Three faces of a cuboid

The diagonal of three faces of a cuboid are 13,√281, and 20 units. Then the total surface area of the cuboid is.

Correct answer:

S =  664

Step-by-step explanation:

$$\begin{gathered} d_1=13 \\ {d}_2=\sqrt{281}=\sqrt{281}\doteq 16.7631 \\ {d}_3=20 \\ {d}_1^2=a^2+b^2 \\ {d}_2^2=b^2+c^2 \\ {d}_3^2=a^2+c^2 \\ {a}^2=d_1^2-b^2=d_1^2-d_2^2+c^2=d_1^2-d_2^2+d_3^2-a^2 \\ 2a^2=d_1^2-d_2^2+d_3^2 \\ a=\sqrt{{\displaystyle \frac{d_1^2-d_2^2+d_3^2}{2}}}=\sqrt{{\displaystyle \frac{13^2-16.7631^2+20^2}{2}}}=12 \\ \dots \\ b=\sqrt{{\displaystyle \frac{d_1^2+d_2^2-d_3^2}{2}}}=\sqrt{{\displaystyle \frac{13^2+16.7631^2-20^2}{2}}}=5 \\ \dots \\ c=\sqrt{{\displaystyle \frac{-d_1^2+d_2^2+d_3^2}{2}}}=\sqrt{{\displaystyle \frac{-13^2+16.7631^2+20^2}{2}}}=16 \\ S=2\cdot (a\cdot b+b\cdot c+c\cdot a)=2\cdot (12\cdot 5+5\cdot 16+16\cdot 12)=664 \end{gathered}$$

Try another example

Related math problems and questions:

• Solid cuboid
  A solid cuboid has a volume of 40 cm³. The cuboid has a total surface area of 100 cm squared. One edge of the cuboid has a length of 2 cm. Find the length of a diagonal of the cuboid. Give your answer correct to 3 sig. Fig.
• Faces diagonals
  If a cuboid's diagonals are x, y, and z (wall diagonals or three faces), then find the cuboid volume. Solve for x=1.3, y=1, z=1.2
• Nice prism
  Calculate the cuboid's surface if the sum of its edges is a + b + c = 19 cm and the body diagonal size u = 13 cm.
• Body diagonal
  The cuboid has a volume of 32 cm³. Its side surface area is double as one of the square bases. What is the length of the body diagonal?
• Regular square prism
  The volume of a regular square prism is 192 cm³. The size of its base edge and the body height is 1: 3. Calculate the surface of the prism.
• Block or cuboid
  The wall diagonals of the block have sizes of √29cm, √34cm, √13cm. Calculate the surface and volume of the block.
• Ratio of edges
  The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid.
• Cuboid diagonals
  The cuboid has dimensions of 15, 20 and 40 cm. Calculate its volume and surface, the length of the body diagonal and the lengths of all three wall diagonals.
• Wall diagonal
  Calculate the length of wall diagonal of the cube whose surface is 384 cm square.
• Ratio-cuboid
  The lengths of the edges of the cuboid are in the ratio 2: 3: 6. Its body diagonal is 14 cm long. Calculate the volume and surface area of the cuboid.
• Diamond perimeter
  Calculate the diamond circumference which area is 288 cm square and one diagonal has a size of 124 cm.
• Quadrangular prism
  Calculate the volume and surface area of a regular quadrangular prism 35 cm high and the base diagonal of 22 cm.
• Cube wall
  The perimeter of one cube wall is 120 meters. Calculate the surface area and the body diagonal of this cube.
• Cuboid diagonal
  Calculate the volume and surface area of the cuboid ABCDEFGH, which sides a, b, c has dimensions in the ratio of 9:3:8. If you know that the diagonal wall AC is 86 cm, and the angle between AC and space diagonal AG is 25 degrees.
• Support colum
  Calculate the volume and surface of the support column that is shaped as a perpendicular quadrangular prism whose base is a rhombus with a diagonals u1 = 102 cm u2 = 64 cm. Column height is 1. 5m.
• Moivre 2
  Find the cube roots of 125(cos 288° + i sin 288°).
• Diagonals of a prism
  The base of the square prism is a rectangle with dimensions of 3 dm and 4 dm. The height of the prism is 1 m. Find out the angle between the body diagonal with the diagonal of the base.