Similar figures drills

Use similarity and scale factors to find missing lengths in similar triangles/polygons.

Free, timed SHSAT practice sets on similar figures— every question includes a full worked explanation, and there's no account required.

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How to solve similar figures problems

Similar figures have the same shape but a different size. Every length is multiplied by the same scale factor \(k\).

  1. Find \(k\) from one pair of matching sides: new \(\div\) original. If a side of 4 matches a side of 6, then \(k = \dfrac{6}{4} = \dfrac{3}{2}\).
  2. Multiply any other length by \(k\): a side of 10 becomes \(10 \times \dfrac{3}{2} = 15\). Perimeter scales by \(k\) too.
  3. Area scales by \(k^2\) and volume by \(k^3\). With \(k = \dfrac{3}{2}\), area is multiplied by \(\dfrac{9}{4}\).

For shadow problems, set up a proportion: \(\dfrac{\text{height}}{\text{shadow}} = \dfrac{\text{height}}{\text{shadow}}\).

Common traps: flipping the ratio (check that the bigger figure gets the bigger number), using \(k\) for area instead of \(k^2\), and matching sides that do not correspond.

Worked examples

Example 1.
Two figures are similar. A side of 15 m in the smaller figure corresponds to a side of 20 m in the larger figure. If a different side of the smaller figure is 21 m, what is the length of the corresponding side in the larger figure?
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Answer:
A

Scale factor \(= 20 \div 15 = 4/3\). Corresponding side \(= 21 \times 4/3 = 28\) m. (Dividing by the scale factor instead of multiplying gives 15.75 m. Adding the difference \(20 - 15 = 5\) instead of multiplying gives 26 m. Matching the wrong sides, \(15 \times 20 \div 21\), gives about 14.29 m.)

Example 2.
A person 5 ft tall casts a shadow 8 ft long. At the same time, a nearby tree casts a shadow 16 ft long. Using similar triangles, how tall is the tree?
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Answer:
C

Height and shadow are in the same ratio for both: \(\dfrac{\text{tree}}{16} = \dfrac{5}{8}\), so the tree \(= 5 \times (16 \div 8) = 10\) ft. (Multiplying by \(8 \div 16\) instead gives 2.5 ft. Adding the shadow difference \(16 - 8\) to 5 gives 13 ft. Flipping only one side of the proportion, \(16 \times 8 \div 5\), gives 25.6 ft.)

Example 3.
Two triangles are similar. The ratio of their corresponding sides is 1 : 3. The larger triangle has an area of 54 square cm. What is the area of the smaller triangle?
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Answer:
A

Areas of similar figures are in the ratio of the squares of the sides, \(1 : 9\). Smaller area \(= 54 \times 1/9 = 6\) square cm. (Using the side ratio 1 : 3 without squaring gives 18 square cm. Multiplying by \(9/1\) instead of \(1/9\) gives 486 square cm. Multiplying by \(3/1\) (flipped and not squared) gives 162 square cm.)