Trapezoid - hard example

Base of the trapezoid are: 24, 16 cm.
Diagonal 22, 26 cm.
Calculate its area and perimeter.

Result

S =  264 cm2
o =  67.966 cm

Solution:

a=24 cm c=16 cm  u=22 cm v=26 cm  w=a+c=24+16=40 cm  s=u+v+w2=22+26+402=44 cm  S=s⋅ (s−w)⋅ (s−u)⋅ (s−v)=44⋅ (44−40)⋅ (44−22)⋅ (44−26)=264 cm2a=24 \ \text{cm} \ \\ c=16 \ \text{cm} \ \\ \ \\ u=22 \ \text{cm} \ \\ v=26 \ \text{cm} \ \\ \ \\ w=a+c=24+16=40 \ \text{cm} \ \\ \ \\ s=\dfrac{ u+v+w }{ 2 }=\dfrac{ 22+26+40 }{ 2 }=44 \ \text{cm} \ \\ \ \\ S=\sqrt{ s \cdot \ (s-w) \cdot \ (s-u) \cdot \ (s-v) }=\sqrt{ 44 \cdot \ (44-40) \cdot \ (44-22) \cdot \ (44-26) }=264 \ \text{cm}^2

Try calculation via our triangle calculator.

S=a+c2⋅ h=w2⋅ h  h=2⋅ S/w=2⋅ 264/40=665=13.2 cm  u2=h2+a12=13.22+17.62=435625=174.24  a1=u2−h2=222−13.22=885=17.6 cm a2=a−a1=24−17.6=325=6.4 cm  b2=h2+a22  b=h2+a22=13.22+6.42≐14.6697 cm  c1=v2−h2=262−13.22=1125=22.4 cm c2=a−c1=24−22.4=85=1.6 cm  d2=h2+c22  d=h2+c22=13.22+1.62≐13.2966 cm   o=a+b+c+d=24+14.6697+16+13.2966≐67.9663≐67.966 cmS=\dfrac{ a+c }{ 2 } \cdot \ h=\dfrac{ w }{ 2 } \cdot \ h \ \\ \ \\ h=2 \cdot \ S/w=2 \cdot \ 264/40=\dfrac{ 66 }{ 5 }=13.2 \ \text{cm} \ \\ \ \\ u^2=h^2 +a_{1}^2=13.2^2 +17.6^2=\dfrac{ 4356 }{ 25 }=174.24 \ \\ \ \\ a_{1}=\sqrt{ u^2 - h^2 }=\sqrt{ 22^2 - 13.2^2 }=\dfrac{ 88 }{ 5 }=17.6 \ \text{cm} \ \\ a_{2}=a - a_{1}=24 - 17.6=\dfrac{ 32 }{ 5 }=6.4 \ \text{cm} \ \\ \ \\ b^2=h^2 + a_{2}^2 \ \\ \ \\ b=\sqrt{ h^2 + a_{2}^2 }=\sqrt{ 13.2^2 + 6.4^2 } \doteq 14.6697 \ \text{cm} \ \\ \ \\ c_{1}=\sqrt{ v^2 - h^2 }=\sqrt{ 26^2 - 13.2^2 }=\dfrac{ 112 }{ 5 }=22.4 \ \text{cm} \ \\ c_{2}=a - c_{1}=24 - 22.4=\dfrac{ 8 }{ 5 }=1.6 \ \text{cm} \ \\ \ \\ d^2=h^2 + c_{2}^2 \ \\ \ \\ d=\sqrt{ h^2 + c_{2}^2 }=\sqrt{ 13.2^2 + 1.6^2 } \doteq 13.2966 \ \text{cm} \ \\ \ \\ \ \\ o=a+b+c+d=24+14.6697+16+13.2966 \doteq 67.9663 \doteq 67.966 \ \text{cm}

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