The idea, step by step
Use the angle, not the appearance
Pythagoras’ theorem applies to a right-angled triangle: one interior angle is exactly 90 degrees. The hypotenuse is the side opposite that angle and is the longest side. The two sides meeting at the right angle are called legs.
A right-angle square on a diagram is information. A triangle that merely looks right-angled is not enough, and rotating a diagram does not change which side is the hypotenuse.
Name the sides before calculating
Letters a and b can name the two legs in either order. Use c for the hypotenuse when writing the familiar formula. The equation relates squared lengths, all expressed in the same unit.
Mark the right angle and identify the opposite side first. This prevents using a shorter side in the hypotenuse position.
A common trap
Choosing the sloping side as the hypotenuse in every drawing.
Instead: Choose the side opposite the right angle; in a rotated drawing it may be horizontal or vertical.
See the thinking
Read the angle marker
The legs in the supplied right triangle are 6 cm and 8 cm. Which side is the hypotenuse?
- The 6 cm and 8 cm sides meet at the right angle.
- The remaining side lies opposite that angle.
- It must be longer than either leg.
Do not trust a sketch
A triangle has side lengths 5 cm, 6 cm and 8 cm. Can you assume it is right-angled because 8 is the longest side?
- Every triangle has a longest side.
- A right triangle must also satisfy the square-length relationship.
- Here , while .
Check your understanding
The right angle is between sides p and q. Which side is opposite it?
Show answer
The third side, the hypotenuse.
The legs meet at the right angle; the hypotenuse does not.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- In the supplied triangle, identify its two legs and its hypotenuse.2 marks
- A right triangle has legs a and b and hypotenuse c. Write its squared-length relationship.2 marks
- A right angle is opposite side AB in triangle ABC. Which side is the hypotenuse?2 marks
- Could a right triangle have hypotenuse 7 cm and a leg of 9 cm? Explain.2 marks
- Can you apply the theorem to a triangle known only to have an angle of 60 degrees?2 marks
- The same right triangle is turned upside down. Does its hypotenuse change? Explain.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- In the supplied triangle, identify its two legs and its hypotenuse.2 marks
The labelled legs meet at the square marker.
The side across from the marker is h.
Legs 5 cm and 12 cm; hypotenuse h. - A right triangle has legs a and b and hypotenuse c. Write its squared-length relationship.2 marks
Square each leg length and add.
The result is the square of the hypotenuse.
. - A right angle is opposite side AB in triangle ABC. Which side is the hypotenuse?2 marks
The hypotenuse is opposite the right angle.
The right angle is therefore at C.
AB. - Could a right triangle have hypotenuse 7 cm and a leg of 9 cm? Explain.2 marks
The hypotenuse must be the longest side.
A 9 cm leg would be longer than 7 cm.
No. - Can you apply the theorem to a triangle known only to have an angle of 60 degrees?2 marks
A 60-degree angle does not establish a right angle.
You need a right angle or enough information to test for one.
Not from that information alone. - The same right triangle is turned upside down. Does its hypotenuse change? Explain.2 marks
Rotation keeps side lengths and angles unchanged.
The same side remains opposite the right angle.
No.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics (2026)
Hypotenuse and right-angled triangles; why the theorem works; finding the hypotenuse; finding a shorter side; exact roots versus rounded answers; checking whether a triangle is right-angled; practical length problems.
Official curriculum source ↗