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Determine the total shaded area of the 'kissing triangles'.


Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.


Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Isometric Areas

Age 11 to 14
Challenge Level

Why do this problem?

This problem invites students to look at area in a slightly different way from usual, using a triangle as the basic unit of area rather than a square, to reinforce the concept that area is about the space enclosed within a shape.

Along the way students have the opportunity to derive and justify a formula that they will not have met before.


Possible approach

Display an isometric dotty grid, and create a shape. This dotty grid environment might be useful, and you can print off isometric paper for students.
"On squared paper, it's easy to work out area by counting squares, but on isometric paper it's a bit different. If we want to compare areas of shapes drawn on isometric paper, we can use a small triangle as our unit of area, instead of a square!"

Create some simple shapes and work out the area as a class, perhaps inviting students to write their answers on mini whiteboards. Once everyone is confident at using the triangular unit of area, show students this slide with the image from the problem. "For each parallelogram, can you find its area, and then find a relationship between the length of the sides of the parallelogram and its area, measured in triangular units?"

Give students some time to work in pairs on the problem, sharing any useful strategies and conjectures that they come up with. Then, once students begin to spot a relationship, invite them to consider why the relationship might occur and how they can justify it.

Students who finish quickly could be challenged to draw some trapeziums (in which all four lengths are whole numbers) and come up with a relationship between the side lengths and the area in triangular units.

Finish off by bringing the class together to share their justifications for the area rule they have found. A follow-up lesson could look at More Isometric Areas.


Key questions

How many triangles are there in a parallelogram with side lengths 1 and 1? 2 and 1? 3 and 1?...
If you double one of the lengths, what would happen to the area?
If you double both of the lengths, what would happen to the area?

Can you work out a formula for the area of parallelograms in triangular units?


Possible support

Torn Shapes works with areas based on counting squares, so would be a useful task for students who are not confident with area.


Possible extension

More Isometric Areas invites students to consider the area of triangles using triangular units.
Of All the Areas looks at the area of equilateral triangles with sides that are not whole numbers.