Area of any polygon

The idea

Any polygon can be divided into triangles, so its area is found by adding the areas of those triangles.

A crooked field

A field has four straight sides, but it is not a rectangle: no two sides are parallel and no corner is a right angle. We know how to find the area of a rectangle and of a triangle.

How can we find the area of the field?

• Multiply two of its sides
• Cut it into triangles by joining two opposite corners, and add their areas
• Add the lengths of its four sides

Cut into triangles

Tap one corner, then another, to draw a line inside the shape. When the shape is cut into triangles, the toy writes the area of each and adds them. Cut every shape two different ways.

Cut and add

Any polygon can be divided into triangles, so its area is found by adding the areas of those triangles.

The quadrilateral was cut by one line into 2 triangles: 14 + 11 = 25 one way and 12 + 13 = 25 the other way. The pentagon (area 30) needed 2 lines and made 3 triangles, and the L-shaped hexagon (area 27) needed 3 lines and made 4 triangles.

Every way of cutting gave the same total. That total is the area of the shape, so we can choose the cut that gives triangles with a base and a height that are easy to find.

Look at the count: 4 sides made 2 triangles, 5 sides made 3, 6 sides made 4. Each extra side adds one more triangle.

Notes

Any polygon can be divided into triangles, so its area is found by adding the areas of those triangles.

Check yourself

This house-shaped plot has a rectangle 8 m wide and 3 m tall under a triangular roof 3 m high. What is the area of the whole plot, in m²?

Answer: 36

Rectangle: 8 × 3 = 24. Triangle: ½ × 8 × 3 = 12. Total: 24 + 12 = 36 m².

Anil cuts a field into three triangles of 10 m², 12 m² and 14 m². Bina cuts the same field in a different way. What could the total of Bina's triangles be?

The diagonal AC of the field ABCD is 10 m long. Corner B is 4 m from AC and corner D is 6 m from AC, on the other side. What is the area of the field, in m²?

Answer: 50

Triangle ABC = ½ × 10 × 4 = 20 m². Triangle ACD = ½ × 10 × 6 = 30 m². Total = 50 m².

A 4-sided shape made 2 triangles, a 5-sided one made 3 and a 6-sided one made 4. Following the pattern, an 8-sided polygon is cut into how many triangles?

• 32 m². Bina's triangles also fill the whole field, and Anil's total is 10 + 12 + 14 = 36 m².
• 34 m². The total must be the area of the field. Anil's triangles add up to 10 + 12 + 14 = 36 m², so Bina's do too.
• 36 m² — correct. Yes! Both sets of triangles fill the same field, so both add up to the area of the field: 10 + 12 + 14 = 36 m².
• 38 m². Bina's triangles fill the same field, so they cannot add up to more than Anil's 36 m².
• 5. That is the number for 7 sides. Count on: 4 sides 2, 5 sides 3, 6 sides 4, 7 sides 5, 8 sides 6.
• 6 — correct. Yes! Each extra side adds one triangle: 4 sides 2, 5 sides 3, 6 sides 4, 7 sides 5, 8 sides 6.
• 7. Always two fewer triangles than sides, so 8 sides gives 6, not 7.
• 8. There are two fewer triangles than sides: 4 sides made 2 triangles, not 4.