What it does
Take a rectangle and push its top sideways, keeping the top the same height. You get a parallelogram. Did the area change? This page lets you try it on a grid of unit squares:
- Parallelogram: drag the round handle (or the s slider) to slide the top. The area stays base × height, even though the slanted sides get longer.
- Cut and slide: cut the triangle off one end and slide it to the other. The parallelogram turns into a rectangle with the same base and height.
- Triangle: slide the top corner along a line parallel to the base. The area stays ½ × base × height. Add a turned copy and two triangles make a parallelogram.
- Trapezium: two parallel sides a and b. A turned copy makes a parallelogram with base a + b, so the trapezium’s area is ½ × (a + b) × h.
Sliding a shape like this, with every horizontal strip moving sideways but keeping its length, is called a shear. It never changes area.
It follows the Class 8 maths chapter Area. Counting area with dots on a grid is in Pick’s theorem, the area of a circle by slicing is in circle area, and a triangle from three sides is in heron's formula explorer.
How to use it
- Choose Parallelogram, Triangle or Trapezium.
- Set the base b, the height h and, for a trapezium, the top side a.
- Drag the round handle sideways, or set s (how far along the top starts). The red dashed line is the height.
- For a parallelogram, move Cut and slide to the right. For a triangle or trapezium, tick Add a turned copy.
- Read the area, the formula with your numbers, and the perimeter, which does change.
A worked example
The page starts with a parallelogram of base 6 and height 4, its top slid 2 squares along.
- Area = 6 × 4 = 24 square units.
- Slide s to 6: the slanted side grows from √(2² + 4²) ≈ 4.47 to √(6² + 4²) ≈ 7.21, the perimeter from 20.94 to 26.42, but the area is still 24.
- Back at s = 2, move Cut and slide to the end: the triangle with corners (0, 0), (2, 0), (2, 4) moves 6 along and the shape is a 6 × 4 rectangle.
Switch to Trapezium with a = 3, b = 6, h = 4: area = ½ × (3 + 6) × 4 = 18. With the turned copy, the parallelogram has base 9 and area 36, twice the trapezium.
Tips
- The height is always measured at right angles to the base, never along the slanted side.
- If the top slides more than a whole base length, one cut is not enough, but thin strips still slide across: the area is unchanged.
- Count the whole squares and pair up the half squares to check the formula by eye.
Check yourself
Four fresh questions: parallelogram and triangle areas, a trapezium, and a height from an area. Nothing is saved.
FAQ
Why doesn’t the slanted side matter? Area counts the squares inside. Sliding the top moves every row of squares sideways, but each row keeps its length, so the total stays the same.
Is this the NCERT book? No. It covers the same topic with fibub’s own pictures and questions. fibub is not part of NCERT.
Where the numbers come from
- Area of a parallelogram, triangle and trapezium, and the shear (Cavalieri) argument, are standard school geometry. Pictures and questions written for fibub.
- Checked: the formula areas equal the shoelace area of the drawn corners for 2 000 random shapes (tests/sci-tools/area-shear.mjs) and by hand for the worked examples.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.