Mensuration
158. The Triangle's Secret · half base times height, every time
The area formula holds regardless of how you stretch the base or shift the apex.
Area of a triangle = ½ × base × height, where height is the perpendicular distance from the apex to the base. Works for every triangle — equilateral, isosceles, scalene, right, acute, obtuse.
What this lesson covers
Try to break it
Drag the apex C around. Base AB is fixed; the height h is the perpendicular drop from C to AB. As C moves, h and the area change together, but area always equals ½ × base × height. Slide C parallel to AB and watch — h stays the same, so the area stays the same.
How you build it
Construct a triangle with altitude.
- Mark point A — the left end of the base.
- Mark point B — the right end of the base.
- Draw segment AB — this is the base b.
- Mark point C — the apex of the triangle, anywhere above AB.
- Drop a perpendicular from C to AB — this is the height h.
- Draw segment AC — the left side of the triangle.
- Draw segment BC — Area = ½ × AB × h.
The proof, step by step
Prove that the area of a triangle is half × base × height.
- Consider a rectangle with base b and height h.
- The area of this rectangle is b × h.
- The triangle occupies exactly half of the rectangle's area.
- Therefore, Area of triangle = 1/2 × b × h.
Worked example
A triangle has a base of 14 cm and a corresponding height of 6 cm. What is its area?
Using the formula Area = ½ × base × height, we get ½ × 14 × 6 = 42 cm².
- 42 cm² — correct
- 84 cm²
- 20 cm²
- 28 cm²