How Tall Is the Tree? Similar Triangles and Shadows

0:00How tall is this tree? Its shadow can tell us.

0:05At the same moment, a person 1.6 metres tall casts a shadow 2 metres long. The drawing is not to scale.

0:14The tree casts a shadow 15 metres long. We call its unknown height h.

0:21The person, their shadow and the sun's ray form a right triangle, with one 90 degree angle. The tree forms one too.

0:31The sun's rays are parallel, so they meet the ground at the same angle in both.

0:36Angles in a triangle add to 180 degrees, so the third angles match too.

0:43Triangles with the same angles are called similar. One is a scaled-up copy of the other.

0:50In similar triangles, corresponding sides, the matching sides, are in the same ratio. So height over shadow is the same for both.

1:00For the person, that is 1.6 over 2. For the tree, it is h over 15.

1:07An equation saying two ratios are equal is called a proportion.

1:13A common mistake is to write one ratio upside down, like 2 over 1.6. Keep height over shadow on both sides.

1:23To get h on its own, multiply both sides by 15. So h equals 1.6 times 15, divided by 2.

1:331.6 times 15 is 24, and 24 divided by 2 is 12. So the tree is 12 metres tall.

1:44Check: 1.6 divided by 2 is 0.8, and 12 divided by 15 is also 0.8. The ratios match.

1:55Recap: in similar triangles, corresponding sides are in the same ratio, so write both ratios in the same order and solve.