Right Triangle worksheet

1. Leg a: 6.89

1. sin A = opposite ÷ hypotenuse, so the leg opposite A is the hypotenuse times sin A

   $a$$=13\times \text{sin}32\text{°}$$=13\times 0.52991926$$=6.88895$

2. cos A = adjacent ÷ hypotenuse, so the leg next to A is the hypotenuse times cos A

   $b$$=13\times \text{cos}32\text{°}$$=13\times 0.8480481$$=11.024625$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-32\text{°}$$=58\text{°}$

4. Area: half the product of the legs

   $\frac{6.88895\times 11.024625}{2}$$=37.974048$

5. Perimeter: add the three sides

   $6.88895+11.024625+13$$=30.913576$

6. So the leg a is

   $a$$=6.88895$

2. Hypotenuse c: 12.69

1. The height h cuts off a smaller right triangle with angle A and hypotenuse b, so h = b × sin A and b = h ÷ sin A

   $b$$=\frac{5}{\text{sin}26\text{°}}$$=11.40586$

2. On the other side the angle is B = 90° − A, and h = a × sin B = a × cos A, so a = h ÷ cos A

   $a$$=\frac{5}{\text{cos}26\text{°}}$$=5.56301$

3. Find the hypotenuse by dividing leg a by sin A, since sin A = a ÷ c

   $c$$=\frac{5.56301}{\text{sin}26\text{°}}$$=12.690182$

4. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-26\text{°}$$=64\text{°}$

5. Area: half the product of the legs

   $\frac{5.56301\times 11.40586}{2}$$=31.725455$

6. Perimeter: add the three sides

   $5.56301+11.40586+12.690182$$=29.659052$

7. So the hypotenuse c is

   $c$$=12.690182$

3. Hypotenuse c: 8.94

1. sin A = opposite ÷ hypotenuse, so the hypotenuse is the opposite leg divided by sin A

   $c$$=\frac{5}{\text{sin}34\text{°}}$$=\frac{5}{0.5591929}$$=8.941458$

2. tan A = opposite ÷ adjacent, so the adjacent leg is the opposite leg divided by tan A

   $b$$=\frac{5}{\text{tan}34\text{°}}$$=\frac{5}{0.67450852}$$=7.412805$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-34\text{°}$$=56\text{°}$

4. Area: half the product of the legs

   $\frac{5\times 7.412805}{2}$$=18.532012$

5. Perimeter: add the three sides

   $5+7.412805+8.941458$$=21.354263$

6. So the hypotenuse c is

   $c$$=8.941458$

4. Leg b: 13.75

1. Use the Pythagorean theorem the other way round: the missing leg is the square root of c² minus a²

   $b$$=\sqrt{{15}^2-6^2}$$=13.747727$

2. Angle A from its sine, the opposite leg over the hypotenuse

   $A$$=\text{sin⁻¹}\left(\frac{6}{15}\right)$$=23.578178\text{°}$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-23.578178\text{°}$$=66.421822\text{°}$

4. Area: half the product of the legs

   $\frac{6\times 13.747727}{2}$$=41.243181$

5. Perimeter: add the three sides

   $6+13.747727+15$$=34.747727$

6. So the leg b is

   $b$$=13.747727$

5. Leg b: 9.8

1. Use the Pythagorean theorem the other way round: the missing leg is the square root of c² minus a²

   $b$$=\sqrt{{11}^2-5^2}$$=9.797959$

2. Angle A from its sine, the opposite leg over the hypotenuse

   $A$$=\text{sin⁻¹}\left(\frac{5}{11}\right)$$=27.035692\text{°}$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-27.035692\text{°}$$=62.964308\text{°}$

4. Area: half the product of the legs

   $\frac{5\times 9.797959}{2}$$=24.494897$

5. Perimeter: add the three sides

   $5+9.797959+11$$=25.797959$

6. So the leg b is

   $b$$=9.797959$

6. Hypotenuse c: 14.56

1. Pythagorean theorem: the hypotenuse is the square root of the sum of the squared legs

   $c$$=\sqrt{4^2+{14}^2}$$=14.56022$

2. Angle A from its tangent, the opposite leg over the adjacent leg

   $A$$=\text{tan⁻¹}\left(\frac{4}{14}\right)$$=15.945396\text{°}$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-15.945396\text{°}$$=74.054604\text{°}$

4. Area: half the product of the legs

   $\frac{4\times 14}{2}$$=28$

5. Perimeter: add the three sides

   $4+14+14.56022$$=32.56022$

6. So the hypotenuse c is

   $c$$=14.56022$

7. Hypotenuse c: 10.08

1. sin A = opposite ÷ hypotenuse, so the hypotenuse is the opposite leg divided by sin A

   $c$$=\frac{7}{\text{sin}44\text{°}}$$=\frac{7}{0.69465837}$$=10.076896$

2. tan A = opposite ÷ adjacent, so the adjacent leg is the opposite leg divided by tan A

   $b$$=\frac{7}{\text{tan}44\text{°}}$$=\frac{7}{0.96568877}$$=7.248712$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-44\text{°}$$=46\text{°}$

4. Area: half the product of the legs

   $\frac{7\times 7.248712}{2}$$=25.370493$

5. Perimeter: add the three sides

   $7+7.248712+10.076896$$=24.325608$

6. So the hypotenuse c is

   $c$$=10.076896$

8. Hypotenuse c: 14.32

1. Pythagorean theorem: the hypotenuse is the square root of the sum of the squared legs

   $c$$=\sqrt{6^2+{13}^2}$$=14.317821$

2. Angle A from its tangent, the opposite leg over the adjacent leg

   $A$$=\text{tan⁻¹}\left(\frac{6}{13}\right)$$=24.775141\text{°}$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-24.775141\text{°}$$=65.224859\text{°}$

4. Area: half the product of the legs

   $\frac{6\times 13}{2}$$=39$

5. Perimeter: add the three sides

   $6+13+14.317821$$=33.317821$

6. So the hypotenuse c is

   $c$$=14.317821$

9. Hypotenuse c: 12.06

1. The height h cuts off a smaller right triangle with angle A and hypotenuse b, so h = b × sin A and b = h ÷ sin A

   $b$$=\frac{5}{\text{sin}28\text{°}}$$=10.650272$

2. On the other side the angle is B = 90° − A, and h = a × sin B = a × cos A, so a = h ÷ cos A

   $a$$=\frac{5}{\text{cos}28\text{°}}$$=5.66285$

3. Find the hypotenuse by dividing leg a by sin A, since sin A = a ÷ c

   $c$$=\frac{5.66285}{\text{sin}28\text{°}}$$=12.062179$

4. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-28\text{°}$$=62\text{°}$

5. Area: half the product of the legs

   $\frac{5.66285\times 10.650272}{2}$$=30.155449$

6. Perimeter: add the three sides

   $5.66285+10.650272+12.062179$$=28.375302$

7. So the hypotenuse c is

   $c$$=12.062179$

10. Hypotenuse c: 12.94

1. cos A = adjacent ÷ hypotenuse, so the hypotenuse is the adjacent leg divided by cos A

   $c$$=\frac{12}{\text{cos}22\text{°}}$$=\frac{12}{0.92718385}$$=12.942417$

2. tan A = opposite ÷ adjacent, so the opposite leg is the adjacent leg times tan A

   $a$$=12\times \text{tan}22\text{°}$$=12\times 0.40402623$$=4.848315$

3. The two acute angles add up to 90°, so subtract A from 90° to get B

   $B$$=90-22\text{°}$$=68\text{°}$

4. Area: half the product of the legs

   $\frac{4.848315\times 12}{2}$$=29.089888$

5. Perimeter: add the three sides

   $4.848315+12+12.942417$$=29.790732$

6. So the hypotenuse c is

   $c$$=12.942417$