Library / Geometry, coordinates, surveying and trigonometry / GEO-13
Geometry, coordinates, surveying and trigonometry
Median length from three sides
Length of the line from a corner to the midpoint of the opposite side a, from the three sides.
- Ref
- GEO-13
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- ma [cm]
A median joins a corner to the midpoint of the opposite side. Apollonius’s theorem gives its length as mₐ = ½√(2b² + 2c² − a²) for the median to side a. The three medians meet at the centroid, which divides each in the ratio 2 : 1 from the corner. In an isosceles triangle the median to the base is also the height.
Length of the line from a corner to the midpoint of the opposite side a, from the three sides. The median is not in general perpendicular to the side.
Calculator
- Distance from the corner to the centroid, 2mₐ/32.667 cm
- Distance from the centroid to the midpoint, mₐ/31.333 cm
- Height to side a from Heron’s area4 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With a = 6 cm and b = c = 5 cm, the median to side a is 4 cm, which is also the height of the isosceles triangle when b equals c.
- The centroid lies 2.667 cm from the corner along it, two thirds of the way to the midpoint.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Side b | 1.5 to 10.5 cm | 2.151 | 7.657 | 5.506 | +10% | +7.89 % |
| c Side c | 1.5 to 10.5 cm | 2.151 | 7.657 | 5.506 | +10% | +7.89 % |
| a Side being bisected | 0.5 to 9.9 cm | 0.7053 | 4.994 | 4.288 | +10% | −6.09 % |
Sweep of Side being bisected
| Side being bisected (cm) | Median to side a (cm) |
|---|---|
| 0.5 | 4.994 |
| 1.675 | 4.929 |
| 2.85 | 4.793 |
| 4.025 | 4.577 |
| 5.2 | 4.271 |
| 6.375 | 3.852 |
| 7.55 | 3.279 |
| 8.725 | 2.443 |
| 9.9 | 0.7053 |
Sweep one variable
Variables and how each one contributes
Median to side a
What the equation gives you. Everything else feeds it.
Side being bisected
The side that the median meets, in cm. A longer side lowers the median, because the midpoint moves further from the opposite corner. Raising it by 10 % (to 6.6) moves the result by −6.09 %, from 4 to 3.756 cm.
Side b
One of the two sides that meet at the corner, in cm. A longer side raises the median. Raising it by 10 % (to 5.5) moves the result by +7.89 %, from 4 to 4.316 cm.
Side c
The other side that meets at the corner, in cm. A longer side raises the median. Raising it by 10 % (to 5.5) moves the result by +7.89 %, from 4 to 4.316 cm.
Worked example with the values above
- Compute 2b² + 2c² − a² = 50 + 50 − 36 = 64.
- Take the square root: 8.
- mₐ = half of that = 4 cm.
Assumptions and limits
- A non-degenerate triangle: the three sides satisfy the strict triangle inequality.
- All lengths and the area use one consistent unit.
Common mistakes
- Mixing up which side is bisected: the median goes to the side called a.
- Forgetting the factor of one half outside the root.
- Using sides that fail the triangle inequality, which gives a negative radicand.
Statistics and uncertainty
The median uses all three sides, but the result changes slowly when one of them is changed a little, so it is robust to small measurement errors.
Where it is used
- Fitting the largest circle in a triangle and the circle through three points.
- Centre lines, balance points and bracing positions of triangular frames and plates.
Maths you need
- Triangle inequality: any two sides add to more than the third
- Heron’s formula: area from three sides
- Midpoint: the point halfway along a segment