Similar triangles: same shape, different size, one scale factor

How to use this simulation

1. Start in “Scale and ratios”. The blue triangle ABC is fixed and the second one, DEF, is a copy enlarged by the scale factor k (it starts at 1.50). Press Play and watch the copy grow from the same size to its final size while every matching angle keeps its arcs.
2. Drag the “Scale factor” slider from 0.5 to 3. Below 1 the copy shrinks, at 1 it is the same triangle, above 1 it grows. Watch the three side ratios in the readings: they are all equal to k. The perimeter ratio is k too, but the area ratio is k².
3. Change “Angle A”. This reshapes the whole family: a different starting triangle, a different copy. The angles of ABC and DEF still match one by one, which is exactly what “similar” promises.
4. Switch to “Similarity tests” and pick SSS, SAS or AA. A reference triangle sits on the left. Build your own triangle on the right with the sliders and the verdict tells you, in exact whole-number arithmetic, whether the two are similar.
5. Try to break each test. In SSS, change one side by 1 and the three ratios stop agreeing. In SAS, keep the two ratios equal but nudge the angle. In AA, change one angle and see the third one move with it.
6. Finish in “Height from a shadow”. Slide the sun lower and both shadows stretch together. The ratio of the two shadows never changes, so the tree height read from the person’s height and the two shadows stays the same.

Zoom a photo and the shape survives

Open a photo on your phone and zoom it to 150%. Everything gets bigger, but nobody’s face gets longer or wider. The tower in the background is still as tall compared to the door in front of it. That is the whole idea of similar shapes: a perfect enlargement or reduction that keeps the proportions. If you printed the zoomed photo, every length would be 1.50 times the original, and the paper you would need would be 2.25 times as big.

Triangles are the simplest shapes with this property, and similar triangles are one of the most useful tools in school geometry. Surveyors measure mountains with them, architects scale drawings with them and your own eyes use them: a far-away car looks small because the triangle from its top and bottom to your eye is a tiny copy of the one a nearby car makes.

What “similar” really means

Two triangles ABC and DEF are similar when two things are true at once. First, their angles match in pairs: ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Second, their matching sides are in the same ratio: DE ÷ AB = EF ÷ BC = FD ÷ CA. That common ratio is the scale factor k. We write ABC ~ DEF, and the order of the letters matters because it says which vertex matches which.

Sides that face equal angles are called corresponding sides. In the simulation the corresponding vertices are A and D, B and E, C and F, so AB matches DE, BC matches EF and CA matches FD. Always match sides by the angle they sit opposite to, not by how they look on the page, because a similar triangle may be turned around or flipped.

Congruent triangles are the special case where k = 1: same shape and same size. If two triangles are congruent they are certainly similar, but similar triangles are almost never congruent.

The scale factor and the k, k, k² rule

Everything about similar triangles hangs on one number, the scale factor k. If DEF is k times as big as ABC, then each side of DEF is k times the matching side of ABC. Put the three sides together and the perimeter is also k times as big, because it is just the sum of three lengths each multiplied by k.

Area is different. Area measures flat space, which has two directions, so both length and width get multiplied by k and the area picks up a factor k × k. The simulation prints the area ratio in the readings, and it always lands exactly on k².

The three tests: you do not need to check everything

The definition asks for three equal angle pairs and three equal side ratios. That is a lot of checking. Luckily a triangle’s shape is so rigid that a few facts fix the rest. Three tests are enough, and they are the ones your exam will ask for.

Try it in the simulation

Five small experiments turn the tests into something you can feel. Do them with the sliders, not just by reading.

• In “Scale and ratios”, set k to 2.50. Read the area ratio: it is 6.25, which is 2.50 × 2.50. Now set k to 3 and it becomes 9.
• Set k to 1. The copy sits exactly on the same size as the original, so the triangles are congruent. That is the “k = 1” special case.
• In “Similarity tests” choose SSS and set d, e, f to 8, 12 and 16. The verdict says similar with k = 2. Change f to 15 and the verdict flips to not similar.
• Choose SAS and make d and e proportional to 4 and 6, for example 8 and 12. The verdict is similar only if the included angle is exactly 50°. Move it to 51° and it fails.
• Choose AA. Set D = 50° and E = 60° and the triangle is similar. Set D = 60° and E = 50° and it is still similar, just turned round, because the same three angles appear in a different order.

Writing the correspondence correctly

Most lost marks in similarity problems come from matching the wrong sides. Here is a safe routine. First, find the equal angles. Second, write the similarity with the vertices in matching order, for example ABC ~ DEF when A = D, B = E and C = F. Third, read the proportion straight from the letters: AB ÷ DE = BC ÷ EF = CA ÷ FD.

If the picture looks different, redraw one triangle so that it sits in the same orientation as the other. Rotate it or flip it on paper. Similar triangles are still similar after a turn, so this costs nothing and makes the matching sides line up visually.

• The longest side always matches the longest side, and the shortest the shortest.
• A side sits opposite its angle, so equal angles lead you to matching sides.
• Write ratios as “small triangle over big triangle” or the reverse, but keep the same direction in every fraction.

Parallel lines inside a triangle

Draw a line DE parallel to the side BC of triangle ABC, with D on AB and E on AC. The small triangle ADE and the big triangle ABC share the angle at A, and the parallel lines give ∠ADE = ∠ABC and ∠AED = ∠ACB. That is AA, so ADE ~ ABC.

This simple picture is the basic proportionality theorem (also called Thales’ theorem): a line parallel to one side of a triangle cuts the other two sides in the same ratio. The proof is just the similarity you have already seen.

Measuring a tall tree with a shadow

Nobody wants to climb a tree with a tape measure. Instead stand next to it on a sunny day. The sun’s rays are practically parallel, so they hit the tree and you at the same angle. The tree and its shadow make a right triangle, and you and your shadow make another one with the same angles. By AA those triangles are similar.

That means the ratio of height to shadow is the same for both. Measure your height, your shadow and the tree’s shadow, and the tree height drops out of a proportion. Switch the simulation to “Height from a shadow” and move the sun: the shadows stretch, the ratio of the two shadows does not change, and the estimate of the tree height does not move.

Solved problems

Each number below is computed from the same formulas the simulation uses.

Common mistakes

These slips cost marks in every exam. Check for them before you hand in a paper.

• Matching sides by position on the page instead of by the angle opposite them. Redraw if you must.
• Using the area ratio as if it were the side ratio. If areas are in the ratio 9 : 25, the sides are in the ratio 3 : 5, not 9 : 25.
• Checking only two side ratios for SSS. All three must agree.
• Using SAS with an angle that is not between the two chosen sides.
• Believing that equal angles alone prove congruence. Equal angles give similarity; you also need one equal side for congruence.

Where similar triangles show up in real life

Map makers rely on similarity: a map is a reduced copy of the land with scale factor perhaps 1 : 50,000. Distances multiply by 50,000 but areas by 50,000 × 50,000. Architects draw plans, and model makers build cars, planes and buildings in scale, all using k.

Photographers use it every time they crop and resize. Painters use it when they enlarge a sketch onto a wall with a grid. Surveyors measure a river’s width without crossing it by building a small triangle on their side that is similar to the big one that spans the river.

Even your pocket calculator’s “zoom” and the shrinking of a far-off object in a video game use the same triangles. Wherever shape must survive a change of size, similar triangles are nearby.

Exam corner

Class 8 to 10 questions on similar triangles are mostly one of five types. Learn to recognise the type before you start writing.

• Prove two triangles similar: find two equal angle pairs (AA) or three proportional sides (SSS).
• Find a missing side: write the proportion from the matching letters and cross-multiply.
• Areas and perimeters: square the ratio for areas, use the plain ratio for perimeters.
• Parallel line in a triangle: use AD ÷ DB = AE ÷ EC and DE ÷ BC = AD ÷ AB.
• Heights and shadows: build a proportion between height and shadow.

Revision summary

If you remember nothing else from this page, remember these lines.

• k = 1 gives congruent triangles.
• Match sides by the angles they face.
• Shadows at the same moment give similar right triangles.

Frequently asked questions