Worksheet: Areas of Similar Polygons

In this worksheet, we will practice finding an unknown length or scale factor when the areas are known or finding an unknown area or scale factor when the lengths are known.

Q1:

Given the following figure, find the area of a similar polygon 𝐴𝐵𝐶𝐷 in which 𝐴𝐵=6.

Q2:

Given the figure shown, determine the area of a similar polygon, 𝐴𝐵𝐶, in which 𝐴𝐵=3.

Q3:

Given the graph, determine the area of similar polygon 𝐴𝐵𝐶𝐷 in which 𝐵𝐶=6.

Q4:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 89. If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.

  • A 4 9
  • B 1 6 9
  • C 3 2 8 1
  • D 6 4 8 1
  • E 1 2 8 8 1

Q5:

Two corresponding sides of two similar polygons have lengths of 54 and 57 centimeters. Given that the area of the smaller polygon is 324 cm2, determine the area of the bigger polygon.

Q6:

Two similar polygons have areas of 20 in2 and 80 in2. Find the scale factor of the first polygon to the second.

  • A 1 5
  • B 1 2
  • C 1 4
  • D 4 1
  • E 2 1

Q7:

𝐴 𝐵 𝐶 𝐷 is a square where 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴 are divided by the points 𝑋, 𝑌, 𝑍, and 𝐿, respectively, by the ratio of 41. Find the ratio of the area of 𝑋𝑌𝑍𝐿 to that of 𝐴𝐵𝐶𝐷.

  • A 3 5
  • B 2 5 1 7
  • C 4 8 9 5 9 3
  • D 1 7 2 5
  • E 5 9 3 4 8 9

Q8:

In the given figure, 𝐻𝐴𝐷 is an equilateral triangle with a perimeter of 45 cm. Given that 𝐴𝐷𝐴𝐵=37, determine the area of rectangle 𝐴𝐵𝐶𝐷.

Q9:

Using the figure below, find the ratio between the area of the parallelogram 𝑋𝑌𝑍𝐿 and the area of the triangle 𝐴𝐵𝐶 in its simplest form.

  • A 5 3
  • B 1 2 5
  • C 5 6
  • D 2 2 1 5
  • E 2 1

Q10:

Given that 𝐴𝐷𝐷𝐶=32 and the area of 𝐴𝐵𝐶=695cm, find the area of trapezium 𝐷𝐶𝐵𝐸.

Q11:

If 𝐴𝐵𝐶𝑋𝑌𝑍 and 𝐴𝐵=95𝑋𝑌, find areaofareaof𝑋𝑌𝑍𝐴𝐵𝐶.

  • A 3 6 5
  • B 9 5
  • C 1 8 5
  • D 2 5 8 1

Q12:

𝐴 𝐵 𝐶 𝐷 is a parallelogram with 𝐴𝐵=9 and 𝐵𝐶=5. Let 𝑋 be a point on ray 𝐴𝐵 but not on the segment 𝐴𝐵, with 𝐵𝑋=18. Let 𝑌 be a point on ray 𝐶𝐵 but not on the segment 𝐶𝐵, with 𝐵𝑌=10. Let 𝑍 be a point so that 𝑋𝐵𝑌𝑍 is a parallelogram. If the area of 𝐴𝐵𝐶𝐷 is 39, what is the area of 𝑋𝐵𝑌𝑍?

  • A78
  • B9.75
  • C180
  • D156
  • E19.5

Q13:

Triangle 𝐴𝐵𝐶 is right angled at 𝐴, where 𝐴𝐵=20 and 𝐴𝐶=21. Suppose 𝐿, 𝑀, and 𝑁 are similar polygons on corresponding sides 𝐴𝐵, 𝐵𝐶, and 𝐴𝐶. If the area of 𝐿 is 145, what are the areas of 𝑀 and 𝑁 to the nearest hundredth?

  • Aarea of 𝑀=159.86, area of 𝑁=304.86
  • Barea of 𝑀=304.86, area of 𝑁=159.86
  • Carea of 𝑀=210.25, area of 𝑁=152.25
  • Darea of 𝑀=17.46, area of 𝑁=12.64

Q14:

Two corresponding sides of two similar polygons have lengths of 55 and 90 centimeters. Given that the area of the smaller polygon is 121 cm2, determine the area of the bigger polygon.

Q15:

Given the following figure, find the area of a similar polygon 𝐴𝐵𝐶𝐷 in which 𝐴𝐵=9.

Q16:

Given the following figure, find the area of a similar polygon 𝐴𝐵𝐶𝐷 in which 𝐴𝐵=6.

Q17:

Given the following figure, find the area of a similar polygon 𝐴𝐵𝐶𝐷 in which 𝐴𝐵=3.

Q18:

Given the following figure, find the area of a similar polygon 𝐴𝐵𝐶𝐷 in which 𝐴𝐵=2.

Q19:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 53. If the dimensions of each rectangle are tripled, find the ratio of the areas of the larger rectangles.

  • A 5 9
  • B 5 1
  • C 2 5 2 7
  • D 2 5 9
  • E 2 5 3

Q20:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 95. If the dimensions of each rectangle are tripled, find the ratio of the areas of the larger rectangles.

  • A 3 5
  • B 2 7 5
  • C 2 7 2 5
  • D 8 1 2 5
  • E 2 4 3 2 5

Q21:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 47. If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.

  • A 2 7
  • B 8 7
  • C 8 4 9
  • D 1 6 4 9
  • E 3 2 4 9

Q22:

Given the figure shown, determine the area of a similar polygon, 𝐴𝐵𝐶, in which 𝐴𝐵=6.

Q23:

A stage for a music festival was built in the shape of an equilateral triangle with an area of 117 m2. During rehearsals, it was discovered that the stage needed to be smaller so that technical equipment and more music fans could fit into the concert hall. They removed equilateral triangles from each of the corners of the original equilateral triangle, and the stage ended up as a regular hexagon. What was the area of the new hexagonal stage?

Q24:

In triangle 𝐴𝐵𝐶, let 𝐷 be a point on 𝐴𝐵 and 𝐸 a point on 𝐴𝐶. Suppose that 𝐴𝐷=43𝐵𝐷, and that 𝐷𝐸𝐵𝐶. If the area of 𝐴𝐷𝐸 is 48, what is the area of the trapezium 𝐷𝐵𝐶𝐸?

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