Each side is shorter than the other two together
The triangle inequality: when three lengths cannot make a triangle.
Three sticks
You have three sticks: 10 cm, 15 cm and 30 cm. You want to join their ends to make a triangle.
Can these three lengths be the sides of a triangle?
What this lesson covers
The idea
Three lengths satisfy the triangle inequality when each is smaller than the sum of the other two; the sidelengths of any triangle satisfy it, so lengths failing it cannot form a triangle.
Three sticks
You have three sticks: 10 cm, 15 cm and 30 cm. You want to join their ends to make a triangle.
Can these three lengths be the sides of a triangle?
- Yes, any three lengths make a triangle
- No, these three lengths cannot make a triangle
- We must draw it very carefully to find out
Direct or roundabout?
In the rough triangle ABC, every side is the direct way between two corners (the red bar). The other two sides together are the roundabout way (the yellow bar). Tap a row to see both paths on the triangle.
A direct side must be the shorter way
A direct path is shorter than a roundabout path. So in a triangle, each side is shorter than the other two sides together. We say the lengths satisfy the triangle inequality.
Only the longest side can fail. So it is enough to compare the longest side with the sum of the other two.
Three lengths satisfy the triangle inequality when each is smaller than the sum of the other two. The sidelengths of any triangle satisfy it, so lengths that fail it cannot form a triangle.
- Lengths | Longest side check
- 3, 4, 5 | 5 < 3 + 4 = 7 ✓
- 10, 15, 30 | 30 > 10 + 15 = 25 ✗
Notes
Three lengths satisfy the triangle inequality when each is smaller than the sum of the other two. The sidelengths of any triangle satisfy it, so lengths that fail it cannot form a triangle.
Check yourself
Which three lengths can be the sides of a triangle?
A triangle has two sides of 5 cm and 7 cm. The third side is a whole number of cm. What is the longest it can be?
Answer: 11
The third side must be smaller than 5 + 7 = 12 cm, and the biggest whole number below 12 is 11 cm.
Ravi checks the lengths 8 cm, 9 cm and 20 cm. He finds 8 < 9 + 20 and 9 < 8 + 20, so he says they make a triangle. What did he forget?
A triangle has two sides of 7 cm and 12 cm. The third side is a whole number of cm. What is the shortest it can be?
Answer: 6
The 12 cm side must be shorter than 7 + the third side. With 5 cm, 7 + 5 = 12 is only equal. With 6 cm, 7 + 6 = 13 is more than 12, so the shortest is 6 cm.
- 6 cm, 7 cm, 12 cm — correct. Yes! The longest side is 12 cm, and 6 + 7 = 13 is more than 12.
- 4 cm, 5 cm, 9 cm. Here 4 + 5 = 9, which is equal to the longest side, not more. The longest side must be shorter than the sum.
- 3 cm, 4 cm, 8 cm. The longest side is 8 cm but 3 + 4 = 7 is less than 8. The roundabout way would be shorter than the direct way.
- Nothing. Two checks are enough.. All three checks must pass, and the check that matters most is the longest side against the sum of the other two.
- He should have added all three lengths.. The check compares one side with the sum of the other two, not with the sum of all three.
- He did not check the longest side: 20 is not smaller than 8 + 9 = 17. — correct. Yes! The longest side is the one that can fail, and here it does. These lengths cannot make a triangle.