Area of a Trapezium and a Polygon
190. The Triangle's Secret · Area depends only on base and height
Dragging C horizontally keeps base and height constant, so area stays the same.
Triangles on the same base and between the same parallels have equal areas. Slide the apex along a line parallel to the base — base × height stays constant, so area = ½ × base × height stays the same too.
What this lesson covers
Try to break it
Drag C up or down to change the height; drag C left or right to slide it without changing the height. The area follows the height (h) but ignores horizontal shifts. Try to change the area while keeping h fixed — you can't. Area = ½ × base × height, period.
How you build it
Construct a triangle with base and height.
- Draw segment AB — the base of the triangle.
- Draw the altitude from C: use the Perp tool to drop a perpendicular from C to AB. This is the height of the triangle.
- Draw segment AC — one side of the triangle.
- Draw segment BC — the other side of the triangle.
The proof, step by step
Prove that the area of a triangle stays the same as the apex slides along a line parallel to the base.
- Take a rectangle with the same base 'b' and height 'h' as the triangle.
- The triangle occupies exactly half of this rectangle's area.
- Therefore, Area of Triangle = 1/2 × Rectangle Area = 1/2 × b × h.
Worked example
The base of a triangle is 12 cm and its corresponding height is 8 cm. Find its area.
Area = 1/2 × base × height = 1/2 × 12 × 8 = 48 cm².
- 48 cm² — correct
- 96 cm²
- 20 cm²
- 32 cm²