Lesson Worksheet: Areas of Similar Polygons Mathematics

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 𝐴′𝐡′𝐢′𝐷′ in which 𝐴′𝐡′=6.

Q2:

Given the figure shown, determine the area of a similar polygon, 𝐴′𝐡′𝐢′, in which 𝐴′𝐡′=3.

Q3:

Given the graph, determine the area of similar polygon 𝐴′𝐡′𝐢′𝐷′ in which 𝐡′𝐢′=6.

Q4:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 8∢9. If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.

  • A4∢9
  • B16∢9
  • C32∢81
  • D64∢81
  • E128∢81

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.

  • A1∢5
  • B1∢2
  • C1∢4
  • D4∢1
  • E2∢1

Q7:

𝐴𝐡𝐢𝐷 is a square where 𝐴𝐡, 𝐡𝐢, 𝐢𝐷, and 𝐷𝐴 are divided by the points 𝑋, π‘Œ, 𝑍, and 𝐿, respectively, by the ratio of 4∢1. Find the ratio of the area of π‘‹π‘Œπ‘πΏ to that of 𝐴𝐡𝐢𝐷.

  • A3∢5
  • B25∢17
  • C489∢593
  • D17∢25
  • E593∢489

Q8:

Given that 𝐴𝐷𝐷𝐢=32 and the area of △𝐴𝐡𝐢=695cm, find the area of trapezoid 𝐷𝐢𝐡𝐸.

Q9:

If β–³π΄π΅πΆβˆΌβ–³π‘‹π‘Œπ‘ and 𝐴𝐡=95π‘‹π‘Œ, find areaofareaofπ‘‹π‘Œπ‘π΄π΅πΆ.

  • A365
  • B95
  • C185
  • D2581

Q10:

𝐴𝐡𝐢𝐷 is a parallelogram with 𝐴𝐡=9 and 𝐡𝐢=5. Let 𝑋 be a point on ray 𝐴𝐡 but not on the segment 𝐴𝐡, with 𝐡𝑋=18. Let π‘Œ be a point on ray οƒͺ𝐢𝐡 but not on the segment 𝐢𝐡, with π΅π‘Œ=10. Let 𝑍 be a point so that π‘‹π΅π‘Œπ‘ is a parallelogram. If the area of 𝐴𝐡𝐢𝐷 is 39, what is the area of π‘‹π΅π‘Œπ‘?

Q11:

Triangle 𝐴𝐡𝐢 is right angled at 𝐴, where 𝐴𝐡=20 and 𝐴𝐢=21. Suppose 𝐿, 𝑀, and 𝑁 are similar polygons on corresponding sides 𝐴𝐡, 𝐡𝐢, and 𝐴𝐢. 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

Q12:

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.

Q13:

Given the following figure, find the area of a similar polygon 𝐴′𝐡′𝐢′𝐷′ in which 𝐴′𝐡′=9.

Q14:

Given the following figure, find the area of a similar polygon 𝐴′𝐡′𝐢′𝐷′ in which 𝐴′𝐡′=6.

Q15:

Given the following figure, find the area of a similar polygon 𝐴′𝐡′𝐢′𝐷′ in which 𝐴′𝐡′=3.

Q16:

Given the following figure, find the area of a similar polygon 𝐴′𝐡′𝐢′𝐷′ in which 𝐴′𝐡′=2.

Q17:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 5∢3. If the dimensions of each rectangle are tripled, find the ratio of the areas of the larger rectangles.

  • A5∢9
  • B5∢1
  • C25∢27
  • D25∢9
  • E25∢3

Q18:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 9∢5. If the dimensions of each rectangle are tripled, find the ratio of the areas of the larger rectangles.

  • A3∢5
  • B27∢5
  • C27∢25
  • D81∢25
  • E243∢25

Q19:

Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀 with their sides having a ratio of 4∢7. If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.

  • A2∢7
  • B8∢7
  • C8∢49
  • D16∢49
  • E32∢49

Q20:

Given the figure shown, determine the area of a similar polygon, 𝐴′𝐡′𝐢′, in which 𝐴′𝐡′=6.

Q21:

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?

Q22:

In triangle 𝐴𝐡𝐢, let 𝐷 be a point on 𝐴𝐡 and 𝐸 a point on 𝐴𝐢. Suppose that 𝐴𝐷=43𝐡𝐷 and that 𝐷𝐸βˆ₯𝐡𝐢. If the area of △𝐴𝐷𝐸 is 48, what is the area of the trapezoid 𝐷𝐡𝐢𝐸?

Q23:

These two rectangles are similar. Given that the area of the yellow one is 69.3 cm2, find the area of the green rectangle.

Q24:

Given that 𝐴𝐷𝐷𝐢=37 and the area of △𝐴𝐡𝐢=484cm, find the area of △𝐴𝐷𝐸.

Q25:

π΄π΅πΆπ·πΈβˆΌπΉπΊπ»πΎπ‘€ where 𝐴𝐢=46cm and 𝐹𝐻=11.5cm. If the area of 𝐴𝐡𝐢𝐷𝐸=3,036cm, what is the area of 𝐹𝐺𝐻𝐾𝑀?

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