Equilateral Triangles

All 3 sides of an equilateral triangle have the same length. All 3 angles have the same measure.

There is a special formula for the area of an equilateral triangle. A = where

s is the length of a side.

How to know when a triangle is an equilateral triangle:

When information given about an isosceles triangle implies that all the angles must be 600. In the figure below, and C must be equal because they correspond to the equal sides of an isosceles triangle. If we call those angles x, then x + x + 60 = 180. Solving for x, we get x = 60.

When a triangle drawn inside a circle has two radii as sides and an angle measuring 600.

A triangle with two radii as sides must have at least two equal sides and therefore two equal angles. If one of the angles measures 600, then the other angles must measure 600 also and the triangle must be equilateral.

A

B C

60

A

BC

60

Example 1: The circle shown below has a radius of 4. Find the area of the triangle.

Because AC and AB are both radii, they must have the same length. Therefore, A and B must have the same measure, which is 60. Therefore, the triangle is equilateral and

we can use the formula for finding the area of an equilateral triangle.

A =

Substituting 4 for s, we get

A =

A

BC

60r = 4

Try the following problems.

1. An equilateral triangle has a side of length 3. Find the area of the triangle.

2. Is the triangle below an equilateral triangle? If so, find the area.

3. A rhombus with side lengths of 5 and an angle, A, measuring 600 is shown below. If a diagonal, DB, is drawn into the figure, how do you know that the resulting triangle is equilateral?

4. What is the area of in problem 3?

5. Review problems. Use your knowledge of special triangles to find the length of diagonal AC in the rhombus above. Remember that the rhombus is a special case of a parallelogram and that the diagonals of a rhombus bisect the angles. You will need to draw an altitude to AC to get a 30-60-90 triangle.

A

B

C 60

r = 5

A B

CD

60

5

6. Find the area of the figure below. Hint: draw a line to separate the figure into a triangle and a rectangle. How do you know the triangle is equilateral?

7. Δ ABC is an equilateral triangle with sides of length 4. D, E, and F are midpoints of the triangle. The points D, E, and F are connected to form a new triangle. What is the area of the new triangle? Hint: How do we know that the new triangle is also equilateral?

A

B

C

D

E

F

A

BC

D

E

5 5

5

150

4

1. . Substitute into the formula

2. Yes, it is equilateral because all the angles must measure 600. The area is equal to

.

3. A rhombus has 4 sides of equal length. All sides in this figure must equal 5. The diagonal, DB, creates Δ ADB. Because AD and AB both equal 5, the angles ADB and ABD must be equal by the theorems about isosceles triangles. If we set up an equation to find the measures of those angles, we get x + x + 60 = 180. Solving for x gives us 60. Since all angles of the triangle equal 600, the triangle is equilateral.

4.

5. The diagonal AC will bisect DAB and DCB into angles of 300 each.

If we draw a height from D to the diagonal AC, we will get two 30-60-90 triangles.

Follow the rules for special triangles. CE and AE will both equal . Adding these pieces together, we get

A B

CD

30

5

30

A B

CD

E30

5

30

6. Draw a line through BC to create a rectangle and a triangle. Since the original angles are 150 degrees and a rectangle has 90-degree angles, the two base angles of the triangle will be 60 degrees. Hence, the angle at E must be 60 degrees also. Since all the angles of the triangle are 60 degrees, the triangle is equilateral and we can use the formula to find the area of an equilateral triangle. To find the area of the figure add the area of the rectangle plus the area of the triangle.

Area of rectangle + area of Triangle

L x W +

5 x 4 +

A

BC

D

E

5 5

5

150

7.

Look at Triangle DEF. Because D and F are midpoints, DC and CF must have a length of 2. Because < C = 60 degrees and the other two sides must be equal, the other angles also measure 60 degrees. Because all the angles measure 60 degrees, the triangle must be equilateral. Hence, DF = 2. Similar reasoning applies to the other triangles. We can thus show that Triangle DEF is equilateral with sides lengths of 2. Hence, we can use the formula for the area of an equilateral triangle.