Problem
Count how many unit shapes make the whole
Two congruent equilateral triangles overlap into a six-pointed star whose total area is known. Three of the six points are shaded, and we need their combined area.
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 divide the star into 12 congruent small equilateral triangles: 6 of them fill the hexagon in the middle and the other 6 are the points.
6 + 6 = 12
One shape, repeated twelve times.
6.G.A.1Draw A Diagram2STEP 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
Each small triangle is 3/5 cm².
6.G.A.1Identify SubproblemsEach of the twelve small triangles covers .
Why?
The twelve small triangles fill the star with no gaps and no overlaps, so their areas add back to the star's area.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
They are all the same size and shape, so each covers the same amount and twelve equal amounts make the whole.
Why?
One can be slid, turned or flipped exactly onto another, and moving a shape does not change how much it covers.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
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
The shaded area is 9/5 = 1 4/5 cm².
5.NF.B.4Look For A PatternAnswer
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