Area of a Triangle: Why It’s Half of Base × Height

The area of a triangle formula is A = ½ × b × h: half of the base times the height. The ½ is not a random rule. Every triangle is exactly half of a rectangle or parallelogram, and this post shows you why.

Why there is a ½

Draw a rectangle around a right triangle, using the base and the height as its sides. The diagonal cuts the rectangle into two identical triangles, so each one has half the area: ½ × b × h. For any other triangle, copy it, turn the copy upside down and join them: you get a parallelogram with area b × h, so one triangle is again ½ × b × h.

Which height do you use?

• The height must be perpendicular (at 90°) to the base you chose.
• It is not the slanted side, unless the triangle has a right angle there.
• In an obtuse triangle, the height can fall outside the triangle. Extend the base with a dotted line and measure to it.

Worked example

A triangle has a base of 10 cm and a height of 6 cm. Find its area.

• A = ½ × b × h
• A = ½ × 10 × 6
• A = ½ × 60 = 30 cm²

When you only know the three sides: Heron’s formula

If there is no height, use Heron’s formula. First find s, half the perimeter: s = (a + b + c) ÷ 2. Then A = √(s(s − a)(s − b)(s − c)).

• Sides 7, 8 and 9 cm
• s = (7 + 8 + 9) ÷ 2 = 12
• A = √(12 × 5 × 4 × 3) = √720
• A = 26.83 cm²

Area of a right triangle

In a right triangle, the two sides that meet at the right angle are already a base and a height, so you multiply them and halve the result. A 3-4-5 right triangle has an area of ½ × 3 × 4 = 6 square units. The longest side, the hypotenuse, is never used as the height.

Example in inches and feet

A triangular pennant is 9 inches wide along its top edge and 12 inches long from that edge to its point. Its area is ½ × 9 × 12 = 54 in².

A triangular corner of a garden has a base of 12 ft and a height of 5 ft, so its area is ½ × 12 × 5 = 30 ft². Always put both measurements in the same unit before you multiply. To change square inches into square feet, divide by 144, because 1 ft² is 12 in × 12 in.

Common mistakes

• Using a slanted side as the height. The height has to meet the base at 90°.
• Forgetting the ½. Base × height alone gives the area of the whole rectangle.
• Mixing units. A base in feet and a height in inches gives a wrong answer.
• Leaving out the square. Area is always in square units: cm², in² or ft².

All triangle formulas, including perimeter and Heron’s formula, are on the triangle formulas page.