Half a square: the 45° triangle

The idea

A right triangle with a 45° angle has equal legs a and hypotenuse a√2, so sin 45° = cos 45° = 1/√2, tan 45° = cot 45° = 1 and cosec 45° = sec 45° = √2.

A square cut in half

Take a square ABCD with side a. Cut it along the diagonal AC. You get a right triangle ABC with the angle at B equal to 90°.

The diagonal splits each corner angle of the square in half. So the angle at A is 45° and the angle at C is 45°. The two legs are both equal to a.

The legs are a and a. What is the hypotenuse AC, from Pythagoras?

• 2a
• a√2
• 2a²

Cut and read

Cut the square. Pick the length of the diagonal AC. Then tap each ratio to see how it is worked out. Change a and watch what stays the same.

The 45° triangle

A right triangle with a 45° angle has two equal legs a and a hypotenuse a√2.

sin 45° = a / a√2 = 1/√2

cos 45° = a / a√2 = 1/√2

tan 45° = a / a = 1

So cosec 45° = √2, sec 45° = √2 and cot 45° = 1. The size a cancels, so these values are the same for every square.

Notes

In the 45° right triangle the legs are equal (a, a) and the hypotenuse is a√2. So sin 45° = cos 45° = 1/√2, tan 45° = cot 45° = 1, cosec 45° = sec 45° = √2.

Check yourself

A square has side 5 cm. How long is its diagonal?

What is the value of tan 45°?

Answer: 1

tan 45° = a / a = 1.

Give sin 45° as a decimal, to two places.

Answer: 0.71

sin 45° = 1/√2 ≈ 0.71.

In a right triangle with a 45° angle, which pair is always equal?

• 10 cm. That adds the two sides. Pythagoras adds the squares: 25 + 25 = 50.
• 5√2 cm — correct. Yes! AC² = 5² + 5² = 50, so AC = √50 = 5√2 cm.
• 50 cm. 50 is AC squared. Take the square root.
• sin 45° and cos 45° — correct. Yes! Both are a / a√2 = 1/√2, because the two legs are equal.
• sin 45° and tan 45°. sin 45° = 1/√2 but tan 45° = 1. They are different.
• sec 45° and cot 45°. sec 45° = √2 but cot 45° = 1. They are different.