Trigonometry

Solving a right-angled triangle

1. ?
   ?

   In a right-angled triangle trigonometric functions can be used:
   
   $\sin \alpha =\frac{opposite}{hypotenuse}$
   
   $\cos \alpha =\frac{adjacent}{hypotenuse}$
   
   $\tan \alpha =\frac{opposite}{adjacent}$
   In a right-angled triangle $ABC$,   $A\hat{C}B={90}^{\circ },A\hat{B}C={37}^{\circ }$ and $a=BC=14\mathrm{cm}$.
   

   (a)   Find the lengths of the other two sides $b=AC$ and $c=AB$.

   (b)   Calculate the perimeter $P$ of the triangle.

   Solutions:    (a)  $b\approx 10.5\mathrm{cm},c\approx 17.5\mathrm{cm}$;     (b)  $P=a+b+c\approx 42.1\mathrm{cm}$
2. In a right-angled triangle $ABC$,   $A\hat{C}B={90}^{\circ },A\hat{B}C={30}^{\circ }$ and $a=BC=12\mathrm{cm}$.
   

   (a)   Find the lengths of the other two sides $b=AC$ and $c=AB$.

   (b)   Calculate the perimeter $P$ of the triangle.

   Solutions:    (a)  $b=4\sqrt{3}\mathrm{cm}\approx 6.93\mathrm{cm}$, $c=8\sqrt{3}\mathrm{cm}\approx 13.9\mathrm{cm}$;     (b)  $P=a+b+c=12+12\sqrt{3}\mathrm{cm}\approx 32.8\mathrm{cm}$
3. In a right-angled triangle $ABC$,   $A\hat{C}B={90}^{\circ },B\hat{A}C={52.5}^{\circ }$ and $c=AB=17\mathrm{cm}$.
   

   (a)   Find the lengths of the other two sides.

   (b)   Calculate the perimeter $P$ of the triangle.

   Solutions:    (a)  $a\approx 13.5\mathrm{cm},b\approx 10.3\mathrm{cm}$;     (b)  $P=a+b+c\approx 40.8\mathrm{cm}$
4. In a right-angled triangle $ABC$,   $A\hat{C}B={90}^{\circ },a=BC=15\mathrm{cm}$ and $b=AC=8\mathrm{cm}$.
   

   (a)   Find the length of the hypotenuse.

   (b)   Calculate the angles $\alpha =B\hat{A}C$ and $\beta =A\hat{B}C$.

   Solutions:    (a)  $c=AB=17\mathrm{cm}$;     (b)  $\alpha \approx {61.9}^{\circ },\beta \approx {28.1}^{\circ }$
5. In a right-angled triangle $ABC$ the hypotenuse is $c=25\mathrm{cm}$ and one of the catheti is $b=7\mathrm{cm}$.
   

   (a)   Find the perimeter of this triangle.

   (b)   Calculate the angles $\alpha$ and $\beta$.

   Solutions:    (a)  $a=24\mathrm{cm},P=56\mathrm{cm}$;     (b)  $\alpha \approx {73.7}^{\circ },\beta \approx {16.3}^{\circ }$
6. In an isosceles triangle $\mathrm{△}ABC$,   $a=b=7\mathrm{cm}$ and $\alpha =B\hat{A}C={68}^{\circ }$.
   

   (a)   Find the length of the side $c$.

   (b)   Find the length of the height $h_c$.

   Solutions:    (a)  $c\approx 5.24\mathrm{cm}$;     (b)  $h_c\approx 6.49\mathrm{cm}$
7. Isosceles triangle $\mathrm{△}ABC$ $(a=b)$ has the base $c=AB=24\mathrm{cm}$ and the height $h_c=35\mathrm{cm}$.
   

   (a)   Find the perimeter of this triangle.

   (b)   Calculate the angles of this triangle.

   Solutions:    (a)  $P=24+37+37=98\mathrm{cm}$;     (b)  $\alpha =\beta \approx {71.1}^{\circ },\gamma \approx {37.8}^{\circ }$
8. Rectangle $ABCD$ has the diagonal $e=AC=20\mathrm{cm}$. The angle between this diagonal and the side $a=AB$ is $C\hat{A}B={27}^{\circ }{45}^{\prime }$.
   

   (a)   Find the sides of this rectangle.

   (b)   Calculate the acute angle between the diagonals.

   Solutions:    (a)  $a\approx 17.7\mathrm{cm},b\approx 9.31\mathrm{cm}$;     (b)  $\phi ={55.5}^{\circ }={55}^{\circ }{30}^{\prime }$
9. Rectangle $ABCD$ has the side $a=AB=18\mathrm{cm}$. The angle between the diagonal $d=AC$ and the side $a$ is $C\hat{A}B={30}^{\circ }$. Find the side $b=BC$ and the diagonal $d=AC$. Write your results in the exact form.
   Solutions:    $b=6\sqrt{3}\mathrm{cm},d=12\sqrt{3}\mathrm{cm}$
10. The diagonals of the rhombus $ABCD$ have lengths $e=AC=30\mathrm{cm}$ and $f=BD=16\mathrm{cm}$.
    

    (a)   Find the side $a=AB$.

    (b)   Calculate the angle $\alpha =D\hat{A}B$.

    Solutions:    (a)  $a=17\mathrm{cm}$;     (b)  $\alpha \approx {56.1}^{\circ }$
11. A regular pentagon $ABCDE$ is inscribed in the circle with the radius $r=10\mathrm{cm}$. Find the side $a=AB$ of this pentagon.
    Solution:    $a\approx 11.8\mathrm{cm}$
12. A regular hexagon $ABCDEF$ has the side $a=7\mathrm{cm}$. Find the length of the diagonal $d=AE$.
    Solution:    $d=7\sqrt{3}\mathrm{cm}\approx 12.1\mathrm{cm}$
13. In a triangle $\mathrm{△}ABC$, $\alpha =C\hat{A}B={60}^{\circ },\beta =A\hat{B}C={45}^{\circ }$ and the height $h_c=4\sqrt{3}\mathrm{cm}$. Find the lengths of the sides $a,b$ and $c$. Write the exact values.
    Solutions:    $a=4\sqrt{6}\mathrm{cm},b=8\mathrm{cm},c=(4+4\sqrt{3})\mathrm{cm}$
14. Annie has a 3 m long ladder. She will place it against a vertical wall so that the bottom part of the ladder is 75 cm apart from the wall (and the top of the ladder touches the wall).
    

    (a)   Find the height to which the top of the ladder will reach.

    (b)   Calculate the angle between the wall and the ladder.

    Solutions:    (a)  $a\approx 2.90\mathrm{m}=290\mathrm{cm}$;     (b)  $\alpha \approx {14.5}^{\circ }$
15. We are looking at a tree from the distance of 60 m. From a point on the ground the angle of elevation of the top of this tree is 25°. Find the height of this tree.
    Solution:    Height = 28 m.

Solving a general triangle

16. In a triangle $ABC$, $b=AC=6\mathrm{cm},c=AB=10\mathrm{cm}$ and $\alpha =C\hat{A}B={70}^{\circ }$. The height $h_c$ has the endpoints $C$ and $H$.
    

    (a)   Find the height $h_c$.

    (b)   Find the distances $AH$ and $HB$.

    (c)   Hence, calculate the side $a=BC$.

    Solutions:    (a)  $h_c\approx 5.64\mathrm{cm}$;     (b)  $AH\approx 2.05\mathrm{cm},HB\approx 7.95\mathrm{cm}$;     (c)  $a\approx 9.74\mathrm{cm}$
17. ?
    ?

    The cosine rule can be used in any triangle:
    
    $a^2=b^2+c^2-2bc\cos \hat{A}$
    
    $b^2=a^2+c^2-2ac\cos \hat{B}$
    
    $c^2=a^2+b^2-2ab\cos \hat{C}$
    
    
    In a triangle $ABC$, $b=AC=5\mathrm{cm},c=AB=8\mathrm{cm}$ and $\alpha =C\hat{A}B={60}^{\circ }$. Find the length of the side $a$.
    Solution:    $a=7\mathrm{cm}$
18. In a triangle $ABC$, $a=BC=13\mathrm{cm},b=AC=19\mathrm{cm}$ and $\gamma =A\hat{C}B={64}^{\circ }{30}^{\prime }$. Find the length of the side $c$.
    Solution:    $c\approx 17.8\mathrm{cm}$
19. In a triangle $ABC$, $a=BC=9\mathrm{cm},b=AC=14\mathrm{cm}$ and $c=AB=13\mathrm{cm}$. Calculate the angles $\alpha ,\beta$ and $\gamma$.
    Solutions:    $\alpha \approx {38.7}^{\circ },\beta \approx {76.7}^{\circ },\gamma \approx {64.6}^{\circ }$
20. In a triangle $ABC$, $a=BC=13\mathrm{cm},b=AC=8\mathrm{cm}$ and $c=AB=7\mathrm{cm}$. Calculate the angles $\alpha ,\beta$ and $\gamma$. Write your results in degrees and minutes.
    Solutions:    $\alpha ={120}^{\circ },\beta \approx {32}^{\circ }{12}^{\prime },\gamma \approx {27}^{\circ }{48}^{\prime }$
21. In a triangle $\mathrm{△}ABC$, $a=BC=2\mathrm{cm},b=AC=7\mathrm{cm}$ and $c=AB=3\sqrt{3}\mathrm{cm}$. Calculate the angle $\beta$.
    Solution:    $\beta ={150}^{\circ }$
22. In a triangle $\mathrm{△}ABC$, $a=23\mathrm{cm},c=17\mathrm{cm}$ and $\beta ={107}^{\circ }$.
    

    (a)   Find the side $b$.

    (b)   Calculate the angles $\alpha$ and $\gamma$.

    Solutions:    (a)  $b\approx 32.4\mathrm{cm}$;     (b)  $\alpha \approx {42.8}^{\circ },\gamma \approx {30.2}^{\circ }$
23. In a triangle $\mathrm{△}ABC$, $\alpha ={30}^{\circ },\beta ={71}^{\circ }$ and $b=16\mathrm{cm}$.
    

    (a)   Find the height $h_c$.

    (b)   Calculate the side $a$.

    Solutions:    (a)  $h_c=8\mathrm{cm}$;     (b)  $a\approx 8.46\mathrm{cm}$
24. ?
    ?

    The sine rule can be used in any triangle:
    
    $\frac{a}{\sin \hat{A}}=\frac{b}{\sin \hat{B}}=\frac{c}{\sin \hat{C}}$
    In a triangle $\mathrm{△}ABC$, $\alpha ={57}^{\circ },\beta ={71}^{\circ }$ and $c=15\mathrm{cm}$.
    

    (a)   Find the angle $\gamma$.

    (b)   Find the sides $a$ and $b$.

    Solutions:    (a)  $\gamma ={52}^{\circ }$;     (b)  $a\approx 16.0\mathrm{cm},b\approx 18.0\mathrm{cm}$
25. In a triangle $\mathrm{△}ABC$, $a=23\mathrm{cm},b=39\mathrm{cm}$ and $\beta ={83}^{\circ }$.
    

    (a)   Find the angles $\alpha$ and $\gamma$.

    (b)   Find the side $c$.

    Solutions:    (a)  $\alpha \approx {35.8}^{\circ },\gamma \approx {61.2}^{\circ }$;     (b)  $c\approx 34.4\mathrm{cm}$
26. We watch the peak of a mountain from two different points in the plain. When viewed from the first point, the angle of elevation of the peak is $\alpha ={35}^{\circ }$. When viewed from the second point, the angle of elevation of the peak is $\beta ={50}^{\circ }$. The distance between the two points is $d=430\mathrm{m}$. Calculate the height of this mountain above the plain.
    
    Solutions:    $h\approx 730\mathrm{m}$

Area

27. ?
    ?

    Area of a triangle can be calculated using the formula:
    
    $A=\frac{base\times height}{2}$
    
    or also using:
    
    $A=\frac{1}{2}ab\sin \hat{C}$
    
    $A=\frac{1}{2}ac\sin \hat{B}$
    
    $A=\frac{1}{2}bc\sin \hat{A}$
    In a triangle $\mathrm{△}ABC$, $b=6\mathrm{cm},c=8\mathrm{cm}$ and $\alpha ={30}^{\circ }$.
    

    (a)   Find the height $h_c$.

    (b)   Find the area $A$.

    Solutions:    (a)  $h_c=3\mathrm{cm}$;     (b)  $A=12{\mathrm{cm}}^2$
28. In a triangle $\mathrm{△}ABC$, $a=14\mathrm{cm},c=11\mathrm{cm}$ and $\beta ={77}^{\circ }{33}^{\prime }$.
    

    (a)   Find the side $b$.

    (b)   Find the area $A$.

    Solutions:    (a)  $b\approx 15.8\mathrm{cm}$;     (b)  $A\approx 75.2{\mathrm{cm}}^2$
29. In a triangle $\mathrm{△}ABC$, $a=5\mathrm{cm},b=2\mathrm{cm}$ and $\gamma ={58}^{\circ }$. Find the area of this triangle.
    Solution:    $A\approx 4.24{\mathrm{cm}}^2$
30. In a triangle $\mathrm{△}ABC$, $a=7\mathrm{cm},b=8\mathrm{cm}$ and $c=5\mathrm{cm}$.
    

    (a)   Find the angle $\alpha =C\hat{A}B$.

    (b)   Find the area $A$.

    Solutions:    (a)  $\alpha ={60}^{\circ }$;     (b)  $A=10\sqrt{3}\mathrm{cm}\approx 17.3{\mathrm{cm}}^2$
31. ?
    ?

    Area of a triangle can be calculated using the Heron formula if you know all three sides:
    
    $A=\sqrt{s(s-a)(s-b)(s-c)}$
    
    where $s=\frac{a+b+c}{2}$
    A triangle $\mathrm{△}ABC$ has sides: $a=26\mathrm{cm},b=15\mathrm{cm}$ and $c=37\mathrm{cm}$. Find the area of this triangle.
    Solution:    $A=156{\mathrm{cm}}^2$
32. A triangle $\mathrm{△}ABC$ has sides: $a=14\mathrm{cm},b=19\mathrm{cm}$ and $c=7\mathrm{cm}$. Find the area of this triangle.
    Solution:    $A\approx 39.5{\mathrm{cm}}^2$
33. A triangle $\mathrm{△}ABC$ has sides: $a=6\mathrm{cm},b=8\mathrm{cm}$ and $c=17\mathrm{cm}$. Find the area of this triangle.
    Solution:    A triangle with these sides doesn't exist.
34. A triangle $\mathrm{△}ABC$ has the sides: $a=17\mathrm{cm},b=10\mathrm{cm}$ and $c=21\mathrm{cm}$.
    

    (a)   Find the area.

    (b)   Find the height $h_c$.

    Solutions:    (a)  $A=84{\mathrm{cm}}^2$;     (b)  $h_c=8\mathrm{cm}$
35. In a triangle $\mathrm{△}ABC$: $\alpha ={39}^{\circ },\beta ={68}^{\circ }$ and $c=14\mathrm{cm}$.
    

    (a)   Find the angle $\gamma$.

    (b)   Find the sides $a$ and $b$.

    (c)   Find the perimeter and area.

    Solutions:    (a)  $\gamma ={73}^{\circ }$;     (b)  $a\approx 9.21\mathrm{cm},b\approx 13.6\mathrm{cm}$;     (c)  $P\approx 36.8\mathrm{cm},A\approx 59.8{\mathrm{cm}}^2$
36. In a parallelogram $ABCD$: $a=AB=15\mathrm{cm},b=BC=7\mathrm{cm}$ and $\alpha =B\hat{A}D={60}^{\circ }$.
    

    (a)   Find the diagonal $f=BD$.

    (b)   Find the area of this parallelogram.

    Solutions:    (a)  $f=13\mathrm{cm}$;     (b)  $A\approx 90.9{\mathrm{cm}}^2$
37. A rhombus $ABCD$ has the side $a=7\mathrm{cm}$ and the angle $\alpha ={69}^{\circ }$.
    

    (a)   Find the diagonal $f=BD$.

    (b)   Find the height $h$.

    (c)   Find the area of this rhombus.

    Solutions:    (a)  $f\approx 7.93\mathrm{cm}$;     (b)  $h\approx 6.54\mathrm{cm}$;     (c)  $A\approx 45.7{\mathrm{cm}}^2$
38. Farmer has a field in form of a quadrilateral $ABCD$. He measured the sides: $AB=140\mathrm{m}$, $BC=85\mathrm{m}$, $CD=125\mathrm{m}$, $DA=80\mathrm{m}$. Finally, he measured one of the diagonals: $AC=195\mathrm{m}$. Determine the area of this field.
    Solution:    $A\approx 8250{\mathrm{m}}^2$
39. A regular pentagon is inscribed in the circle with the radius $r=8\mathrm{cm}$.
    

    (a)   Find the side of this pentagon.

    (b)   Find the diagonal.

    (c)   Find the area.

    Solutions:    (a)  $a\approx 9.40\mathrm{cm}$;     (b)  $d\approx 15.2\mathrm{cm}$;     (c)  $A\approx 152{\mathrm{cm}}^2$
40. A regular octagon $ABCDEFGH$ has the side $a=10\mathrm{cm}$.
    

    (a)   Find the diagonal $AC$.

    (b)   Find the inradius $r$ (radius of the inscribed circle).

    (c)   Find the area of this octagon.

    Solutions:    (a)  $AC\approx 18.5\mathrm{cm}$;     (b)  $r\approx 12.1\mathrm{cm}$;     (c)  $A\approx 483{\mathrm{cm}}^2$
41. A regular nonagon has the side $a=6\mathrm{cm}$.
    

    (a)   Find the circumradius $R$ (radius of the circumscribed circle).

    (b)   Find the length of the longest diagonal.

    (c)   Find the area of this nonagon.

    Solutions:    (a)  $R\approx 8.77\mathrm{cm}$;     (b)  $d\approx 17.3\mathrm{cm}$;     (c)  $A\approx 223{\mathrm{cm}}^2$
42. A regular hexagon has the side $a=16\mathrm{cm}$.
    

    (a)   Find the area of this hexagon.

    (b)   Find the circumradius $R$ (radius of the circumscribed circle).

    (c)   Find the inradius $r$ (radius of the inscribed circle).

    (d)   By what percentage is the circumradius greater than the inradius?

    Solutions:    (a)  $A\approx 665{\mathrm{cm}}^2$;     (b)  $R=a=16\mathrm{cm}$;     (c)  $r\approx 13.9\mathrm{cm}$;     (d)  By $15.5\mathrm{\%}$

Circle, arc, circular sector and circular segment

43. A circle has the radius $r=5\mathrm{cm}$. Find the area and perimeter of this circle.
    Solutions:    $A=25\pi {\mathrm{cm}}^2\approx 78.5{\mathrm{cm}}^2,P=10\pi \mathrm{cm}\approx 31.4\mathrm{cm}$
44. A circle has the circumference $100\mathrm{cm}$. Find the area of this circle.
    Solutions:    $r\approx 15.9\mathrm{cm},A\approx 796{\mathrm{cm}}^2$
45. A circle has the area $A=95{\mathrm{cm}}^2$. Calculate the radius and the perimeter of this circle.
    Solutions:    $r\approx 5.50\mathrm{cm},P\approx 34.6\mathrm{cm}$
46. A circle has the radius $r=10\mathrm{cm}$,
    

    (a)   Find the circumference.

    (b)   Find the length of the arc with the central angle ${90}^{\circ }$.

    (c)   Find the area of this circle.

    (d)   Find the area of the circular sector with the central angle ${90}^{\circ }$.

    Solutions:    (a)  $P\approx 62.8\mathrm{cm}$;     (b)  $L\approx 15.7\mathrm{cm}$;     (c)  $A_0\approx 314{\mathrm{cm}}^2$;     (d)  $A_1\approx 78.5{\mathrm{cm}}^2$
47. ?
    ?

    Length of an arc can be calculated using the formula:
    
    $L=2\pi r\frac{\theta }{{360}^{\circ }}$     or
    
    $L=\pi r\frac{\theta }{{180}^{\circ }}$
    
    Area of a circular sector can be calculated using the formula:
    
    $A=\pi r^2\frac{\theta }{{360}^{\circ }}$
    A circle has the radius $r=12\mathrm{cm}$. Find the length of the arc with the central angle $\theta ={40}^{\circ }$.
    Solution:    $L=\frac{8}{3}\pi \mathrm{cm}\approx 8.38\mathrm{cm}$
48. Find the area of the sector of the circle with the radius $r=18\mathrm{cm}$ and the central angle $\theta ={10}^{\circ }$.
    Solution:    $A=9\pi {\mathrm{cm}}^2\approx 28.3{\mathrm{cm}}^2$
49. In a circle with the radius $r=15\mathrm{cm}$, arc $L$ subtends the angle ${69}^{\circ }$ at the centre.
    

    (a)   Find the length of the arc $L$.

    (b)   Find the area of the corresponding circular sector.

    Solutions:    (a)  $L\approx 18.1\mathrm{cm}$;     (b)  $A\approx 135{\mathrm{cm}}^2$
50. A circular sector has the radius $r=18\mathrm{cm}$. The area of this sector is $A=288{\mathrm{cm}}^2$.
    

    (a)   Find the corresponding central angle.

    (b)   Find the length of the corresponding circular arc.

    Solutions:    (a)  $\theta \approx {102}^{\circ }$;     (b)  $L=32\mathrm{cm}$
51. Given the radius $r=8\mathrm{cm}$ and central angle $\theta ={90}^{\circ }$,
    

    (a)   find the area of the circular sector,

    (b)   find the area of the corresponding circular segment.

    Solutions:    (a)  $A_1\approx 50.3{\mathrm{cm}}^2$;     (b)  $A_2\approx 18.3{\mathrm{cm}}^2$
52. A circle has the radius $r=26\mathrm{cm}$. Calculate the area of the circular segment corresponding to the central angle $\theta ={125}^{\circ }$.
    Solution:    $A\approx 461{\mathrm{cm}}^2$
53. Circular segment is a part of the circle with $r=13\mathrm{cm}$. The arc corresponding to this circular segment has the length $L=15\mathrm{cm}$. Calculate the area of this circular segment.
    Solution:    $A\approx 20.2{\mathrm{cm}}^2$

Measuring angles in radians

54. Write the following angles in radians:
    

    (a)   ${30}^{\circ }$

    (b)   ${45}^{\circ }$

    (c)   ${120}^{\circ }$

    (d)   ${135}^{\circ }$

    Solutions:    (a)  ${30}^{\circ }=\frac{\pi }{6}$;     (b)  ${45}^{\circ }=\frac{\pi }{4}$;     (c)  ${120}^{\circ }=\frac{2\pi }{3}$;     (d)  ${135}^{\circ }=\frac{3\pi }{4}$
55. Convert the following angles to radians. Round the results to three significant figures.
    

    (a)   ${75}^{\circ }$

    (b)   ${20}^{\circ }$

    (c)   ${10}^{\circ }$

    (d)   $1^{\circ }$

    Solutions:    (a)  ${75}^{\circ }=\frac{5\pi }{12}\approx 1.31$;     (b)  ${20}^{\circ }=\frac{\pi }{9}\approx 0.349$;     (c)  ${10}^{\circ }=\frac{\pi }{18}\approx 0.175$;     (d)  $1^{\circ }=\frac{\pi }{180}\approx 0.0175$
56. Convert the following angles to degrees:
    

    (a)   $\frac{\pi }{3}$

    (b)   $\frac{\pi }{2}$

    (c)   $\frac{5\pi }{6}$

    (d)   $\frac{5\pi }{4}$

    Solutions:    (a)  $\frac{\pi }{3}={60}^{\circ }$;     (b)  $\frac{\pi }{2}={90}^{\circ }$;     (c)  $\frac{5\pi }{6}={150}^{\circ }$;     (d)  $\frac{5\pi }{4}={225}^{\circ }$
57. Convert the following angles to degrees:
    

    (a)   $1$

    (b)   $0.25$

    (c)   $1.85$

    (d)   $3.14$

    Solutions:    (a)  $1\mathrm{rad}\approx {57.3}^{\circ }$;     (b)  $0.25\mathrm{rad}\approx {14.3}^{\circ }$;     (c)  $1.85\mathrm{rad}\approx {106}^{\circ }$;     (d)  $3.14\mathrm{rad}\approx {180}^{\circ }$
58. A circle has the radius $r=9\mathrm{cm}$. Find the length of the arc which subtends the angle $\theta =2\mathrm{radians}$.
    Solution:    $L=18\mathrm{cm}$
59. Given the radius $r=6\mathrm{cm}$ and central angle $\theta =\frac{1}{8}\pi$,
    

    (a)   find the area of the circular sector,

    (b)   find the length of the arc.

    Solutions:    (a)  $A\approx 7.07{\mathrm{cm}}^2$;     (b)  $L\approx 2.36\mathrm{cm}$
60. Find the area of the circular segment of the radius $r=14\mathrm{cm}$ and central angle $\theta =1$.
    Solution:    $A\approx 15.5{\mathrm{cm}}^2$

 Index