Explanation
Triangle measurements
Choose the information you have: base and perpendicular height for area, three side lengths for perimeter, or area and base to recover the perpendicular altitude. These operations do not assume a right triangle.
A = b × h ÷ 2
| b | Base (m) |
|---|---|
| h | Height (m) |
Area with base 3 m and height 4 m
| Base | 3 |
|---|---|
| Height | 4 |
| Result | 6 m² |
A = b × h ÷ 2 = 3 m × 4 m ÷ 2 = 6 m²
P = a + b + c
| a | Side a (m) |
|---|---|
| b | Side b (m) |
| c | Side c (m) |
Perimeter of a 3–4–5 triangle
| Side a | 3 |
|---|---|
| Side b | 4 |
| Side c | 5 |
| Result | 12 m |
P = a + b + c = 3 m + 4 m + 5 m = 12 m
h = 2A ÷ b
| A | Area (m²) |
|---|---|
| b | Base (m) |
Height from an area of 6 m² on a 3 m base
| Area | 6 |
|---|---|
| Base | 3 |
| Result | 4 m |
h = 2A ÷ b = 2 × 6 m² ÷ 3 m = 4 m
Diagram of this result
The triangle is drawn in the proportions you entered. Base and height sit on dimension lines and the dashed altitude meets the base at a right angle; for perimeter, each side is labelled.
The triangle is drawn in the proportions you entered. Base and height sit on dimension lines and the dashed altitude meets the base at a right angle; for perimeter, each side is labelled.
Common mistakes
A sloping edge is generally not the height. The altitude meets the base at a right angle and may fall outside an obtuse triangle. Perimeter adds all three edges and cannot be found from base and altitude alone.
Assumptions and limits
Every measurement must be positive. Area supplied to the altitude operation is in m²; base and sides are metres. Perimeter requires three sides that satisfy the strict triangle inequality. Base/height operations define area but do not infer a unique triangle or edge lengths. Canonical inputs are limited to 30 significant digits, 256 characters and adjusted exponent magnitude 100. Computation uses 50 significant digits without intermediate presentation rounding; output overflow fails explicitly. Approximate fixtures use absolute tolerance 1e-12 plus relative tolerance 1e-12.
How do the three measurements relate?
A base of 3 m with perpendicular height 4 m gives area 6 m². Area 6 m² with base 3 m recovers height 4 m. A separate triangle with side lengths 3 m, 4 m and 5 m has perimeter 12 m.
Sources
Standard Euclidean measurement formulas: OpenStax Prealgebra 2e, measurement and geometry topics, https://openstax.org/details/books/prealgebra-2e. Worked examples use independent geometric identities.