Geometry

Pythagorean theorem

In a right triangle, the two shorter sides (legs) and the longest side (hypotenuse) are linked by one equation. München names Pythagoras for its T-Kurs test.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• a² + b² = c², where c is the hypotenuse opposite the right angle.
• Missing leg: a = √(c² − b²).
• Common triples save time: 3-4-5, 5-12-13, 8-15-17 and their multiples.
• Diagonal of a rectangle a × b: d = √(a² + b²).

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. A cuboid measures 4 cm × 20 cm × 22 cm. How long is its space diagonal (in cm)?

   • a31
   • b20
   • c30
   • d46
   Show solution

   Answer: 30

   Pythagoras twice: first the face diagonal, then with the third edge: d² = a² + b² + c².

   d² = 4² + 20² + 22² = 900 ⇒ d = 30.

2. 2. A right triangle has hypotenuse 25 and one leg 7. How long is the other leg?

   • a24
   • b25
   • c23
   • d18
   Show solution

   Answer: 24

   c² − a² = b².

   √(25² − 7²) = √576 = 24.

3. 3. A right triangle has legs 9 and 12. How long is the hypotenuse?

   • a16
   • b21
   • c15
   • d14
   Show solution

   Answer: 15

   a² + b² = c².

   √(9² + 12²) = √225 = 15.

4. 4. An isosceles triangle has two legs of 10 cm and a base of 12 cm. What is its area (in cm²)?

   • a60
   • b96
   • c120
   • d48
   Show solution

   Answer: 48

   The height to the base halves it; Pythagoras in the half triangle gives the height.

   h² = 10² − 6² = 64 ⇒ h = 8; A = ½ · 12 · 8 = 48.

5. 5. A right triangle has hypotenuse 17 and one leg 8. How long is the other leg?

   • a15
   • b16
   • c14
   • d9
   Show solution

   Answer: 15

   c² − a² = b².

   √(17² − 8²) = √225 = 15.

6. 6. A right triangle has hypotenuse 13 and one leg 5. How long is the other leg?

   • a12
   • b13
   • c8
   • d11
   Show solution

   Answer: 12

   c² − a² = b².

   √(13² − 5²) = √144 = 12.

Common mistakes

• Using the hypotenuse as a leg.
• Adding instead of subtracting when finding a leg.
• Forgetting to take the square root at the end.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs