Geometry of solids

Volume and surface area

1. ?
   ?

   Volume of a cuboid with length \(\ell\), width \(w\) and height \(h\) is:
   
   \(V=\ell\,w\,h\)
   A cuboid has the length \(\ell=18~\mathrm{cm}\), width \(w=10~\mathrm{cm}\) and heigth \(h=25~\mathrm{cm}\).
   

   (a)   Find the volume.

   (b)   Find the surface area.

   (c)   Calculate the space diagonal.

   Solutions:    (a)  \(V=4500~\mathrm{cm}^3\);     (b)  \(A=1760~\mathrm{cm}^2\);     (c)  \(D\approx32.4~\mathrm{cm}\)
2. A cuboid has \(\ell=w=4~\mathrm{cm}\), \(h=7~\mathrm{cm}\).
   

   (a)   Find the volume.

   (b)   Find the space diagonal.

   (c)   Calculate the angle between the space diagonal and the diagonal of the base.

   Solutions:    (a)  \(V=112~\mathrm{cm}^3\);     (b)  \(D=9~\mathrm{cm}\);     (c)  \(\varphi\approx51.1^\circ\)
3. A cube has \(\ell=w=h=10~\mathrm{cm}\).
   

   (a)   Find the volume.

   (b)   Find the surface area.

   Solutions:    (a)  \(V=1000~\mathrm{cm}^3\);     (b)  \(A=600~\mathrm{cm}^2\)
4. A cube has the volume \(V=3375~\mathrm{cm}^3\).
   

   (a)   Find the edge.

   (b)   Find the surface area.

   Solutions:    (a)  \(\ell=w=h=15~\mathrm{cm}^3\);     (b)  \(A=1350~\mathrm{cm}^2\)
5. ?
   ?

   Volume of a prism with base area \(A\) and height \(h\) is:
   
   \(V=A\,h\)
   A three-sided right prism has the height \(h=12~\mathrm{cm}\). The base of this prism is a triangle with sides \(a=13~\mathrm{cm}\), \(b=11~\mathrm{cm}\), \(c=20~\mathrm{cm}\).
   

   (a)   Find the area of the base.

   (b)   Find the volume of this prism.

   (c)   Find the surface area of this prism.

   Solutions:    (a)  \(A_{BASE}=66~\mathrm{cm}^2\);     (b)  \(V=792~\mathrm{cm}^3\);     (c)  \(A_{TOTAL}=660~\mathrm{cm}^2\)
6. A three-sided right prism has the height \(h=9~\mathrm{cm}\). The base of this prism is a triangle with sides \(a=8~\mathrm{cm}\), \(b=15~\mathrm{cm}\) and \(c=17~\mathrm{cm}\).
   

   (a)   Find the volume of this prism.

   (b)   Find the total surface area of this prism.

   Solutions:    (a)  \(V=540~\mathrm{cm}^3\);     (b)  \(A_{TOTAL}=480~\mathrm{cm}^2\)
7. A three-sided right prism has the height \(h=11~\mathrm{cm}\). The base of this prism is an isosceles triangle with sides \(a=b=15~\mathrm{cm}\) and \(c=18~\mathrm{cm}\).
   

   (a)   Find the area of the base.

   (b)   Find the volume of this prism.

   (c)   Find the total surface area of this prism.

   Solutions:    (a)  \(A_{BASE}=108~\mathrm{cm}^2\);     (b)  \(V=1188~\mathrm{cm}^3\);     (c)  \(A_{TOTAL}=744~\mathrm{cm}^2\)
8. A regular three-sided prism has the height \(h=7~\mathrm{cm}\). The base of this prism is an equilateral triangle with sides equal to \(a=15~\mathrm{cm}\).
   

   (a)   Find the area of the base.

   (b)   Find the volume of this prism.

   (c)   Find the total surface area of this prism.

   Solutions:    (a)  \(A_{BASE}\approx97.4~\mathrm{cm}^2\);     (b)  \(V\approx682~\mathrm{cm}^3\);     (c)  \(A_{TOTAL}\approx510~\mathrm{cm}^2\)
9. A regular three-sided prism has the height \(h=7.3~\mathrm{cm}\) and the base edge \(a=8.4~\mathrm{cm}\).
   

   (a)   Find the volume of this prism.

   (b)   Find the total surface area of this prism.

   Solutions:    (a)  \(V\approx223~\mathrm{cm}^3\);     (b)  \(A_{TOTAL}\approx245~\mathrm{cm}^2\)
10. A regular six-sided prism has the height \(h=10~\mathrm{cm}\) and the base edge \(a=8~\mathrm{cm}\).
    

    (a)   Find the volume of this prism.

    (b)   Find the total surface area of this prism.

    Solutions:    (a)  \(V\approx1660~\mathrm{cm}^3\);     (b)  \(A_{TOTAL}\approx813~\mathrm{cm}^2\)
11. An uniform regular six-sided prism has the edge \(a=7~\mathrm{cm}\).
    

    (a)   Find the volume of this prism.

    (b)   Find the surface area of this prism.

    Solutions:    (a)  \(V\approx891~\mathrm{cm}^3\);     (b)  \(A_{TOTAL}\approx549~\mathrm{cm}^2\)
12. A uniform regular six-sided prism has the volume of 1 litre. Find the base edge of this prism.
    Solution:    \(a\approx7.27~\mathrm{cm}\)
13. ?
    ?

    Volume of a cylinder with radius \(r\) and height \(h\) is:
    
    \(V=\pi\,r^2\,h\)
    
    Area of the curved surface of a cylinder with radius \(r\) and height \(h\) is:
    
    \(A=2\,\pi\,r\,h\)
    A cylinder has the radius \(r=7~\mathrm{cm}\) and height \(h=9~\mathrm{cm}\).
    

    (a)   Find the volume of this cylinder.

    (b)   Find the area of the base.

    (c)   Find the area of the curved surface.

    (d)   Find the total surface area of this cylinder.

    Solutions:    (a)  \(V\approx1385~\mathrm{cm}^3\);     (b)  \(A_{BASE}\approx154~\mathrm{cm}^2\);     (c)  \(A_{CURVED}\approx396~\mathrm{cm}^2\);     (d)  \(A_{TOTAL}\approx704~\mathrm{cm}^2\)
14. A cylinder has the radius \(r=2~\mathrm{cm}\) and height \(h=19~\mathrm{cm}\).
    

    (a)   Find the volume of this cylinder.

    (b)   Find the total surface area of this cylinder.

    Solutions:    (a)  \(V\approx239~\mathrm{cm}^3\);     (b)  \(A_{TOTAL}\approx264~\mathrm{cm}^2\)
15. ?
    ?

    Volume of a right pyramid with base area \(A\) and height \(h\) is:
    
    \(V=\frac{1}{3}\,A\,h\)
    A right pyramid with a square base has the base edge \(a=12~\mathrm{cm}\) and the height \(h=15~\mathrm{cm}\). Find the volume of this pyramid.
    Solution:    \(V=720~\mathrm{cm}^3\)
16. A right pyramid has the height \(h=18~\mathrm{cm}\). The base of this pyramid is a triangle with sides \(a=13~\mathrm{cm}\), \(b=14~\mathrm{cm}\), \(c=15~\mathrm{cm}\). Find the volume of this pyramid.
    Solution:    \(V=504~\mathrm{cm}^3\)
17. A three-sided pyramid has the height \(h=20~\mathrm{cm}\). The base of this pyramid is an isosceles triangle with sides \(a=b=25~\mathrm{cm}\), \(c=14~\mathrm{cm}\). Find the volume of this pyramid.
    Solution:    \(V=1120~\mathrm{cm}^3\)
18. A four-sided pyramid has the height \(h=16~\mathrm{cm}\). The base of this pyramid is a rectangle with sides \(a=12~\mathrm{cm}\), \(b=8~\mathrm{cm}\). Find the volume of this pyramid.
    Solution:    \(V=512~\mathrm{cm}^3\)
19. A square right pyramid has the base edge \(a=10~\mathrm{cm}\) and the height \(h=12~\mathrm{cm}\).
    

    (a)   Find the area of the base.

    (b)   Find the volume of this pyramid.

    (c)   Find the slant height (height of a lateral face).

    (d)   Find the area of a lateral face.

    (e)   Find the total surface area of this pyramid.

    Solutions:    (a)  \(A_{BASE}=100~\mathrm{cm}^2\);     (b)  \(V=400~\mathrm{cm}^3\);     (c)  \(h_{SLANT}=13~\mathrm{cm}\);     (d)  \(A_{LATERAL}=65~\mathrm{cm}^2\);     (e)  \(A_{TOTAL}=360~\mathrm{cm}^2\)
20. A regular four-sided right pyramid has the base edge \(a=14~\mathrm{cm}\) and the height \(h=24~\mathrm{cm}\).
    

    (a)   Find the volume of this pyramid.

    (b)   Find the area of a lateral face.

    (c)   Find the total surface area of this pyramid.

    Solutions:    (a)  \(V=1568~\mathrm{cm}^3\);     (b)  \(A_{LATERAL}=175~\mathrm{cm}^2\);     (c)  \(A_{TOTAL}=896~\mathrm{cm}^2\)
21. A square right pyramid has the base edge \(a=32~\mathrm{cm}\) and the lateral edge \(e=34~\mathrm{cm}\).
    

    (a)   Find the area of a lateral face.

    (b)   Find the total surface area of this pyramid.

    (c)   Find the height of this pyramid.

    (d)   Find the volume of this pyramid. Express the volume in litres.

    Solutions:    (a)  \(A_{LATERAL}=480~\mathrm{cm}^2\);     (b)  \(A_{TOTAL}=2944~\mathrm{cm}^2\);     (c)  \(h\approx25.4~\mathrm{cm}\);     (d)  \(V\approx8.66~\ell\)
22. A regular four-sided pyramid has the base edge \(a=18~\mathrm{cm}\) and the lateral edge \(e=41~\mathrm{cm}\). Find the surface area of this pyramid.
    Solution:    \(A_{TOTAL}=1764~\mathrm{cm}^2\)
23. An uniform regular four-sided pyramid has the edge \(a=10~\mathrm{cm}\). Find the surface area of this pyramid.
    Solution:    \(A_{TOTAL}\approx273~\mathrm{cm}^2\)
24. An uniform regular three-sided pyramid has the edges \(a=e=12~\mathrm{cm}\). Find the surface area of this pyramid.
    Solution:    \(A_{TOTAL}\approx249~\mathrm{cm}^2\)
25. ?
    ?

    Volume of a cone with radius \(r\) and height \(h\) is:
    
    \(V=\frac{1}{3}\,\pi\,r^2\,h\)
    
    Area of the curved surface of a cone with radius \(r\), height \(h\) and slant height \(\ell\) is:
    
    \(A=\pi\,r\,\ell\)     where   \(\ell=\sqrt{r^2+h^2}\)
    A cone has the radius \(r=5~\mathrm{cm}\) and the height \(h=12~\mathrm{cm}\).
    

    (a)   Find the volume of this cone.

    (b)   Find the slant height of this cone.

    (c)   Find the total surface area of this cone.

    Solutions:    (a)  \(V=100\pi~\mathrm{cm}^3\approx314~\mathrm{cm}^3\);     (b)  \(\ell=13~\mathrm{cm}\);     (c)  \(A_{TOTAL}=90\pi~\mathrm{cm}^2\approx283~\mathrm{cm}^2\)
26. A cone has the radius \(r=5~\mathrm{cm}\) and the slant height \(\ell=12~\mathrm{cm}\).
    

    (a)   Find the height of this cone.

    (b)   Find the volume of this cone.

    (c)   Find the total surface area of this cone.

    Solutions:    (a)  \(h\approx10.9~\mathrm{cm}\);     (b)  \(V\approx286~\mathrm{cm}^3\);     (c)  \(A_{TOTAL}=85\pi~\mathrm{cm}^2\approx267~\mathrm{cm}^2\)
27. ?
    ?

    Volume of a sphere with radius \(r\) is:
    
    \(V=\frac{4}{3}\,\pi\,r^3\)
    
    Surface area of a sphere with radius \(r\) is:
    
    \(A=4\,\pi\,r^2\)
    A sphere has the radius \(r=3~\mathrm{cm}\).
    

    (a)   Find the volume of this sphere.

    (b)   Find the surface area of this sphere.

    Solutions:    (a)  \(V=36\pi~\mathrm{cm}^3\approx113~\mathrm{cm}^3\);     (b)  \(A=36\pi~\mathrm{cm}^2\approx113~\mathrm{cm}^2\)
28. A sphere has the volume \(V=1~\ell\) (one litre \(=1000~\mathrm{cm}^3\)).
    

    (a)   Find the raduis.

    (b)   Find the surface area of this sphere.

    Solutions:    (a)  \(r\approx6.20~\mathrm{cm}\);     (b)  \(A\approx484~\mathrm{cm}^2\)
29. Earth is (approximately) a sphere. Its circumference is the length of equator \(=40\,000~\mathrm{km}\).
    

    (a)   Find the radius of Earth.

    (b)   Find the surface area of Earth.

    Russia is the largest state in the world and its area is \(17\,098\,246~\mathrm{km}^2\).

    (c)   Calculate how many percent of the surface of Earth lies in Russia.

    Solutions:    (a)  \(r\approx6366~\mathrm{km}\);     (b)  \(A\approx5.09\cdot10^8~\mathrm{km}^2\);     (c)  \(\approx3.36\%\)

Angles in solids

30. A cuboid has \(\ell=w=6~\mathrm{cm}\), \(h=17~\mathrm{cm}\).
    

    (a)   Find the space diagonal.

    (b)   Find the angle between the space diagonal and lateral edge.

    (c)   Find the angle between the space diagonal and the base of this cuboid.

    Solutions:    (a)  \(D=19~\mathrm{cm}\);     (b)  \(\alpha\approx26.5^\circ\);     (c)  \(\varphi\approx63.5^\circ\)
31. Calculate the angle between the space diagonal and the base of a cube.
    Solution:    \(\varphi\approx35.3^\circ\)
32. A square right pyramid has the base edge \(a=12~\mathrm{cm}\) and the lateral edge \(e=11~\mathrm{cm}\).
    

    (a)   Find the height of this pyramid.

    (b)   Find the angle between the lateral edge and the base.

    (c)   Find the angle between two adjacent lateral edges.

    Solutions:    (a)  \(h=7~\mathrm{cm}\);     (b)  \(\varphi\approx39.5^\circ\);     (c)  \(\psi\approx66.1^\circ\)
33. A square pyramid has all base edges and lateral edges of the same length: \(a=e=10~\mathrm{cm}\).
    

    (a)   Find the angle between the lateral edge and the base edge of the pyramid.

    (b)   Find the angle between the lateral edge and the base of the pyramid.

    Solutions:    (a)  \(\alpha=60^\circ\);     (b)  \(\varphi=45^\circ\)
34. A square right pyramid has the base edge \(a=14~\mathrm{cm}\) and the height \(h=24~\mathrm{cm}\).
    

    (a)   Find the angle between the slant height and the height.

    (b)   Find the angle between the lateral edge and the height.

    Solutions:    (a)  \(\alpha\approx16.3^\circ\);     (b)  \(\beta\approx22.4^\circ\)
35. A cone has the radius \(r=9~\mathrm{cm}\) and height \(h=12~\mathrm{cm}\).
    

    (a)   Find the slant height of this cone.

    (b)   Find the angle between the slant height and the base of this cone.

    (c)   Find the opening angle (the angle between two opposite slant heights).

    Solutions:    (a)  \(\ell=15~\mathrm{cm}\);     (b)  \(\varphi\approx53.1^\circ\);     (c)  \(\theta\approx73.7^\circ\)

Solids in real life situations

36. Our town is building a swimming pool which has the shape of a cuboid with length of 50 m, width of 25 m and depth of 2.5 m. The bottom and lateral sides are covered in ceramic tiles. One square meter of these tiles costs 7.20 €.
    

    (a)   Find the surface area of the bottom and lateral sides.

    (b)   Calculate the cost of the ceramic tiles.

    When the pool is complete, they start filling it with water. They use a hose with water flow of 15 litres per second. They stop filling it when the water level is 25 cm below the upper edge of the pool.

    (c)   Find the volume of water in litres.

    (d)   Calculate the time necessary to fill the pool up to this level.

    Solutions:    (a)  \(A=1625~\mathrm{m}^2\);     (b)  cost = 11 700 €     (c)  \(V=2812.5~\mathrm{m}^3=2\,812\,500~\ell\);     (d)  52 hours (approximately)
37. Farmer has a silo for storing corn. This silo has the form of a vertical cylinder with the diameter of 4 m and height of 5 m. The silo is filled up to one half of its height.
    

    (a)   Find the total volume of this silo.

    (b)   Calculate the mass of corn (in kg), if you know that one liter of corn has the weight of 0.91 kg.

    Farmer has a ladder, which is 515 cm long. He puts it to the silo, so that the top of the ladder is aligned with the top of the silo.

    (c)   How far apart are the bottom of the silo and the bottom of the ladder?

    (d)   Calculate the angle of elevation of this ladder.

    Solutions:    (a)  \(V=62.832~\mathrm{m}^3=62\,832~\ell\);     (b)  \(m=28\,588~\mathrm{kg}\);     (c)  \(x=123.4~\mathrm{cm}\);     (d)  \(\alpha=76.1^\circ\)
38. A town is planning the Christmas decorations. They bought 480 plastic balls with radius of 7 cm and they want to colour them with gold colour. Now they are planning to buy several cans of gold colour. Each can contains 1 kg of gold colour. One gram of gold colour covers \(30~\mathrm{cm}^2\).
    

    (a)   Find the surface of one ball.

    (b)   Calculate, how much colour is required for one ball.

    (c)   Find out, how many cans will they need for all 480 balls.

    Solutions:    (a)  \(A=615.75~\mathrm{cm}^2\);     (b)  \(20.5~\mathrm{g}\);     (c)  \(9\,852~\mathrm{g}\) means 10 cans at 1kg
39. Green peas are sold in a can which has the form of a cylinder. The diameter of this cylinder is 7 cm and the height is 11 cm. Each single pea has the form of a sphere with the diameter 7 mm.
    

    (a)   Find the volume of the can.

    (b)   Find the volume of a single pea.

    (c)   Find out how many peas are there in the can, knowing that peas represent 85% of the volume of the can.

    Solutions:    (a)  \(V_{CAN}=423~\mathrm{cm}^3\);     (b)  \(V_{PEA}=0.180~\mathrm{cm}^3\);     (c)  this can contains 2000 peas (approximately)

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