Area and Perimeter of Plane Figures
264. Area of a Triangle Formula · Base, height, and the magic of 1/2
The altitude AD is always perpendicular to the base BC.
The area of a triangle is ½ × base × height, where the height is the perpendicular distance from the chosen base to the opposite vertex. Any side may serve as the base, as long as the matching altitude is used.
What this lesson covers
Try to break it
Drag the apex A. The base BC stays the same; only the perpendicular height h from A to BC changes. The area always equals ½ × base × height. Slide A horizontally and the height (and area) stays exactly the same; lift A vertically and both grow in step.
How you build it
Draw the base BC, mark the apex A, and join AB and AC.
- Point tool: drop B at the left end of the base.
- Point tool: drop C to the right of B — BC is the base.
- Segment tool: join B to C — the base.
- Point tool: drop the apex A somewhere above the base BC.
- Segment tool: join A to B.
- Segment tool: join A to C — the triangle is closed.
- Perp tool: click A, then click on the base BC — the altitude drops square to BC, with a right angle at its foot. This is the height h.
- Point tool: mark D where the altitude meets BC. AD is the height; Area = ½ × base BC × height AD.
The proof, step by step
Prove that the area of a triangle is half × base × height.
- Consider a rectangle with the same base `b` and height `h` as the triangle.
- The area of this rectangle is `b * h`.
- The triangle divides the rectangle into two equal areas (or can be rearranged to fill half the rectangle).
- Therefore, the area of the triangle is `1/2 * b * h`.
Worked example
A triangle has a base of 10 cm and a height of 6 cm. What is its area?
Area = 1/2 * base * height = 1/2 * 10 * 6 = 30 cm².
- 30 cm² — correct
- 60 cm²
- 16 cm²
- 120 cm²