Triangle

Properties

• The three angles add up to 180°.
• The sum of any two sides is greater than the third side.
• The longest side is opposite the largest angle.
• By sides, a triangle is scalene, isosceles or equilateral. By angles, it is acute, right or obtuse.

Why the formulas work

Draw a rectangle with the same base and height as the triangle. The triangle fills exactly half of it, which you can see by cutting along the height and matching the two small triangles with the two spare pieces. So the area is half of base times height. When the height is not known but all three sides are, Heron's formula gives the area from the sides alone.

Worked example

Take a triangle with base 14 cm and height 9 cm.

1. Area. Area = ½ × b × h. Here: ½ × 14 × 9 = 63 cm².

When only the three sides are known: Heron's formula

• Perimeter = a + b + c
• Area = √(s(s − a)(s − b)(s − c)), where s is half the perimeter

Here a, b and c are the sides and s is half of their sum. The formula is named after Heron of Alexandria. It needs no height, which makes it the natural choice for a plot of land measured along its three boundaries.

Take a triangle with side a 13 cm, side b 14 cm and side c 15 cm. First, half the perimeter s = 21.

1. Perimeter. Perimeter = a + b + c. Here: 13 + 14 + 15 = 42 cm.
2. Area (Heron). Area = √(s(s − a)(s − b)(s − c)), where s is half the perimeter. Here: √(21 × (21 − 13) × (21 − 14) × (21 − 15)) = 84 cm².

Finding a missing measurement

Height = 2 × area ÷ base. A triangle of area 63 cm² on a base of 14 cm has a height of 9 cm. In the same way, base = 2 × area ÷ height.

Mistakes to avoid

• Using a slanting side as the height. The height must meet the base at a right angle.
• Forgetting the half. Base times height is the area of the surrounding parallelogram.
• In an obtuse triangle the height may fall outside the triangle. The formula still works.

Where you see it

Roof trusses, bridge frames, road warning signs, set squares and samosas are triangles. Engineers use triangles because a triangle cannot be pushed out of shape without bending a side.

Try these

Work each one out, then tap to check.

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