What are similar figures
Read,3 minutes
Similar figures have the same shape. Their corresponding angles are equal, and their corresponding side lengths are proportional.
The number that multiplies one figure to get the matching lengths in another figure is the scale factor. If a side grows from \(4\) cm to \(10\) cm, the scale factor is \(10 \div 4 = 2.5\).
For similar figures, lengths scale by the scale factor, areas scale by the square of the scale factor, and volumes scale by the cube of the scale factor. Most ACT-style problems start by matching corresponding sides and writing a proportion.
Similar Figures
Think of this lesson as more than a rule to memorize. Similar Figures is about comparisons, scaling, and equal ratios. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
A ratio compares quantities, and a proportion says two ratios are equal. Cross products help because \(\frac{a}{b}=\frac{c}{d}\) implies \(ad=bc\).
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Read what is given and what is being asked.
- Choose the rule that connects them.
- Substitute carefully and simplify in small steps.
- Check the final answer against the original question.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Similar Figures
1) Two rectangles are similar. The first is \(4\) cm wide and \(6\) cm long. The second is \(8\) cm wide. Find its length.
2) A triangle has sides \(3,4,5\). A similar triangle has shortest side \(9\). Find the other two sides.
3) Two similar figures have scale factor \(5\) from small to large. A small side is \(7\) inches. Find the matching large side.
4) Two similar polygons have matching sides \(10\) and \(25\). Find the scale factor from the smaller polygon to the larger polygon.
5) A photo is \(5\) inches by \(7\) inches. It is enlarged so the shorter side is \(15\) inches. Find the longer side.
6) A building casts a \(30\)-foot shadow while a \(6\)-foot person casts a \(4\)-foot shadow. Find the building height.
7) A map scale is \(1\) inch to \(12\) miles. Two cities are \(3.5\) inches apart on the map. Find the actual distance.
8) Two similar triangles have corresponding sides \(12\) and \(18\). If another side in the smaller triangle is \(20\), find the matching larger side.
9) A rectangle \(9\) cm by \(12\) cm is similar to a rectangle with width \(15\) cm. Find the length.
10) A model has length \(8\) inches and height \(3\) inches. The real object is \(56\) inches long. Find the real height.
11) Determine whether sides \(6,10,14\) and \(9,15,21\) could be corresponding sides of similar triangles.
12) Determine whether sides \(8,12,18\) and \(12,18,30\) could be corresponding sides of similar triangles.
13) Two similar pentagons have perimeters \(36\) cm and \(54\) cm. A side of the smaller pentagon is \(8\) cm. Find the matching larger side.
14) A small triangle has area \(20\) square units. A similar large triangle has scale factor \(3\). Find the large area.
15) A similar figure has scale factor \(\frac{2}{5}\) from original to image. An original side is \(35\) cm. Find the image side.
16) Two similar solids have linear scale factor \(4\) from small to large. The small volume is \(10\) cubic units. Find the large volume.
17) A ladder and a wall form a right triangle with height \(12\) ft and ground distance \(5\) ft. A similar drawing has ground distance \(2.5\) in. Find the drawing height.
18) A blueprint scale is \(\frac{1}{4}\) inch to \(3\) feet. A room is \(2.5\) inches long on the blueprint. Find the real length.
19) Two similar cylinders have radii \(3\) cm and \(9\) cm. The smaller cylinder has surface area \(48\pi\) square cm. Find the larger surface area.
20) A triangle with sides \(x, x+4, 20\) is similar to a triangle with corresponding sides \(9, 15, 30\). Find \(x\).
1) The scale factor from the first rectangle to the second is \(8 \div 4=2\). Multiply the matching length by \(2\): \(6\cdot 2=12\). The second length is \(\color{red}{12}\) cm.
2) The scale factor is \(9 \div 3=3\). Multiply each matching side by \(3\): \(4\cdot 3=12\) and \(5\cdot 3=15\). The other sides are \(\color{red}{12}\) and \(\color{red}{15}\).
3) A scale factor of \(5\) means multiply small lengths by \(5\). The matching side is \(7\cdot 5=35\). The large side is \(\color{red}{35}\) inches.
4) Divide the larger matching side by the smaller matching side: \(25 \div 10=2.5\). The scale factor is \(\color{red}{2.5}\).
5) The scale factor is \(15 \div 5=3\). Multiply the longer side by \(3\): \(7\cdot 3=21\). The longer side is \(\color{red}{21}\) inches.
6) Similar triangles give \(\frac{h}{30}=\frac{6}{4}\). Cross multiply: \(4h=180\). Divide by \(4\): \(h=45\). The building is \(\color{red}{45}\) feet tall.
7) Use the scale factor \(12\) miles per inch. Multiply: \(3.5\cdot 12=42\). The actual distance is \(\color{red}{42}\) miles.
8) The scale factor from smaller to larger is \(18 \div 12=1.5\). Multiply the other small side: \(20\cdot 1.5=30\). The matching larger side is \(\color{red}{30}\).
9) The scale factor is \(15 \div 9=\frac{5}{3}\). Multiply the length: \(12\cdot \frac{5}{3}=20\). The length is \(\color{red}{20}\) cm.
10) The scale factor is \(56 \div 8=7\). Multiply the model height by \(7\): \(3\cdot 7=21\). The real height is \(\color{red}{21}\) inches.
11) Compare matching ratios: \(\frac{9}{6}=1.5\), \(\frac{15}{10}=1.5\), and \(\frac{21}{14}=1.5\). All scale factors match, so the triangles are \(\color{red}{similar}\).
12) Compare ratios: \(\frac{12}{8}=1.5\), \(\frac{18}{12}=1.5\), but \(\frac{30}{18}=\frac{5}{3}\). The scale factors are not all equal, so the triangles are \(\color{red}{\text{not similar}}\).
13) For similar figures, the ratio of perimeters equals the scale factor. The scale factor is \(54 \div 36=1.5\). Multiply the side: \(8\cdot 1.5=12\). The larger side is \(\color{red}{12}\) cm.
14) Area changes by the square of the scale factor. Since \(3^2=9\), multiply the small area by \(9\): \(20\cdot 9=180\). The large area is \(\color{red}{180}\) square units.
15) Multiply the original side by the scale factor: \(35\cdot \frac{2}{5}=7\cdot 2=14\). The image side is \(\color{red}{14}\) cm.
16) Volume changes by the cube of the linear scale factor. Since \(4^3=64\), multiply: \(10\cdot 64=640\). The large volume is \(\color{red}{640}\) cubic units.
17) The scale factor from real ground distance to drawing ground distance is \(2.5 \div 5=0.5\) inches per foot in this comparison. Multiply the matching height: \(12\cdot 0.5=6\). The drawing height is \(\color{red}{6}\) inches.
18) Set \(\frac{0.25}{3}=\frac{2.5}{x}\). Cross multiply: \(0.25x=7.5\). Divide by \(0.25\): \(x=30\). The real length is \(\color{red}{30}\) feet.
19) The linear scale factor is \(9 \div 3=3\). Surface area changes by \(3^2=9\). Multiply: \(48\pi\cdot 9=432\pi\). The larger surface area is \(\color{red}{432\pi}\) square cm.
20) Use the known corresponding sides \(20\) and \(30\), so the scale factor from the first triangle to the second is \(30 \div 20=\frac{3}{2}\). Then \(x\cdot \frac{3}{2}=9\). Multiply by \(\frac{2}{3}\): \(x=6\). Check: \((x+4)\cdot \frac{3}{2}=10\cdot \frac{3}{2}=15\). Thus \(x=\color{red}{6}\).
Similar Figures Quiz
