Applied solid geometry (applied practice)

{"questions":[{"content":"The aquarium has the shape of a cube with side $40\\space cm.$ How many liters is the volume of the aquarium? In one liter $1\\space000$ cubic centimeters.[[fill_choice_big-1]][[image-34]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$64$","$66$","$68$"],"placeholder":0,"answer":0},"image-34":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n1k.svg","width":"300"}},"hints":["Calculate volume in cubic centimeters.<br />The volume of a cube is calculated using the formula: <br />$$V=a^3$$where $a$ is the length of the side of the cube.<br /><br />Substitute the value of the side: $$V=40\\cdot40\\cdot40=64\\space000\\space cm^3$$ Convert the volume into liters: $$V=\\frac{64\\space000}{1\\space000}=64\\space L$$ The volume of the aquarium is $64$ liters.<br />"]},{"content":"The Sneferu pyramid has the shape of a regular quadrangular pyramid, the side of the base is equal to $220\\space m,$ and the height is $104\\space m.$ The side of the base of the exact museum copy of this pyramid is equal to $44\\space cm.$ Find the height of the museum copy. Give your answer in centimeters.[[fill_choice_big-7]][[image-83]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$20.8$","$21.8$","$20.2$"],"placeholder":0,"answer":0},"image-83":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n2k.svg","width":"400"}},"hints":["Convert the side of the base of the museum copy into meters for convenience of calculations: $$44\\space cm=0.44\\space m$$ Find how many times the side of the base of the museum copy is smaller than the side of the base of the original pyramid: $$\\frac{220}{0.44}=500$$ Calculation of the height of the museum copy: $$\\frac{104}{500}=0.208\\space m=20.8\\space cm$$ The height of the museum copy of Snofru's pyramid - $20.8\\space cm.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-22]][[image-149]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$18$","$22$","$24$"],"placeholder":0,"answer":0},"image-149":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n3k.svg","width":"300"}},"hints":["The polyhedron can be represented as a rectangular prism with edges $2,3,1$, from which two rectangles with sides $2$ and $1.$ are cut<br /><br />Calculation of the surface area of a parallelepiped: $$S=2\\cdot(ab+bc+ac)$$where $a=2, b=3, c=1.$<br /><br />Substitute the values: $$S = 2\\cdot(2 \\cdot 3 + 2 \\cdot 1 + 3 \\cdot 1) = 22$$ Area of the cut out rectangles: $$S=2\\cdot1\\cdot2=4$$ Subtract the area of the cut out parts from the surface area of the parallelepiped: $$S=22-4=18$$ The surface area of the polyhedron is $18.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-40]][[image-206]]","widgets":{"fill_choice_big-40":{"type":"fill_choice_big","options":["$76$","$78$","$74$"],"placeholder":0,"answer":0},"fill_choice_big-73":{"type":"fill_choice_big","options":["$92$","$94$","$90$"],"placeholder":0,"answer":0},"image-206":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n4k.svg","width":"300"}},"hints":["The polyhedron can be represented as a rectangular prism with edges $3,3,5$, from which two squares with side $1.$ are cut<br /><br />Calculation of the surface area of a parallelepiped: $$S=2\\cdot(ab+bc+ac)$$where $a=3, b=3, c=5.$<br /><br />Substitute the values: $$S=2\\cdot(3\\cdot3+3\\cdot5+3\\cdot5)=78$$Area of the cut out squares:$$S=1\\cdot1\\cdot2=2$$Subtract the area of the cut out parts from the surface area of the parallelepiped:$$S=78-2=76$$The surface area of the polyhedron is $76.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-120]][[image-267]]","widgets":{"fill_choice_big-120":{"type":"fill_choice_big","options":["$92$","$94$","$90$"],"placeholder":0,"answer":0},"image-267":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n5k.svg","width":"300"}},"hints":["The polyhedron can be represented as a rectangular prism with edges $3,4,5$, from which two squares with side $1.$ are cut<br /><br />Calculation of the surface area of a parallelepiped: $$S=2\\cdot(ab+bc+ac)$$where $a=3, b=4, c=5.$<br /><br />Substitute the values: $$S=2\\cdot(3\\cdot4+3\\cdot5+4\\cdot5)=94$$Area of the cut out squares:$$S=1\\cdot1\\cdot2=2$$Subtract the area of the cut out parts from the surface area of the parallelepiped:$$S=94-2=92$$The surface area of the polyhedron is $92.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-171]][[image-332]]","widgets":{"fill_choice_big-171":{"type":"fill_choice_big","options":["$110$","$112$","$108$"],"placeholder":0,"answer":0},"image-332":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n6k.svg","width":"300"}},"hints":["The surface area of a given polyhedron is equal to the surface area of a rectangular prism with edges $3,5,5.$<br /><br />Calculation of the surface area of a parallelepiped: $$S=2\\cdot(ab+bc+ac)$$where $a=3, b=5, c=5.$<br /><br />Substitute the values: $$S=2\\cdot(3\\cdot5+3\\cdot5+5\\cdot5)=110$$ The surface area of the polyhedron is equal to $110.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-178]][[image-401]]","widgets":{"fill_choice_big-178":{"type":"fill_choice_big","options":["$94$","$92$","$90$"],"placeholder":0,"answer":0},"image-401":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n7kkk.svg","width":"300"}},"hints":["A polyhedron can be represented as a rectangular prism with edges $3,5\\space and\\space4.$ The surface area of a rectangular prism is calculated by the formula: $$S=2(ab+bc+ac)$$ Substitute the values: $$S = 2\\cdot(3 \\cdot 5 + 5 \\cdot 4 + 3 \\cdot 4)= 94$$ The surface area of a polyhedron is $94.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-240]][[image-510]]","widgets":{"fill_choice_big-240":{"type":"fill_choice_big","options":["$132$","$130$","$138$"],"placeholder":0,"answer":0},"image-510":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n8k.svg","width":"300"}},"hints":["The surface area of a given polyhedron can be found by adding the surface area of a rectangular prism with edges $6, 4,4$ and the area of two additional rectangles with sides $1\\space and\\space 4,$ and then subtracting the areas of two “cut out” rectangles with sides $1\\space and\\space2.$<br /><br />The surface area of a rectangular prism is calculated by the formula: $$S=2(ab+bc+ac)$$ Substitute the values: $$S=2\\cdot(6\\cdot4+4\\cdot4+6\\cdot4)=128$$ Add two rectangles: $$S=(1\\cdot4\\cdot2)+128=136$$ Subtract two “cut out” rectangles: $$S=136-(1\\cdot2\\cdot2)=132$$ The surface area of the polyhedron is $132.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-319]][[image-587]]","widgets":{"fill_choice_big-319":{"type":"fill_choice_big","options":["$114$","$108$","$118$"],"placeholder":0,"answer":0},"image-587":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n9k.svg","width":"300"}},"hints":["The surface area of a rectangular prism with edges $4, 4, 5$ is calculated by the formula:<br />$$S = 2(ab + bc + ac)$$ Substitute the values:<br />$$S = 2\\cdot(4 \\cdot 4 + 4 \\cdot 5 + 4 \\cdot 5) = 112$$Add two rectangles with sides $1 \\space and \\space4{:}$<br />$$S = 112 + (1 \\cdot 4 \\cdot 2) =  120$$Subtract two “cut out” rectangles with sides $1\\space and \\space3{:}$<br />$$S = 120-(1 \\cdot 3 \\cdot 2) =  114$$The surface area of the polyhedron is $114.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-397]][[image-668]]","widgets":{"fill_choice_big-397":{"type":"fill_choice_big","options":["$48$","$42$","$46$"],"placeholder":0,"answer":0},"image-668":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver1n10k.svg","width":"300"}},"hints":["The sum of the surface areas of two rectangular prisms with sides $1, 3, 4\\space and\\space1, 2, 3.$<br /><br />The surface area of a rectangular prism is calculated by the formula:<br />$$S = 2(ab + bc + ac)$$For the first parallelepiped $(1, 3, 4){:}$ $$S=2\\cdot(4\\cdot1+3\\cdot1+4\\cdot3)=38$$For the second parallelepiped $(1, 2, 3){:}$ $$S=2\\cdot(2\\cdot1+1\\cdot3+2\\cdot3)=22$$Total amount:$$S=38+22=60$$The double area of the rectangle with sides $2\\space and\\space3,$ is subtracted, which is the common face for the two parallelepipeds:$$S=60-(2\\cdot2\\cdot3)=48$$Surface area polyhedron is equal to $48.$"]}]}

{"questions":[{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-1]][[image-31]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$84$","$80$","$86$"],"placeholder":0,"answer":0},"image-31":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n1.svg","width":"250"}},"hints":["Surface area formula:$$S=2(ab+bc+ac)$$Calculate the surface area of the first parallelepiped with edges $1, 6, 4{:}$ $$S=2\\cdot(4\\cdot1+6\\cdot1+4\\cdot6)=68$$Calculate the surface area of the second parallelepiped with edges $1, 4, 4{:}$ $$S=2\\cdot(4\\cdot1+4\\cdot1+4\\cdot4)=48$$Sum up the surface areas of both parallelepipeds:$$S=68+48=116$$Twice the area of a square with a side $4$ (since this is a common face for both parallelepipeds): $$S=4\\cdot4\\cdot2=32$$ Subtract twice the area of the square: $$S=116-32=84$$ The surface area of the polyhedron is $84.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-7]][[image-87]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$96$","$90$","$94$"],"placeholder":0,"answer":0},"image-87":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n2.svg","width":"400"}},"hints":["Formula for the surface area of a rectangular prism: $$S = 2(ab + bc + ac)$$ Area of a large parallelepiped with edges $1, 5, 7$: $$S = 2 \\cdot (5 \\cdot 1 + 7 \\cdot 1 + 7 \\cdot 5) =94$$ Area of a small parallelepiped with edges $1, 1, 2$: $$S = 2 \\cdot (1 \\cdot 1 + 2 \\cdot 1 + 2 \\cdot 1) =10$$ Sum of areas: $$S = 94 + 10 = 104$$ Subtract the overly taken into account area of the front face of the small one parallelepiped $($ $4$ area of the rectangle $1 \\times 2$ $){:}$ $$S = 104-4 \\cdot (2 \\cdot 1) = 96$$ surface area of the polyhedron is $96.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-21]][[image-148]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$124$","$122$","$126$"],"placeholder":0,"answer":0},"image-148":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n3.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$Area of the first parallelepiped with sides $2, 3, 3$:$$S = 2 \\cdot (3 \\cdot 2 + 3 \\cdot 3 + 2 \\cdot 3) = 42$$Area of the second parallelepiped with sides $5, 4, 3$:<br />$$S = 2 \\cdot (4 \\cdot 5 + 4 \\cdot 3 + 5 \\cdot 3) = 94$$Sum of areas:$$S = 42 + 94 = 136$$Subtract the double area of the general rectangle $3 \\times 2$:$$S = 136-2 \\cdot (2 \\cdot 3) = 124$$The surface area of the polyhedron is $124$."]},{"content":"Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right.[[fill_choice_big-43]][[image-201]]","widgets":{"fill_choice_big-43":{"type":"fill_choice_big","options":["$14$","$12$","$18$"],"placeholder":0,"answer":0},"image-201":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n4.svg","width":"300"}},"hints":["Squares with side $1$: $$S=4\\cdot1=4$$Rectangles with sides $1\\space and\\space2{:}$ $$S=2\\cdot2=4$$Front and back faces (composite):$$S=2\\cdot3=6$$Total surface area: $$S=4+4+6=14$$The surface area of the polyhedron is $14.$"]},{"content":"Find the surface area of the spatial cross shown in the figure and composed of unit cubes.[[fill_choice_big-73]][[image-258]]","widgets":{"fill_choice_big-73":{"type":"fill_choice_big","options":["$30$","$25$","$28$"],"placeholder":0,"answer":0},"image-258":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n5kkk.svg","width":"300"}},"hints":["The surface of the cross is made up of six surfaces of cubes, each of which is missing one face.<br />Area of one cube without one face:<br />$$S=6 \\cdot (1 \\cdot 1)-(1 \\cdot 1) = 5$$Total $6$ cubes in the cross.<br />Total surface area:<br />$$S=6\\cdot 5 = 30$$The surface area of the spatial cross is equal to $30$."]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-111]][[image-319]]","widgets":{"fill_choice_big-111":{"type":"fill_choice_big","options":["$162$","$164$","$166$"],"placeholder":0,"answer":0},"image-319":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n6.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$First parallelepiped $($edges $6, 6, 2){:}$<br />$$S = 2 \\cdot (6 \\cdot 6 + 6 \\cdot 2 + 6 \\cdot 2) = 120$$ Second parallelepiped $($ edges $3, 3, 4){:}$<br />$$S = 2 \\cdot (3 \\cdot 3 + 3 \\cdot 4 + 3 \\cdot 4) = 66$$Sum of areas:$$S = 120 + 66 = 186$$Subtract the area of two common faces $(3 \\times 4){:}$<br />$$S = 186-(2 \\cdot 3 \\cdot 4) = 162$$The surface area of the polyhedron is $162.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-167]][[image-384]]","widgets":{"fill_choice_big-167":{"type":"fill_choice_big","options":["$156$","$154$","$158$"],"placeholder":0,"answer":0},"image-384":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n7.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$Area of the first parallelepiped $($edges $2, 5, 6){:}$ $$S = 2 \\cdot (2 \\cdot 5 + 5 \\cdot 6 + 2 \\cdot 6) = 104$$Area of the second parallelepiped $($edges $2, 5, 3){:}$ $$S = 2 \\cdot (2 \\cdot 5 + 5 \\cdot 3 + 2 \\cdot 3) = 62$$Area of the third parallelepiped $($edges $2, 2, 3){:}$ $$S = 2 \\cdot (2 \\cdot 2 + 2 \\cdot 3 + 3 \\cdot 2) = 32$$Sum of areas:$$S = 104 + 62 + 32 = 198$$Subtract the areas of common faces:$$S =198-(2 \\cdot (5 \\cdot 3 + 2 \\cdot 3)) = 156$$The surface area of the polyhedron is $156.$"]},{"content":"Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-221]][[image-453]]","widgets":{"fill_choice_big-221":{"type":"fill_choice_big","options":["$152$","$154$","$158$"],"placeholder":0,"answer":0},"image-453":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n8.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$Area of the first parallelepiped $($ edges $2, 4, 6){:}$<br />$$S = 2 \\cdot (2 \\cdot 4 + 4 \\cdot 6 + 2 \\cdot 6) = 88$$Area of the second parallelepiped $($ edges $1, 6, 2){:}$<br />$$S = 2 \\cdot (1 \\cdot 6 + 6 \\cdot 2 + 1 \\cdot 2) = 40$$Area of the third parallelepiped $($ edges $2, 2, 2){:}$<br />$$S = 2 \\cdot (2 \\cdot 2 + 2 \\cdot 2 + 2 \\cdot 2) = 24$$Sum of areas:$$S = 88 + 40 + 24 = 152$$The surface area of the polyhedron is $152$."]},{"content":"The part has the shape of the polyhedron shown in the figure (all dihedral angles are right). The numbers in the figure indicate the lengths of the edges in centimeters. Find the surface area of this part. Give your answer in square centimeters.[[fill_choice_big-283]][[image-526]]","widgets":{"fill_choice_big-283":{"type":"fill_choice_big","options":["$180$","$178$","$182$"],"placeholder":0,"answer":1},"image-526":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n9.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$Area of the main parallelepiped $($edges $6, 5, 5){:}$ $$S = 2 \\cdot (6 \\cdot 5 + 5 \\cdot 5 + 6 \\cdot 5) = 170\\space cm^ 2$$ Areas of two side rectangles:$$S = 2 \\cdot (3 \\cdot 2) = 12\\space cm^ 2$$Areas of the upper and lower rectangles:$$S = 2 \\cdot (2 \\cdot 5) = 20\\space cm^ 2$$Subtract the areas of the two side rectangles and add the areas of the upper and lower rectangles:$$S = 170-12 + 20 = 178\\space cm^ 2$$The surface area of the part is $178\\space cm^ 2.$"]},{"content":"The part has the shape of the polyhedron shown in the figure (all dihedral angles are right). The numbers in the figure indicate the lengths of the edges in centimeters. Find the surface area of this part. Give your answer in square centimeters.[[fill_choice_big-370]][[image-603]]","widgets":{"fill_choice_big-370":{"type":"fill_choice_big","options":["$146$","$142$","$144$"],"placeholder":0,"answer":0},"image-603":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n10.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism: $$S = 2(ab + bc + ac)$$ Area of the main parallelepiped $($ edges $7, 5, 3){:}$ $$S = 2 \\cdot (7 \\cdot 5 + 5 \\cdot 3 + 7 \\cdot 3) = 142\\space cm^2$$ Subtract the areas of the two side rectangles: $$S = 2 \\cdot (3 \\cdot 1) = 6\\space cm^2$$ Add the areas of the upper and lower rectangles: $$S = 2 \\cdot (1 \\cdot 5) = 10\\space cm^2$$ Final surface area: $$S = 142-6 + 10 = 146\\space cm^2$$The surface area of the part is $146\\space cm^2.$"]}]}

{"questions":[{"content":"The part has the shape of the polyhedron shown in the figure (all dihedral angles are right). The numbers in the figure indicate the lengths of the edges in centimeters. Find the surface area of this part. Give your answer in square centimeters.[[fill_choice_big-1]][[image-31]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$60$","$62$","$68$"],"placeholder":0,"answer":0},"image-31":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n1.svg","width":"350"}},"hints":["Formula for the surface area of a rectangular prism: $$S = 2(ab + bc + ac)$$ Area of the first polyhedron $($ edges $4, 4, 1){:}$ $$S = 2 \\cdot (4 \\cdot 4 + 4 \\cdot 1 + 4 \\cdot 1) = 48\\space cm^2$$ Area of the second polyhedron $($ edges $2, 1, 4){:}$ $$S = 2 \\cdot (2 \\cdot 1 + 2 \\cdot 4 + 1 \\cdot 4) = 28\\space cm^2$$ Sum of areas: $$S = 48 + 28 = 76\\space cm^2$$ Subtract the areas of the total faces:$$S = 76-2 \\cdot (2 \\cdot 4) = 60\\space cm^2$$The surface area of the part is $60\\space cm^2.$"]},{"content":"The part has the shape of the polyhedron shown in the figure (all dihedral angles are right). The numbers in the figure indicate the lengths of the edges in centimeters. Find the surface area of this part. Give your answer in square centimeters.[[fill_choice_big-18]][[image-87]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$50$","$51$","$52$"],"placeholder":0,"answer":0},"image-87":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n2.svg","width":"300"}},"hints":["Formula for the surface area of a rectangular prism:<br />$$S = 2(ab + bc + ac)$$Area of the first polyhedron $($edge $1, 2, 5){:}$ $$S = 2 \\cdot (1 \\cdot 2 + 2 \\cdot 5 + 5 \\cdot 1) = 34\\space\\text{cm}^2$$Area of the second polyhedron $($edge $2, 2, 2){:}$ $$S = 2 \\cdot (2 \\cdot 2 + 2 \\cdot 2 + 2 \\cdot 2) = 24\\space\\text{cm}^2$$Total surface area $($, taking into account the subtraction of common faces $2$ $2\\times2) {:}$ $$S = 34 + 24-2 \\cdot (2 \\cdot 2) = 50\\space\\text{cm}^2$$The surface area of the part is $50\\space\\text{cm}^2.$"]},{"content":"The gable roof of a house with a rectangle at its base must be completely covered with roofing felt. The height of the roof is equal to $4\\,$m, the lengths of the walls of the house are equal to $6\\,m$ and $8\\,m.$ Find how much roofing felt (in square meters) is needed to cover this roof if the roof slopes are equal.[[fill_choice_big-29]][[image-136]]","widgets":{"fill_choice_big-29":{"type":"fill_choice_big","options":["$80$","$82$","$88$"],"placeholder":0,"answer":0},"image-136":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n3kkk.svg","width":"300"}},"hints":["Roof slope length (according to the Pythagorean theorem): $$l = \\sqrt{4^2 + 3^2} = 5$$ Roof area: $$S = 2 \\cdot (8 \\cdot 5) = 80\\space\\text{m}^2$$ Roof covering area is $80\\,\\text{m}^2$."]},{"content":"The five steps of the staircase were painted as shown in the picture (blue and light blue). Find the area of the painted surface if the depth of each step is $40\\space cm,$, the height is $15\\space cm,$ and the width is $90 cm.$ Give your answer in square centimeters.[[fill_choice_big-89]][[image-189]]","widgets":{"fill_choice_big-89":{"type":"fill_choice_big","options":["$21\\space150$","$21\\space350$","$22\\space250$"],"placeholder":0,"answer":0},"image-189":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n4.svg","width":"300"}},"hints":["Formula for the area of a rectangle:<br />$$S = a \\cdot b$$Area of the horizontal part of the step:$$S=40 \\cdot 90 = 3\\space600\\,\\text{cm}^2$$Area of the vertical part of the step:<br />$$S=15 \\cdot 90 = 1\\space350\\,\\text{cm}^2$$Total painted surface area (total $4$ steps and $5$ rises):$$S=4 \\cdot 3\\space600 + 5 \\cdot 1\\space350 = 21\\space150\\,\\text{cm}^2$$The painted surface area is $21\\space150\\,\\text{cm}^2$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles of the polyhedron are right).[[fill_choice_big-186]][[image-288]]","widgets":{"fill_choice_big-186":{"type":"fill_choice_big","options":["$8$","$7$","$9$"],"placeholder":0,"answer":0},"image-288":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n5.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:<br />$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $3, 2, 1){:}$ $$V=3 \\cdot 2 \\cdot 1 = 6$$Volume of the second parallelepiped $($edges $1, 1, 2){:}$ $$V=1 \\cdot 1 \\cdot 2 = 2$$Total volume of the polyhedron:$$V=6 + 2 = 8$$The volume of the polyhedron is $8$.<br />"]},{"content":"Find the volume of the spatial cross shown in the figure and composed of unit cubes.[[fill_choice_big-228]][[image-349]]","widgets":{"fill_choice_big-228":{"type":"fill_choice_big","options":["$7$","$8$","$9$"],"placeholder":0,"answer":0},"image-349":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver2n5kkk.svg","width":"300"}},"hints":["Formula for the volume of a cube: $$V = a^3$$ Volume of one unit cube: $$V=1 \\cdot 1 \\cdot 1 = 1$$ Number of cubes in the cross: $$7$$ Total volume of the spatial cross: $$V=7 \\cdot 1 = 7$$ The volume of the spatial cross is $7$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right angles).<br />[[fill_choice_big-429]][[image-414]]","widgets":{"fill_choice_big-429":{"type":"fill_choice_big","options":["$56$","$65$","$60$"],"placeholder":0,"answer":0},"image-414":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n7.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $5, 4, 2){:}$ $$V=5 \\cdot 4 \\cdot 2 = 40$$Volume of the second parallelepiped $($edges $2, 2, 4){:}$ $$V=2 \\cdot 2 \\cdot 4 = 16$$Total volume of the polyhedron:$$V=40 + 16 = 56$$The volume of the polyhedron is $56$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-369]][[image-483]]","widgets":{"fill_choice_big-369":{"type":"fill_choice_big","options":["$7$","$6$","$8$"],"placeholder":0,"answer":0},"image-483":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n8.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $2, 3, 1){:}$ $$V=2 \\cdot 3 \\cdot 1 = 6$$Volume of the second parallelepiped $($edges $1, 1, 1){:}$ $$V=1 \\cdot 1 \\cdot 1 = 1$$Total volume of the polyhedron:$$V=6 + 1 = 7$$The volume of the polyhedron is $7$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-538]][[image-556]]","widgets":{"fill_choice_big-538":{"type":"fill_choice_big","options":["$40$","$42$","$43$"],"placeholder":0,"answer":0},"image-556":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n9.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $3, 3, 4){:}$ $$V=3 \\cdot 3 \\cdot 4 = 36$$Volume of the second parallelepiped $($edges $1, 1, 4){:}$ $$V=1 \\cdot 1 \\cdot 4 = 4$$Total volume of the polyhedron:$$V=36 + 4 = 40$$The volume of the polyhedron is $40.$"]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-679]][[image-633]]","widgets":{"fill_choice_big-679":{"type":"fill_choice_big","options":["$34$","$36$","$32$"],"placeholder":0,"answer":0},"image-633":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver3n10.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:<br />$$V = a \\cdot b \\cdot c$$Volume of the main parallelepiped $($edges $3, 3, 4){:}$ $$V=3 \\cdot 3 \\cdot 4 = 36$$Volume of the subtractive parallelepiped $($edges $1, 1, 2){:}$ $$V=1 \\cdot 1 \\cdot 2 = 2$$Total volume of the polyhedron:$$V=36-2 = 34$$The volume of the polyhedron is $34$."]}]}

{"questions":[{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-1]][[image-721]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$36$","$32$","$38$"],"placeholder":0,"answer":0},"image-721":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n1.svg","width":"350"}},"hints":["Formula for the volume of a rectangular prism: $$V = a \\cdot b \\cdot c$$ Volume of the main parallelepiped $($ edges $5, 2, 4){:}$ $$V=5 \\cdot 2 \\cdot 4 = 40$$ Volume of the cut part $($ edges $1, 2, 2){:}$ $$V=1 \\cdot 2 \\cdot 2 = 4$$ Total volume of the polyhedron: $$V=40-4 = 36$$ Volume of the polyhedron is $36$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-13]][[image-782]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$90$","$92$","$88$"],"placeholder":0,"answer":0},"image-782":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n2.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism: $$V = a \\cdot b \\cdot c$$ Volume of the main parallelepiped $($ edges $5, 5, 4){:}$ $$V=5 \\cdot 5 \\cdot 4 = 100$$ Volume of the cut part $($ edges $1, 2, 5){:}$ $$V=1 \\cdot 2 \\cdot 5 = 10$$ Total volume of the polyhedron: $$V=100-10 = 90$$ Volume of the polyhedron is $90$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-37]][[image-827]]","widgets":{"fill_choice_big-37":{"type":"fill_choice_big","options":["$18$","$16$","$22$"],"placeholder":0,"answer":0},"image-827":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n3.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $1, 3, 2){:}$ $$V=1 \\cdot 3 \\cdot 2 = 6$$Volume of the second parallelepiped $($edges $1, 3, 4){:}$ $$V=1 \\cdot 3 \\cdot 4 = 12$$Total volume of the polyhedron:$$V=6 + 12 = 18$$The volume of the polyhedron is $18$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-79]][[image-876]]","widgets":{"fill_choice_big-79":{"type":"fill_choice_big","options":["$24$","$21$","$22$"],"placeholder":0,"answer":0},"image-876":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n4.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $2, 3, 2){:}$ $$V=2 \\cdot 3 \\cdot 2 = 12$$Volume of the second parallelepiped $($edges $1, 3, 4){:}$ $$V=1 \\cdot 3 \\cdot 4 = 12$$Total volume of the polyhedron:$$V=12 + 12 = 24$$The volume of the polyhedron is $24$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[image-929]][[fill_choice_big-1345]]","widgets":{"image-929":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n5.svg","width":"250"},"fill_choice_big-1345":{"type":"fill_choice_big","options":["$45$","$44$","$43$"],"placeholder":0,"answer":0}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the main parallelepiped $($edges $1, 8, 6){:}$ $$V=1 \\cdot 8 \\cdot 6 = 48$$Volume of the subtractive parallelepiped $($edges $1, 3, 1){:}$ $$V=1 \\cdot 3 \\cdot 1 = 3$$Total volume of the polyhedron:$$V=48-3 = 45$$Volume polyhedron is equal to $45$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-172]][[image-1000]]","widgets":{"fill_choice_big-172":{"type":"fill_choice_big","options":["$78$","$72$","$76$"],"placeholder":0,"answer":0},"image-1000":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n6.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $2, 3, 3){:}$ $$V=2 \\cdot 3 \\cdot 3 = 18$$Volume of the second parallelepiped $($edges $5, 3, 4){:}$ $$V=5 \\cdot 3 \\cdot 4 = 60$$Total volume of the polyhedron:$$V=18 + 60 = 78$$The volume of the polyhedron is $78$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-234]][[image-1061]]","widgets":{"fill_choice_big-234":{"type":"fill_choice_big","options":["$104$","$98$","$102$"],"placeholder":0,"answer":0},"image-1061":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n7.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $7, 4, 2){:}$ $$V=7 \\cdot 4 \\cdot 2 = 56$$Volume of the second parallelepiped $($edges $4, 3, 4){:}$ $$V=4 \\cdot 3 \\cdot 4 = 48$$Total volume of the polyhedron:$$V=56 + 48 = 104$$The volume of the polyhedron is $104$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-321]][[image-1126]]","widgets":{"fill_choice_big-321":{"type":"fill_choice_big","options":["$87$","$92$","$89$"],"placeholder":0,"answer":0},"image-1126":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n8.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $5, 3, 2){:}$ $$V=5 \\cdot 3 \\cdot 2 = 30$$Volume of the second parallelepiped $($edges $3, 3, 5){:}$ $$V=3 \\cdot 3 \\cdot 5 = 45$$Volume of the third parallelepiped $($edges $2, 3, 2){:}$ $$V=2 \\cdot 3 \\cdot 2 = 12$$Total volume of the polyhedron:$$V=30 + 45 + 12 = 87$$The volume of the polyhedron is $87$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-416]][[image-1195]]","widgets":{"fill_choice_big-416":{"type":"fill_choice_big","options":["$114$","$110$","$112$"],"placeholder":0,"answer":0},"image-1195":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n9.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the first parallelepiped $($edges $5, 3, 3){:}$ $$V=5 \\cdot 3 \\cdot 3 = 45$$Volume of the second parallelepiped $($edges $6, 3, 3){:}$ $$V=6 \\cdot 3 \\cdot 3 = 54$$Volume of the third parallelepiped $($edges $1, 3, 5){:}$ $$V=1 \\cdot 3 \\cdot 5 = 15$$Total volume of the polyhedron:$$V=45 + 54 + 15 = 114$$The volume of the polyhedron is $114$."]},{"content":"Find the volume of the polyhedron shown in the figure (all dihedral angles are right).[[fill_choice_big-89]][[image-1268]]","widgets":{"fill_choice_big-89":{"type":"fill_choice_big","options":["$78$","$72$","$76$"],"placeholder":0,"answer":0},"image-1268":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2025/04/ver4n10.svg","width":"300"}},"hints":["Formula for the volume of a rectangular prism:$$V = a \\cdot b \\cdot c$$Volume of the main parallelepiped $($edges $4, 4, 5){:}$ $$V=4 \\cdot 4 \\cdot 5 = 80$$Volume of the subtractive parallelepiped $($edges $1, 2, 1){:}$ $$V=1 \\cdot 2 \\cdot 1 = 2$$Total volume of the polyhedron:$$V=80-2 = 78$$Volume polyhedron is equal to $78$."]}]}