Triangles

Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

$$a^2 + b^2 = c^2$$

Triangle angle sum

The interior angles of a triangle add to 180 degrees.

$$A + B + C = 180^\circ$$

• Triangle inequality: each side is shorter than the sum of the other two.
• Area $= \frac{1}{2} \times \text{base} \times \text{height}$, with the height perpendicular to the chosen base.
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.
• Similar triangles: if sides scale by $k$, areas scale by $k^2$.

• Using the theorem on a triangle that is not right-angled.
• Treating a leg as the hypotenuse.
• Using a slanted side as the height in the area formula.

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1. 1.Find the hypotenuse of a right triangle with legs 9 and 12.

   Show answer

   $15$

2. 2.Two angles of a triangle are $47^\circ$ and $68^\circ$. Find the third.

   Show answer

   $65^\circ$

3. 3.Find the area of a triangle with base 14 and height 9.

   Show answer

   $63$

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