Objective: Find the area of any triangle given at least three pieces of information.

1. Area of Triangles Objective: Find the area of any triangle given at least three pieces of information. Process: Derive several formulas to allow use of given information (to avoid rounding errors). By: Anthony E. Davis Summer 2003

2. b C a Where to start? B C a If you have two sides and the included angle then click here. If you have two angles and the included side then click here. If you have all three sides then click here. c b a Not Quite Sure? - Try Some Practice Problems.

3. b C a Two sides and the Included angle See derivation. Look at formula. Try an example. Return to choices.

4. Two angles and the included side See derivation. Look at formula. Try an example. Return to choices. B C a

5. All three sides c b a Look at formula. Try an example. Return to choices.

6. Not Quite Sure? Draw a triangle. Label the information given. Match this triangle with one of the three shown. Remember all triangles and all variables are arbitrarily drawn so rotation may be necessary. Return to choices

7. A c b B C a h Derivation We know Area=1/2(base)(height). Let a represent the base. Using right triangle trigonometry, sin C = h/b Solve for h, h = b sin C. Replace values in area formula: Area = 1/2 a (bsin C) Hence the Area given two sides and the included angle is any of the following: Area = 1/2 ab sin C Area = 1/2 bc sin A Area = 1/2 ac sin B

8. Formula forTwo Sides and the Included Angle

9. Example Find the area of ∆DEF, if d = 3 cm, e = 8 cm and F = 35°. Round to the nearest hundredth.

10. A c b B C a h Derivation We know Area = 1/2 (base)(height) Let a represent the base Find the third angle by subtracting the two known from 180. From right triangle trigonometry, sin C = h/b Solve for h, h = b sin C Replacing values, Area = 1/2 a (bsin C).

11. A c b B C a Derivation (cont.) h • However we are only given one side, so we need to substitute the ‘a’ or ‘b’ out. (Let say the ‘b’). • From the Law of Sines, sin A/a = sin B/b. We know A, B and ‘a’ so we will solve for ‘b’. • Solve for ‘b’, b = (a sin B)/(sin A) • Replace, Area = 1/2 a ((a sin B)/(sin A)) sin C • Thus we have the following

12. Formula forTwo Angles and the Included Side

13. Example Find the area of ∆CAB if b = 7 ft., C = 42º, and B = 28º. Round your answer to the nearest tenth.

14. Formula for All Three Sides(Heron’s Formula)

15. Example Find the area of an equilateral triangle having legs of length 3.2 mm. Round your answer to two decimal places.

16. Practice Problems Answer Answer Answer Answer Answer Answer Directions: Find the area of each triangle using the given information. Round only your final answer to the nearest tenth. You may click on the question to see the solution.

17. Answer #1: Return to problems.

18. Answer #2 Return to problems.

19. Answer #3 Return to problems.

20. Answer #4 Return to problems.

21. Answer #5 Return to problems.

22. Answer #6 Return to problems.