Area of a triangle

The idea

Area of a triangle = 1/2 × base × height, since it is half of the rectangle enclosing it with those sidelengths; written as a difference of two such triangles, the formula holds for all types of triangles.

Two corners

Two identical rectangles stand side by side. In each one, a triangle is drawn on the bottom side. The top corner of the triangle is X in the first rectangle and Y in the second.

Which triangle has the greater area, X or Y?

• Triangle X
• Triangle Y
• Both have the same area

Drag the corner

Drag the top corner A, or use the buttons. When A is over the base, the line from A cuts the rectangle in two. Make the three triangles on the list.

Half of the rectangle

Area of a triangle = ½ × base × height, since it is half of the rectangle enclosing it with those sidelengths. Written as a difference of two such triangles, the formula holds for all types of triangles.

With A over the base, the line from A cuts the rectangle into two smaller rectangles. In each one the triangle fills exactly half, because the yellow left-over piece is a copy of the triangle piece beside it.

So the triangle is half of 6 × 5 = 30: ½ × 6 × 5 = 15 wherever A is on the top side. That is why X and Y have equal areas.

When A is beyond the base, no rectangle with BC as a side fits round the triangle. But the big triangle ADC and the small triangle ADB each fit in a rectangle, and ABC is what is left when ADB is taken away. For the areas: ABC = ADC − ADB = ½ × h × DC − ½ × h × DB = ½ × h × (DC − DB) = ½ × h × BC

For example, with DC = 9, DB = 3 and h = 4: 18 − 6 = 12, and ½ × 4 × 6 = 12.

Notes

Area of a triangle = ½ × base × height, since it is half of the rectangle enclosing it with those sidelengths. Written as a difference of two such triangles, the formula holds for all types of triangles.

Check yourself

A triangle has a base of 14 cm and a height of 9 cm. What is its area, in cm²?

Answer: 63

Area = ½ × 14 × 9 = 63 cm².

Triangle ABC has A beyond the base BC. The dotted line AD is its height. Which gives the area of ABC?

A triangle has a right angle between its sides of 6 cm and 8 cm. Its longest side is 10 cm. How long is the height that falls on the 10 cm side, in cm?

Answer: 4.8

Area = ½ × 6 × 8 = 24 cm². Also area = ½ × 10 × height. So 5 × height = 24, and height = 24 ÷ 5 = 4.8 cm.

Triangles P and Q stand on the same base of 6 units and both are 5 units tall. P has its top corner over the middle of the base and Q has its top corner far to the right. Which has the greater area?

• Area (ADC) + Area (ADB). ADB is a part of ADC, so adding them counts that part twice. ABC is what is left when ADB is taken away from ADC.
• ½ × BC × AD — correct. Yes! ADC − ADB = ½ × AD × DC − ½ × AD × DB = ½ × AD × (DC − DB) = ½ × AD × BC.
• ½ × AB × AC. AB and AC are two slanting sides. The formula needs the base and the height at right angles to it.
• The formula does not work, because no rectangle with BC as a side fits round ABC. It does work. Take the difference of the big triangle ADC and the small triangle ADB: each fits in a rectangle.
• P, its corner is over the base. The area does not depend on where the corner is along the top. It depends only on the base and the height: ½ × 6 × 5 = 15 for both.
• Q, it is stretched out wider. Q looks bigger, but the area is ½ × base × height = ½ × 6 × 5 = 15, the same as P.
• Both have the same area — correct. Yes! Same base 6 and same height 5 give ½ × 6 × 5 = 15 for both.