Are all triangles with same area similar?

Answer 1

No. When we say triangles are similar we mean that they are the same shape. So a triangle could have the same shape but be double the size. It would then have an area 4 times the original triangle.

Example: a right angled triangle with sides 3, 4 and 5 cm would be similar to a right angled triangle with sides 6,8 and 10 cm. The area of the first triangle would be (3x4)/2=6sq cm, while the area of the latter would be (6x8)/2+24 sq cm.

When triangles are identical they are called "congruent". They are the same shape and size and the same area.

The formula to calculate the area of a triangle is half(basexheight) or bh/2. Triangles of the same area do not have to be similar. It is just necessary that the product of the base and the height are the same. So a triangle with a base of 4 and a height of 3cm would have the same area area as a triangle with a base of 1 and a height of 12cm.

No. Long thin triangles can have the same area as short fat ones but they will not be similar just as two rectangles may have the same area but different proportions.

Example:

A rectangle 4 * 6 units has an area of 24 square units, so does a rectangle that is 2 * 12 units.

Now the cut these rectangles in half, corner to diametrically opposite corner and you have triangles. The one 6 units long and 4 units high and the other 12 units long by 2 units. These have the same area (the both are 12 square units) but are not similar.