Pythagoras’ Theorem and Trigonometry In 3D (Copy)
1.
A cuboid has edges 6 cm, 8 cm, 10 cm.
Find the length of its space diagonal.
2.
A cube has side 12 cm.
Find the length of its space diagonal.
3.
A cuboid has base 9 cm by 12 cm, height 20 cm.
Find the diagonal of the cuboid.
4.
A tent has the shape of a right triangular prism. Cross-section is a right triangle with sides 6 m, 8 m, 10 m. Length = 12 m.
Find the slant edge along the roof.
5.
A pyramid has square base 10 cm × 10 cm and height 12 cm.
Find the slant height from the apex to the midpoint of a side.
6.
Find the angle between the slant edge of Q5 and the base.
7.
A cuboid has dimensions 5 cm × 12 cm × 84 cm.
Find the space diagonal.
8.
A square-based pyramid has base 8 cm and height 15 cm.
Find the angle between the slant edge and the base.
9.
A room is 6 m long, 5 m wide, and 4 m high.
Find the distance from one bottom corner to the opposite top corner.
10.
In a cube of side 8 cm, find the angle between a diagonal of a face and a space diagonal.
11.
A triangular prism has equilateral triangle base side 6 cm, prism length 12 cm.
Find the length of the diagonal running from one triangle vertex to the farthest vertex.
12.
A cuboid has edges 9 cm, 12 cm, 15 cm.
Find the angle between the space diagonal and the base diagonal of length 15 cm.
13.
A cuboid has edges 3 cm, 4 cm, 12 cm.
Find the space diagonal.
14.
In a square pyramid with base 6 cm and height 8 cm, find the slant edge length.
15.
Find the angle between the slant edge and base in Q14.
16.
A cube has side 10 cm.
Find the angle between a face diagonal and a space diagonal.
17.
A cuboid is 5 m × 12 m × 13 m.
Find the angle between the diagonal across the 5×12 face and the space diagonal.
18.
A triangular prism has right triangle base with sides 5, 12, 13, prism length 10.
Find the distance between a vertex and the farthest vertex.
19.
In a cuboid 7×24×25 cm, find the space diagonal.
20.
A pyramid has base side 20 cm, height 15 cm.
Find the angle between slant edge and base.
21.
A cube has side 4 cm.
Find the angle between a diagonal of one face and an adjacent face diagonal.
22.
A pyramid has square base side 12 cm, height 9 cm.
Find the slant height from apex to midpoint of a side.
23.
Find the angle between the slant height in Q22 and the base.
24.
In a cuboid 10×24×26 cm, find the angle between the longest edge and the space diagonal.
25.
In a cube of side 6 cm, find the angle between two face diagonals meeting at a corner.
26.
A cuboid 8×15×17 cm.
Find its space diagonal.
27.
A right pyramid has square base 14 cm, height 24 cm.
Find the slant edge.
28.
Find the angle between the slant edge in Q27 and the base.
29.
A cuboid 9×40×41 cm.
Find the angle between the vertical edge (41) and the space diagonal.
30.
A cube has side 20 cm.
Find the angle between a line joining opposite corners of one face and a space diagonal.
1.
Diagonal = √(6²+8²+10²) = √(36+64+100) = √200 = 14.1 cm.
2.
Space diagonal = side√3 = 12√3 ≈ 20.8 cm.
3.
Diagonal = √(9²+12²+20²) = √(81+144+400) = √625 = 25 cm.
4.
Roof edge = 10 m (hypotenuse of cross-section).
Already known → 10 m.
5.
Slant height = √((10/2)²+12²) = √(25+144) = √169 = 13 cm.
6.
Angle = tan⁻¹(12/5) = tan⁻¹(2.4) ≈ 67.4°.
7.
Diagonal = √(5²+12²+84²) = √(25+144+7056) = √7225 = 85 cm.
8.
Slant edge = √((8/2)²+15²) = √(16+225)=√241=15.5 cm.
Angle = tan⁻¹(15/4) = 75.0°.
9.
Diagonal = √(6²+5²+4²) = √(36+25+16) = √77 = 8.8 m.
10.
Face diagonal = 8√2 = 11.3.
Space diagonal = 8√3=13.9.
cosθ=11.3×13.9/(11.3×13.9)? Correction: angle between face diagonal (8√2,0,0) and space diagonal (8,8,8).
Dot = 8×8+8×8=128. |FD|=11.3, |SD|=13.9.
cosθ=128/(11.3×13.9)=128/157=0.815.
θ=35.8°.
11.
Base equilateral triangle height=√(6²−3²)=√27=5.2.
Farthest diagonal=√(12²+6²)=√180=13.4 cm.
12.
Space diag=√(9²+12²+15²)=√450=21.2.
Base diagonal=√(9²+12²)=15.
cosθ=15/21.2=0.707.
θ=45°.
13.
Diagonal=√(3²+4²+12²)=√(9+16+144)=√169= 13 cm.
14.
Slant edge=√(8²+(6√2/2)²)=√(64+18)=√82= 9.1 cm.
15.
Angle=tan⁻¹(8/(6/√2))=tan⁻¹(8/4.24)=tan⁻¹(1.89)= 62.7°.
16.
Face diagonal=10√2=14.1.
Space diagonal=10√3=17.3.
cosθ= (100+100)/ (14.1×17.3)? Check vectors.
θ≈ 35.3°.
17.
Face diag=√(5²+12²)=13.
Space diag=√(5²+12²+13²)=√(25+144+169)=√338=18.4.
cosθ=13/18.4=0.707 → θ=45°.
18.
Farthest vertex = √(13²+10²)=√269=16.4 cm.
19.
Diagonal=√(7²+24²+25²)=√(49+576+625)=√1250= 35.4 cm.
20.
Slant edge=√((20/√2)²+15²)=√(200+225)=√425=20.6 cm.
Angle=tan⁻¹(15/(10√2))=tan⁻¹(15/14.1)=46.8°.
21.
Cube side=4. Face diagonal=4√2=5.66. Another face diagonal adjacent. Angle between=60°.
22.
Slant height=√(9²+(12/2)²)=√(81+36)=√117=10.8 cm.
23.
Angle=tan⁻¹(9/6)=tan⁻¹(1.5)=56.3°.
24.
Space diag=√(10²+24²+26²)=√(100+576+676)=√1352=36.8.
cosθ=26/36.8=0.707. θ=45°.
25.
Angle between face diagonals meeting at corner=90°.
26.
Diagonal=√(8²+15²+17²)=√(64+225+289)=√578=24.0 cm.
27.
Slant edge=√(24²+(14/√2)²)=√(576+98)=√674=25.9 cm.
28.
Angle=tan⁻¹(24/(7√2))=tan⁻¹(24/9.9)=tan⁻¹(2.42)=67.5°.
29.
Space diag=√(9²+40²+41²)=√(81+1600+1681)=√3362=58.0.
cosθ=41/58=0.707. θ=45°.
30.
Face diagonal=20√2=28.3. Space diag=20√3=34.6.
cosθ=28.3/34.6=0.817. θ=35.0°.