Perimeter & Area

Depth · one concept from first principles · step 2 of 4

Count how many unit shapes make the whole

Two congruent equilateral triangles are overlapped so that the shared part is a regular hexagon, making the star shown at the right. If the whole star covers 715 cm27\frac{1}{5}\ \text{cm}^2, what is the area of the shaded part, in cm2\text{cm}^2?

GeometryFractions
Your answer
How to solve
Strategy Draw a Diagram — Find a unit shape the whole figure is made of. Once the star is 12 copies of one small triangle, both the whole and the shaded part are just counts of that unit, and the awkward mixed number only has to be divided once.
1STEP 1

Cut the star into one repeated shape

The dotted lines cut the star into 12 congruent small triangles: 6 fill the hexagon in the middle and the other 6 are the points.

6 + 6 = 12
2STEP 2

Find the area of one small triangle

The 12 triangles are congruent, so each takes an equal share of the star's area. Write the mixed number as 36/5 and divide by 12.

36/5 ÷ 12 = 3/5
3STEP 3

Count the shaded triangles and multiply

The shaded part is three of the points, and each point is exactly one small triangle, so multiply by 3.

3/5 × 3 = 9/5
Answer
1451\frac{4}{5} cm²
Three of the twelve triangles is a quarter of the star, and a quarter of 36/5 is 9/5 -- the same answer by a different route.
Takeaway

Find the one small shape a figure repeats, and every area in the picture becomes a count.

  • Cut the star into one repeated shape
  • Find the area of one small triangle
  • Count the shaded triangles and multiply
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Triangle height depends on chosen baseUse shared base to find composite area

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▶ Practice — 12 problems