# Triangle Area — Half a Rectangle
> **Animation coming** — full Sora 2 prompt below. Video will replace this placeholder.
## Plain English first
Draw a rectangle. Now cut it diagonally from corner to corner. You get two identical triangles. Each triangle is exactly half the rectangle.
That's the whole formula. A triangle with the same base and height as a rectangle always contains exactly half the area. The ½ in A = ½bh is not a magic number — it's literally the diagonal cut.
This means to find a triangle's area, you don't need to count unit squares directly. Find what rectangle would surround it, compute that area, then take half.
---
## Standard math notation
```
A = ½ × b × h
Where:
A = area of the triangle
b = base (the bottom edge length)
h = height (perpendicular distance from base to opposite vertex)
Note: h must be perpendicular to b — not the slant side length.
For a right triangle, h is one of the legs.
```
---
## Verbose Python with descriptive names
```python
def compute_triangle_area(
length_of_base,
perpendicular_height_from_base_to_apex
):
"""
A triangle is exactly half of the rectangle that surrounds it.
Draw the rectangle: base × height gives the full rectangle area.
The diagonal cut gives us half of that.
"""
area_of_surrounding_rectangle = length_of_base * perpendicular_height_from_base_to_apex
area_of_triangle = area_of_surrounding_rectangle / 2
return area_of_triangle
area = compute_triangle_area(
length_of_base=6,
perpendicular_height_from_base_to_apex=4
)
print(f"Area: {area} square units") # 12.0
# Works for any triangle — right, acute, obtuse
# As long as h is the perpendicular height, not a slant side
area_obtuse = compute_triangle_area(
length_of_base=8,
perpendicular_height_from_base_to_apex=3
)
print(f"Area: {area_obtuse} square units") # 12.0
```
---
## Sora 2 video prompt
```
8-second animation. A blue rectangle appears labeled base b and height h,
filled with a grid. A diagonal line slices it corner to corner. One triangle
glows orange, the other fades to 30% opacity. The formula A = ½ × b × h builds
onscreen — the ½ highlights as the orange half of the rectangle. Numbers:
b=6, h=4, A=12. Clean white background, warm earth tones, satisfying diagonal
cut motion.
```
---
## The perpendicular height trap
The most common mistake: using a slant side as h.
```python
# RIGHT TRIANGLE — easy, the legs are perpendicular
area = compute_triangle_area(
length_of_base=3,
perpendicular_height_from_base_to_apex=4 # the vertical leg IS the height
)
# = 6
# OBTUSE TRIANGLE — height drops outside the base
# You may need to extend the base line to drop a perpendicular from the apex
# The formula still works, as long as h is truly perpendicular to b
```
---
## Builds on
- [Rectangle Area — Counting the Grid](/articles/rectangle-area)
## See also
- [Pythagorean Theorem — Squares on Sides](/articles/pythagorean-theorem)
- [Visual Triangle Geometry](/articles/794e2e02-16a0-43fc-955a-ab27f8d1de8d)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)