Molly Ireland

Research Assistant, Durham University Maths Department.

Children can experiment with using perspective to help them learn about the world around them. In one nursery, the children were particularly interested in the Angel of the North. They spent time making models of the angel (as shown below), and some had even visited the angel.

Children using found objects to make models of the Angel of the North.

One child had taken several photos with the angel, some with them standing right next to the angel and some with them closer to the camera. The child could use these photos to help them estimate the height of the Angel of the North.

A photo of the Angel of the North.

If we had a photo from a distance where the child and the Angel of the North were the same distance away from the camera, then we could estimate the height of the angel by measuring how many times the child’s height could be stacked up next to the angel. An example of this is illustrated below.

On a grid, we have two columns of oblongs. One is red and one is blue; both columns are the same height. The blue column is made of a single tall oblong. The red column is made of five identical oblongs, which are 1 meter tall each.

In the image above, one red rectangle can represent the height of the child, and the blue rectangle can represent the height of the angel (this is not to scale). As the blue rectangle is the same height as 5 of the red rectangles (which are 1 meter tall), we could predict that the height of the angel is 5 meters.

We could also try to find the height of the angel using perspective. This is illustrated in the image below. Here we can imagine that the camera has been placed at the red dot and through the camera the child (the red rectangle) appears to be the same height as the angel (the blue rectangle). By measuring the distance the child and the angel are away from the camera and by knowing the height of the child, we can utilise the properties of similar triangles to determine the height of the angel. The child and the angel both form right-angled triangles with the same internal angles. This means that the triangles they form must be similar. By working out the ratio between the side lengths of these triangles, we can estimate the height of the angel. The angel is 5 times further away than the child, and so is five times taller than the child, making it 5 meters tall. This is the same as the estimate we got before when the child and the angel were side by side. To read more about similar triangles and ratios of side lengths, please see our article on triangles.

A large triangle with two slim rectangles inside such that they extend up from the base of the triangle until they reach the top edge of the triangle. The oblongs are placed so that one (red) is 2 meters away from the corner of the triangle and one (blue) is 10 meters away. The height of the shorter red oblong (which is closer to the corner) is 1 meter. There is a red circle placed at the corner of the triangle.

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