Week 27 · Area of any triangle and of a trapezium · 3 lessons
1
Acute and obtuse triangles: where the height falls
Discover Mathematics P6: Term 2, Unit 12, Lessons 3–4
GoalFind the area of acute and obtuse triangles by choosing a base and its matching height.
New words
acute triangle · a triangle whose angles are all smaller than a right angleobtuse triangle · a triangle with one angle bigger than a right angleextend · to make a line longer
Explanation
The area of ANY triangle is base × height ÷ 2. The height is the perpendicular distance from a corner to the line that holds the base. In an acute triangle the height falls inside the triangle. In an obtuse triangle the height from the pointed top falls OUTSIDE: we extend the base with a dashed line and drop the perpendicular to the extension. The height is still measured at a right angle, and the area formula does not change. Any side can be the base, but you must use the height that goes with it. If you pick a different base you get a different height, and the area is the same.
★Remember
Area = base × height ÷ 2. The height is perpendicular to the base, even if it falls outside the triangle (extend the base).
Worked examples
Acute triangle: base 8 cm, height 5 cm (inside). Area = 8 × 5 ÷ 2 = 20 cm².
Obtuse triangle: base AB = 6 cm. The height from C falls outside, on the extension of AB, and is 3 cm. Area = 6 × 3 ÷ 2 = 9 cm². We use the 6 cm base, not the slanted side AC.
Two bases, one area. In a triangle, the height on the 10 cm side is 6 cm and the height on the 12 cm side is 5 cm. Area = 10 × 6 ÷ 2 = 30 and 12 × 5 ÷ 2 = 30 cm². The same.
Class activity
Heights outside. Draw an obtuse triangle on squared paper. Lay a ruler along the longest side and use a set square to draw the height from the opposite corner. Where does it land? Measure and find the area.
Practice
1
An acute triangle has base 10 cm and height 7 cm. Find its area.
10 × 7 ÷ 2 = 35 cm².
2
An obtuse triangle has base 12 m and height 4 m (outside). Find its area.
12 × 4 ÷ 2 = 24 m².
3
Is the height of an obtuse triangle always inside it? Explain.
No. From the top corner it falls outside, on the extension of the base.
4
A triangle has a side of 14 cm with height 6 cm on it, and a side of 12 cm. Find the area, then the height on the 12 cm side.
Area = 14 × 6 ÷ 2 = 42 cm². Height = 42 × 2 ÷ 12 = 7 cm.
5
Yara uses the slanted side 5 cm as the height of the obtuse triangle in the picture and finds 6 × 5 ÷ 2 = 15. What is the correct area?
The height is the 3 cm perpendicular: 6 × 3 ÷ 2 = 9 cm².
