Quickly determine a triangle's perimeter by entering: Three sides, Two sides and the included angle or One side and two adjacent angles.
Simply add the lengths of the three sides: \( P = a + b + c \)
Use the law of cosines to calculate the third side, then add them together:
Given two sides \(a\) and \(b\) with the included angle \(C\), the third side \(c\) is: \( c = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)} \) The perimeter is then: \( P = a + b + c \) If it is not an included angle, we need to discuss the side corresponding to the angle, and then use the law of sines to find the corresponding angle and side length. Finally, add the three sides together to get the perimeter of the triangle.
If the known side is \(a\) and its opposite angle is \(A\), and the adjacent angles are \(B\) and \(C\), calculate the third angle as: \( A = 180^\circ - B - C \quad \text{(or } \pi - B - C \text{ for radians)} \) Using the law of sines, find the other two sides \(b\) and \(c\): \( b = \frac{a \cdot \sin(B)}{\sin(A)} \) \( c = \frac{a \cdot \sin(C)}{\sin(A)} \) The perimeter is: P = a + b + c
Solution:
\( P = 5 + 7 + 9 = 21 \)
Result: The perimeter is 21.
Solution:
1. Calculate the third side \(c\):
\( c = \sqrt{6^2 + 8^2 - 2 \times 6 \times 8 \times \cos(60^\circ)} = \sqrt{100 - 48} = \sqrt{52} \approx 7.21 \)
2. Calculate the perimeter:
\( P = 6 + 8 + 7.21 \approx 21.21 \)
Result: The perimeter is approximately 21.21.
Solution:
1. Calculate the third angle \(A\):
\( A = 180^\circ - 45^\circ - 60^\circ = 75^\circ \)
2. Use the law of sines to find \(b\) and \(c\):
\( b = \frac{10 \times \sin(45^\circ)}{\sin(75^\circ)} \approx 7.32 \)
\( c = \frac{10 \times \sin(60^\circ)}{\sin(75^\circ)} \approx 8.97 \)
3. Calculate the perimeter:
\( P = 10 + 7.32 + 8.97 \approx 26.29 \)
Result: The perimeter is approximately 26.29.