The area of a parallelogram and a trapezium

Cut, move, glue — and you have a rectangle

Today two formulas, but only one idea: rearrange a slanted shape into a straight one and calculate the straight one.

Parallelogram: cut a right-angled triangle from one side and glue it on the other side. A rectangle with the same base and height is born. That is why:

S = a × vₐ (base times the height to it).

Watch — the height, not the slanted side! That is the most expensive mistake of this chapter.

A trapezium has two parallel bases: a long a and a short c. Picture it as a rectangle whose width sits somewhere between them — exactly in the middle. That is why you calculate with the average of the bases:

S = (a + c) : 2 × v

Add the bases, divide by two (that is the average width) and multiply by the height. You will see both formulas born today, so you will not have to believe them — you will know them.

Tereza and Filip help grandpa measure a garden bed in the shape of a parallelogram: base 4 m, slanted side 2.5 m, perpendicular distance between the long sides 2 m. Filip counts: “4 times 2.5 is 10 square metres.” Grandpa laughs and stretches a string with a plumb between the long sides: “Measure perpendicular, boy. Two metres.” Right: 4 × 2 = 8 m². Filip does not get it: “And what is the side 2.5 for then?” Tereza: “For the fence. Perimeter walks along the sides, area along the perpendiculars.”

Both formulas with a derivation

Parallelogram: S = a × vₐ.

Derivation with scissors: from the left side of the parallelogram cut off a right-angled triangle and slide it to the right side. The pieces fit and a rectangle is born: base a, height vₐ. Nothing added, nothing taken — the area is the same.

Method: 1) find the base, 2) find the height to it (the perpendicular distance to the opposite side), 3) multiply.

Trapezium: S = (a + c) : 2 × v.

Derivation by gluing: take a second identical trapezium, turn it upside down and glue. A parallelogram with base (a + c) and height v is born. Its area is (a + c) × v — and your trapezium is half.

Method: 1) add both bases (the parallel sides!), 2) divide by two, 3) multiply by the height.

Check questions before plugging in: Are my two bases really the parallel sides? Is my height perpendicular? Do I have everything in the same units?

Reverse questions work as with a triangle: from S = a × vₐ you make vₐ = S : a. With a trapezium, from the area and the height you calculate the sum of the bases: a + c = 2 × S : v.

Parallelogram: base 9 cm, height 4 cm, side 5 cm

Step 1: for area I need the base and the height. The side 5 cm is bait — it belongs to perimeter.

Step 2: S = 9 × 4 = 36 cm².

Result: 36 cm². And while we are here: perimeter = 2 × (9 + 5) = 28 cm. See the difference of roles? Side 5 cm was used in perimeter, height 4 cm in area. Every number in a question has its job — but not every one belongs in every formula.

Trapezium: bases 10 cm and 6 cm, height 5 cm

Step 1: the bases are the parallel sides: a = 10, c = 6.

Step 2: average width: (10 + 6) : 2 = 8 cm.

Step 3: times height: 8 × 5 = 40.

Result: S = 40 cm². A sense check: a rectangle 10 × 5 would have 50 cm², a rectangle 6 × 5 only 30 cm². A trapezium sits somewhere between — and 40 is exactly between. The formula with the average does exactly what intuition says.

Reverse question: a parallelogram has area 54 m² and base 12 m

I look for the height.

Step 1: from the formula S = a × v I make v = S : a.

Step 2: v = 54 : 12 = 4.5.

Result: the height is 4.5 m. Check: 12 × 4.5 = 54 m². That checks out.

A use from life: that much space is left between two parallel fences when you know the plot area and the length along the path. Reverse questions are not a school invention — that is how measuring really works.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Revise the area of a rectangle and a square — a parallelogram is just their rearranged cousin. Write only the number in cm².

Practise the questions

On paper: scissors and formulas

The first question is with scissors — a derivation with your own hands lasts longest in your head.

1. Draw a parallelogram on squared paper, cut it out, cut off the side triangle and rearrange it into a rectangle. Calculate the area by counting squares and by the formula — they must match.

2. Calculate the area of a parallelogram with base 15 cm and height 8 cm. Side 9 cm — write what that one is good for.

3. A trapezium field has bases 60 m and 40 m, height 25 m. How many m² does it measure? And how many such fields fit into a hectare (10 000 m²)?

4. A trapezium has area 42 cm², height 6 cm and one base 9 cm. Calculate the other base and check.

Now you 💪

1. I know S = a × vₐ for a parallelogram and I know why it holds (rearranging into a rectangle).
2. I know S = (a + c) : 2 × v for a trapezium and I recognise its bases.
3. I do not mix a slanted side with a height.

What to take from this lesson

• Parallelogram: S = base × height to it.
• Trapezium: S = average of the bases × height.
• Perimeter walks along the sides, area along the heights.
• Both formulas were born by rearranging into a rectangle — you can derive them again any time.
• A slanted side does not belong in area, a perpendicular is always shorter.

© 2026 Ing. Martin Polak / AlgoRhino · 内容使用条款