Awkward Triangle

Different methods to find the exact area of a unique triangle given its side lengths. Using formulas like the Cosine Rule and Heron's formula, calculations are demonstrated step by step along with visual aids for better understanding.

• Geometry
• Triangle
• Area Calculation
• Herons Formula
• Cosine Rule

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1. Awkward Triangle

2. Awkward Triangle (not to scale) What s the exact area of this triangle? 2 6 3 3 15 3

3. Awkward Triangle (not to scale) One approach: Use the Cosine Rule to determine angle ?: 2 6 3 3 ?2= ?2+ ?2 2??cos? Then use the formula for the area of a triangle: ???? =1 2?? sin? ? 15 3

4. Awkward Triangle 2 2 2 6 3 2+ 3 2 6 3 15 3 15 3 = 3 2 3 cos? 8 3= 3 +5 3 2 5cos? ? 15 3 1 2 cos? = 5 sin? = 5 Area is given by: ???? =1 5 2 5= ? 15 3 2 3 ? 2 1

6. Awkward Triangle (not to scale) Another option would be to use Heron s formula. For triangle with sides ?,?,? 2 6 3 3 Area = ? ? ? ? ? ? ? where ? =1 2? + ? + ? 15 3

7. Awkward Triangle Using Heron s formula with: 3 ? =2 6 15 3 ? =1 3 +2 6 15 3 3 ? = ? = 3+ 2 Area = 1 2 3 +2 6 15 3 1 2 3 +2 6 15 3 1 2 3 2 6 15 3 1 2 3 +2 6 15 3 3+ 3+ 3+ 3 =1 4 3+4 10 4 3+4 10 4 3 3 = ? =1 16 9+160 =1 144 9 9 4 4

8. Awkward Triangle (not to scale) What s the exact area of this triangle? 2 6 3 3 ? 15 3

9. RESOURCES

10. Awkward Triangle (not to scale) What s the exact area of this triangle? 2 6 3 3 15 3 SIC_38

11. A B (not to scale) (not to scale) ? ? ? ? ? ? ? ? ? Awkward Triangle Awkward Triangle ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

12. D C (not to scale) (not to scale) ? ? ? ? ? ? ? ? ?? Awkward Triangle Awkward Triangle ?? ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

13. E F (not to scale) (not to scale) ?? ? ?? ? ? ? ? ? ? ??? Awkward Triangle Awkward Triangle ?? ?? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

14. H G (not to scale) (not to scale) ? ? ? ? ? ? ? ? ?? Awkward Triangle Awkward Triangle ?? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

15. J I (not to scale) (not to scale) ?? ? ? ?? ? ? ??? ?? Awkward Triangle Awkward Triangle ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

16. K L (not to scale) (not to scale) ? ? ? ? ? ? ? ? ?? Awkward Triangle Awkward Triangle ?? ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

17. M N (not to scale) (not to scale) ? ? ? ? ? ? ? ? ??? ?? Awkward Triangle Awkward Triangle ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

18. O P (not to scale) (not to scale) ? ? ? ? ? ? ? ?? Awkward Triangle Awkward Triangle ?? ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

19. W X (not to scale) (not to scale) ? ? ? ?? ? ? ? Awkward Triangle Awkward Triangle ?? ?? ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

20. Z Y (not to scale) (not to scale) ?? ? ? ?? ? ? ??? ?? Awkward Triangle Awkward Triangle ? ? What s the exact What s the exact area of this area of this triangle? triangle? SIC_38 SIC_38

21. Note to Teacher Tasks W, X, Y and Z have angle A >90 and so cosine A is negative. This could throw pupils and so should really be given as an extension, given the way the worked solution goes. If calculators are allowed (to speed the task up, say) then there is no real problem. Tasks W, X, Y and Z are tasks E, K, I and J, respectively, with sides ? and ? swapped over.

22. REFERENCE MATERIAL

23. C a b Area of triangle using Heron's formula B A Upper apex angle c SIC A B C D E F G H I J K L M N O P W X Y Z a b c s s-a s-b 0.79618 s-c 0.20382 0.12132 Area A rad 0.321751 18.43495 0.785398 0.141897 8.130102 0.785398 0.197396 11.30993 0.785398 0.463648 26.56505 0.785398 0.588003 33.69007 0.785398 1.030377 59.03624 0.785398 0.084902 4.864514 0.927295 0.179853 10.30485 0.927295 0.231091 13.24052 0.927295 0.398522 22.83365 0.927295 0.519146 29.74488 0.927295 0.643501 36.8699 0.927295 0.089314 5.117315 0.643501 0.141897 8.130102 0.643501 0.708626 40.60129 0.643501 1.515298 86.82017 0.643501 1.768192 101.3099 0.785398 1.695151 97.12502 0.927295 1.983207 113.6294 0.927295 1.815775 104.0362 0.927295 A deg B rad B deg C rad C deg A+B+C deg 1 2.236068 2.828427 3.032248 2.032248 0.707107 3.535534 0.816497 2.94392 3.464102 3.612259 2.795763 0.668339 0.148158 1.154701 1.825742 2.44949 2.714966 1.560266 0.889224 0.265476 1.264911 1.612452 2.236068 2.556715 1.291804 0.944264 0.320647 1.581139 1.30384 1.788854 2.336917 0.755778 1.033076 0.548062 0.5 4.716991 5 5.108495 4.608495 0.391505 0.108495 0.707107 3.162278 3.535534 3.702459 2.995352 0.540182 0.166925 0.790569 2.76134 3.162278 3.357094 2.566524 0.595753 0.194816 1 2.061553 2.5 2.780776 1.780776 0.719224 0.280776 1.118034 1.802776 2.236068 2.578439 1.460405 0.775663 0.342371 1.224745 1.632993 2.041241 2.44949 1.224745 0.816497 0.408248 0.666667 4.484541 5 5.075604 4.408937 0.591063 0.075604 0.816497 3.464102 4.082483 4.181541 3.365044 0.717439 0.099058 1.490712 1.374369 2.236068 2.550574 1.059862 1.176206 0.314506 2 1.20185 1.666667 2.434259 0.434259 1.232408 0.767592 2.236068 1.612452 1.264911 2.556715 0.320647 0.944264 1.291804 2.236068 1.802776 1.118034 2.578439 0.342371 0.775663 1.460405 3.162278 2.76134 0.790569 3.357094 0.194816 0.595753 2.566524 2.5 2.061553 1 2.780776 0.280776 0.719224 1.780776 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45 2.034444 116.5651 45 2.214297 126.8699 45 2.158799 123.6901 45 1.892547 108.4349 45 1.768192 101.3099 45 1.325818 75.96376 53.1301 2.129396 122.0054 53.1301 2.034444 116.5651 53.1301 1.983207 113.6294 53.1301 1.815775 104.0362 53.1301 1.695151 97.12502 53.1301 1.570796 36.8699 2.408778 138.0128 36.8699 2.356194 36.8699 1.789465 102.5288 36.8699 0.982794 56.30993 45 0.588003 33.69007 53.1301 0.519146 29.74488 53.1301 0.231091 13.24052 53.1301 0.398522 22.83365 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 A B C D E F G H I J K L M N O P W X Y Z 4 4.12132 3.414214 0.585786 90 135

24. SIC A B C D E F G H I J K L M N O P a b c SIC A B C D E F G H I J K L M N O P a b c 1 =SQRT(2)/2 =SQRT(6)/3 =2/3*SQRT(3) =SQRT(5) =SQRT(10)/2 0.5 =SQRT(2)/2 =SQRT(10) 2.5 =SQRT(5) =SQRT(6)/2 =2/3 =SQRT(6)/3 =SQRT(5) 2 =SQRT(5) =5*SQRT(2)/2 =SQRT(78)/3 =SQRT(30)/3 =SQRT(65)/5 =SQRT(170)/10 =SQRT(89)/2 =SQRT(10) =SQRT(122)/4 =SQRT(17)/2 =SQRT(13)/2 =SQRT(24)/3 =SQRT(181)/3 =SQRT(12) =SQRT(17)/3 =SQRT(13)/3 =2*SQRT(2) 4 =2*SQRT(3) =SQRT(6) =2/5*SQRT(10) =0.8*SQRT(5) 5 =5*SQRT(2)/2 =SQRT(10)/4 1 =SQRT(5)/2 =5/6*SQRT(6) 5 =5*SQRT(6)/3 =2/3*SQRT(5) =5/3 1 2.236068 2.828427 0.707107 3.535534 0.816497 2.94392 3.464102 1.154701 1.825742 2.236068 1.612452 1.264911 1.581139 1.30384 1.788854 0.5 4.716991 0.707107 3.162278 3.535534 3.162278 2.76134 0.790569 2.5 2.061553 2.236068 1.802776 1.118034 1.224745 1.632993 2.041241 0.666667 4.484541 0.816497 3.464102 4.082483 2.236068 1.374369 1.490712 2 1.20185 1.666667 4 2.44949 5 1 5 A rad 0.321751 18.43495 0.785398 0.141897 8.130102 0.785398 0.197396 11.30993 0.785398 0.463648 26.56505 0.785398 1.768192 101.3099 0.785398 1.030377 59.03624 0.785398 0.084902 4.864514 0.927295 0.179853 10.30485 0.927295 1.983207 113.6294 0.927295 1.815775 104.0362 0.927295 1.695151 97.12502 0.927295 0.643501 36.8699 0.927295 0.089314 5.117315 0.643501 0.141897 8.130102 0.643501 1.789465 102.5288 0.643501 1.515298 86.82017 0.643501 A deg B rad B deg C rad C deg A+B+C deg 45 2.034444 116.5651 45 2.214297 126.8699 45 2.158799 123.6901 45 1.892547 108.4349 45 0.588003 33.69007 45 1.325818 75.96376 53.1301 2.129396 122.0054 53.1301 2.034444 116.5651 53.1301 0.231091 13.24052 53.1301 0.398522 22.83365 53.1301 0.519146 29.74488 53.1301 1.570796 36.8699 2.408778 138.0128 36.8699 2.356194 36.8699 0.708626 40.60129 36.8699 0.982794 56.30993 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 A B C D E F G H I J K L M N O P C a b B A c 90 135