Similar polygons: splitting into similar triangles, the area ratio k², farms, scale drawings and maps
Mathematics Sec 1 Term 1, Unit 2, Lesson 3
GoalUse the area ratio k² for similar polygons, scale drawings and maps.
New words
area scale · the scale for areas, the square of the length scalehectare · a unit of land area: 1 hectare = 10 000 m²
Explanation
Two similar polygons can be cut by their diagonals from one vertex into the same number of triangles, n − 2 for an n-sided polygon, and the triangles in the first polygon are similar to the matching triangles in the second with the same ratio k. So the area of every triangle is multiplied by k², and the whole area too: the ratio of the areas of two similar polygons is k², where k is the ratio of any two corresponding sides (and also of the perimeters). This is what happens with scale drawings. If a map has the scale 1 cm : 5 km, then 1 cm² on the map stands for 5 × 5 = 25 km². A model of a building at 1 : 50 has a floor area 50² = 2500 times smaller than the real one. Convert all lengths to one unit first, and square the factor, not the area.
Length scale
1 : 2
1 : 50
1 : 200
1 : 1000
Area scale (squared)
1 : 4
1 : 2500
1 : 40 000
1 : 1 000 000
Worked examples
A map has the scale 1 cm : 5 km. A lake covers 6.4 cm² on the map. 1 cm² stands for 5 × 5 = 25 km², so the real area is 6.4 × 25 = 160 km².
Two similar fields have sides in the ratio 5 : 3 and the difference of their areas is 48 hectares. The areas are in the ratio 25 : 9, so the difference is 25 − 9 = 16 parts. One part = 48 ÷ 16 = 3 hectares. The fields have 25 × 3 = 75 hectares and 9 × 3 = 27 hectares. Check: 75 − 27 = 48 ✓.
The perimeters of two similar polygons are in the ratio 2 : 5 and the total area of both is 580 m². The areas are in the ratio 4 : 25, so 4x + 25x = 580, x = 20. The areas are 80 m² and 500 m².
A model of a school building at the scale 1 : 50 has a floor of 0.2 m². The real floor is 0.2 × 50² = 0.2 × 2500 = 500 m².
Class activity
Scale a drawing. On squared paper draw a rectangle 2 squares by 3 squares. Draw a similar one with k = 3. Count the squares in both and compare with 3².
Practice
1
Two similar polygons have k = 2/5 and the smaller has area 36 cm². Find the area of the larger.
The area ratio is 4/25, so the larger is 36 × 25/4 = 225 cm².
2
A map has the scale 1 : 100 000. A park covers 12 cm² on it. Find its real area in km².
1 cm stands for 100 000 cm = 1 km, so 1 cm² stands for 1 km². The park is 12 km².
3
A photo 10 cm × 15 cm is enlarged to 20 cm × 30 cm. How many times bigger is its area?
k = 2, so the area becomes 2² = 4 times bigger (150 cm² → 600 cm²).
4
Two similar fields have areas 200 m² and 450 m². The fence of the smaller is 80 m. Find the fence of the larger.
Area ratio 200 : 450 = 4 : 9, so k = 2 : 3. The fence is 80 × 3/2 = 120 m.
5
Two regular hexagons have sides 4 cm and 6 cm. What is the ratio of their areas?
All regular hexagons are similar, k = 4/6 = 2/3, so the areas are in the ratio 4 : 9.
