# Relationship between height and side of an equilateral triangle

### Equilateral Triangles Geometry calculator for solving the altitude of an equilateral triangle given the length of a side. Is the height of an equilateral triangle equal to its side length. No It is easy to imagine with everything being equal that the height, and the side length would be . An isosceles triangle is a triangle with (at least) two equal sides. The height of the isosceles triangle illustrated above can be found from the Pythagorean there is a surprisingly simple relationship between the area and vertex angle theta.

## Isosceles Triangle

They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three-dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling. Geometric construction[ edit ] Construction of equilateral triangle with compass and straightedge An equilateral triangle is easily constructed using a straightedge and compassbecause 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment An alternative method is to draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius.

### Equilateral triangle - Wikipedia

The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection. Triangle area proofs Video transcript Let's say that this triangle right over here is equilateral, which means all of its sides have the same length. And let's say that that length is s. What I want to do in this video is come up with a way of figuring out the area of this equilateral triangle, as a function of s.

And to do that, I'm just going to split this equilateral in two.

### Isosceles Triangle -- from Wolfram MathWorld

I'm just going to drop an altitude from this top vertex right over here. This is going to be perpendicular to the base. And it's also going to bisect this top angle. So this angle is going to be equal to that angle. And we showed all of this in the video where we proved the relationships between the sides of a triangle. Well, in a regular equilateral triangle, all of the angles are 60 degrees. So this one right over here is going to be 60 degrees, let me do that in a different color. This one down here is going to be 60 degrees. And then this one up here is 60 degrees, but we just split it in two. So this angle is going to be 30 degrees. And then this angle is going to be 30 degrees.

And then the other thing that we know is that this altitude right over here also will bisect this side down here. So that this length is equal to that length. And we showed all of this a little bit more rigorously on that triangle video. So to figure out what the actual altitude is. Then the side opposite the 60 degree angle is going to be square root of 3 times that. So it's going to be square root of 3 s over 2.