·b·sin(γ) from two sides and an angle — when you do not know the height. Both formulas match when h = b·sin(γ), but the input fields differ.
Sources
The base × height formula and Euclidean context below come from established sources.
- Wikipedia — Parallelogram (area formulas)
- MathWorld — Parallelogram
- Wikipedia — Euclidean geometry
- Wikipedia PL — Równoległobok
- Euclid, Elements, Book I (parallelograms, bases and heights; classical Euclidean geometry).
Page updated November 3, 2026.