3D Shapes Volume and Surface Area Formulas Table & Solver
Volume is the space inside a solid and surface area is the total area of its faces. This master matrix lists the formulas for six common 3D shapes, with an instant solver and a worked example for each.
CubeV = a³SA = 6a²
CuboidV = lbhSA = 2(lb + bh + hl)
CylinderV = πr²hTSA = 2πr(r + h)
ConeV = ⅓πr²hTSA = πr(r + l)
SphereV = 4/3 πr³TSA = 4πr²
HemisphereV = 2/3 πr³TSA = 3πr²
Instant 3D Solid Solver
Tip: open this page with ?solid=cylinder (cube, cuboid, cylinder, cone, sphere or hemisphere) to start with that solid and its rows highlighted.
| 3D Shape | Surface Area Formula | Curved/Lateral Area | Volume Formula | Copy Formulas |
|---|---|---|---|---|
| Cube | SA = 6a² | LSA = 4a² | V = a³ | |
| Cuboid | SA = 2(lb + bh + hl) | LSA = 2h(l + b) | V = l × b × h | |
| Cylinder | TSA = 2πr(r + h) | CSA = 2πrh | V = πr²h | |
| Cone | TSA = πr(r + l) | CSA = πrl | V = ⅓ × πr²h | |
| Sphere | TSA = 4πr² | CSA = 4πr² | V = 4/3 × πr³ | |
| Hemisphere | TSA = 3πr² | CSA = 2πr² | V = 2/3 × πr³ |
| 3D Shape | Dimensions | Total Surface Area | Lateral/Curved Area | Volume (V) | Copy |
|---|---|---|---|---|---|
| Cube | a = 5 | 150 | 100 | 125 | |
| Cuboid | l = 6, b = 4, h = 3 | 108 | 60 | 72 | |
| Cylinder | r = 7, h = 10 | 748 | 440 | 1540 | |
| Cone | r = 7, h = 24 (l = 25) | 704 | 550 | 1232 | |
| Sphere | r = 7 | 616 | 616 | 1437.3333 | |
| Hemisphere | r = 7 | 462 | 308 | 718.6667 |
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120 Worked Benchmark Examples (π = 22/7)
| 3D Shape | Dimensions | Total Surface Area | Lateral/Curved Area | Volume (V) | Copy |
|---|---|---|---|---|---|
| Cube | a = 1 cm | 6 cm² | 4 cm² | 1 cm³ | |
| Cube | a = 2 cm | 24 cm² | 16 cm² | 8 cm³ | |
| Cube | a = 3 cm | 54 cm² | 36 cm² | 27 cm³ | |
| Cube | a = 4 cm | 96 cm² | 64 cm² | 64 cm³ | |
| Cube | a = 5 cm | 150 cm² | 100 cm² | 125 cm³ | |
| Cube | a = 6 cm | 216 cm² | 144 cm² | 216 cm³ | |
| Cube | a = 7 cm | 294 cm² | 196 cm² | 343 cm³ | |
| Cube | a = 8 cm | 384 cm² | 256 cm² | 512 cm³ | |
| Cube | a = 9 cm | 486 cm² | 324 cm² | 729 cm³ | |
| Cube | a = 10 cm | 600 cm² | 400 cm² | 1000 cm³ | |
| Cube | a = 12 cm | 864 cm² | 576 cm² | 1728 cm³ | |
| Cube | a = 14 cm | 1176 cm² | 784 cm² | 2744 cm³ | |
| Cube | a = 15 cm | 1350 cm² | 900 cm² | 3375 cm³ | |
| Cube | a = 16 cm | 1536 cm² | 1024 cm² | 4096 cm³ | |
| Cube | a = 18 cm | 1944 cm² | 1296 cm² | 5832 cm³ | |
| Cube | a = 20 cm | 2400 cm² | 1600 cm² | 8000 cm³ | |
| Cube | a = 25 cm | 3750 cm² | 2500 cm² | 15625 cm³ | |
| Cube | a = 30 cm | 5400 cm² | 3600 cm² | 27000 cm³ | |
| Cube | a = 40 cm | 9600 cm² | 6400 cm² | 64000 cm³ | |
| Cube | a = 50 cm | 15000 cm² | 10000 cm² | 125000 cm³ | |
| Cuboid | l = 2 cm, b = 2 cm, h = 2 cm | 24 cm² | 16 cm² | 8 cm³ | |
| Cuboid | l = 3 cm, b = 2 cm, h = 1 cm | 22 cm² | 10 cm² | 6 cm³ | |
| Cuboid | l = 4 cm, b = 3 cm, h = 2 cm | 52 cm² | 28 cm² | 24 cm³ | |
| Cuboid | l = 5 cm, b = 4 cm, h = 3 cm | 94 cm² | 54 cm² | 60 cm³ | |
| Cuboid | l = 6 cm, b = 4 cm, h = 3 cm | 108 cm² | 60 cm² | 72 cm³ | |
| Cuboid | l = 6 cm, b = 5 cm, h = 4 cm | 148 cm² | 88 cm² | 120 cm³ | |
| Cuboid | l = 7 cm, b = 5 cm, h = 3 cm | 142 cm² | 72 cm² | 105 cm³ | |
| Cuboid | l = 8 cm, b = 6 cm, h = 5 cm | 236 cm² | 140 cm² | 240 cm³ | |
| Cuboid | l = 8 cm, b = 5 cm, h = 4 cm | 184 cm² | 104 cm² | 160 cm³ | |
| Cuboid | l = 9 cm, b = 6 cm, h = 4 cm | 228 cm² | 120 cm² | 216 cm³ | |
| Cuboid | l = 10 cm, b = 5 cm, h = 4 cm | 220 cm² | 120 cm² | 200 cm³ | |
| Cuboid | l = 10 cm, b = 8 cm, h = 6 cm | 376 cm² | 216 cm² | 480 cm³ | |
| Cuboid | l = 12 cm, b = 8 cm, h = 5 cm | 392 cm² | 200 cm² | 480 cm³ | |
| Cuboid | l = 12 cm, b = 10 cm, h = 6 cm | 504 cm² | 264 cm² | 720 cm³ | |
| Cuboid | l = 15 cm, b = 10 cm, h = 8 cm | 700 cm² | 400 cm² | 1200 cm³ | |
| Cuboid | l = 15 cm, b = 12 cm, h = 9 cm | 846 cm² | 486 cm² | 1620 cm³ | |
| Cuboid | l = 16 cm, b = 10 cm, h = 8 cm | 736 cm² | 416 cm² | 1280 cm³ | |
| Cuboid | l = 20 cm, b = 12 cm, h = 10 cm | 1120 cm² | 640 cm² | 2400 cm³ | |
| Cuboid | l = 25 cm, b = 15 cm, h = 10 cm | 1550 cm² | 800 cm² | 3750 cm³ | |
| Cuboid | l = 30 cm, b = 20 cm, h = 15 cm | 2700 cm² | 1500 cm² | 9000 cm³ | |
| Cylinder | r = 3.5 cm, h = 10 cm | 297 cm² | 220 cm² | 385 cm³ | |
| Cylinder | r = 3.5 cm, h = 14 cm | 385 cm² | 308 cm² | 539 cm³ | |
| Cylinder | r = 3.5 cm, h = 7 cm | 231 cm² | 154 cm² | 269.5 cm³ | |
| Cylinder | r = 7 cm, h = 10 cm | 748 cm² | 440 cm² | 1540 cm³ | |
| Cylinder | r = 7 cm, h = 14 cm | 924 cm² | 616 cm² | 2156 cm³ | |
| Cylinder | r = 7 cm, h = 21 cm | 1232 cm² | 924 cm² | 3234 cm³ | |
| Cylinder | r = 7 cm, h = 15 cm | 968 cm² | 660 cm² | 2310 cm³ | |
| Cylinder | r = 7 cm, h = 5 cm | 528 cm² | 220 cm² | 770 cm³ | |
| Cylinder | r = 10.5 cm, h = 10 cm | 1353 cm² | 660 cm² | 3465 cm³ | |
| Cylinder | r = 10.5 cm, h = 20 cm | 2013 cm² | 1320 cm² | 6930 cm³ | |
| Cylinder | r = 10.5 cm, h = 14 cm | 1617 cm² | 924 cm² | 4851 cm³ | |
| Cylinder | r = 14 cm, h = 10 cm | 2112 cm² | 880 cm² | 6160 cm³ | |
| Cylinder | r = 14 cm, h = 20 cm | 2992 cm² | 1760 cm² | 12320 cm³ | |
| Cylinder | r = 14 cm, h = 7 cm | 1848 cm² | 616 cm² | 4312 cm³ | |
| Cylinder | r = 14 cm, h = 25 cm | 3432 cm² | 2200 cm² | 15400 cm³ | |
| Cylinder | r = 21 cm, h = 10 cm | 4092 cm² | 1320 cm² | 13860 cm³ | |
| Cylinder | r = 21 cm, h = 14 cm | 4620 cm² | 1848 cm² | 19404 cm³ | |
| Cylinder | r = 21 cm, h = 28 cm | 6468 cm² | 3696 cm² | 38808 cm³ | |
| Cylinder | r = 28 cm, h = 15 cm | 7568 cm² | 2640 cm² | 36960 cm³ | |
| Cylinder | r = 35 cm, h = 20 cm | 12100 cm² | 4400 cm² | 77000 cm³ | |
| Cone | r = 3 cm, h = 4 cm, l = 5 cm | 75.43 cm² | 47.14 cm² | 37.71 cm³ | |
| Cone | r = 4 cm, h = 3 cm, l = 5 cm | 113.14 cm² | 62.86 cm² | 50.29 cm³ | |
| Cone | r = 5 cm, h = 12 cm, l = 13 cm | 282.86 cm² | 204.29 cm² | 314.29 cm³ | |
| Cone | r = 12 cm, h = 5 cm, l = 13 cm | 942.86 cm² | 490.29 cm² | 754.29 cm³ | |
| Cone | r = 6 cm, h = 8 cm, l = 10 cm | 301.71 cm² | 188.57 cm² | 301.71 cm³ | |
| Cone | r = 8 cm, h = 6 cm, l = 10 cm | 452.57 cm² | 251.43 cm² | 402.29 cm³ | |
| Cone | r = 8 cm, h = 15 cm, l = 17 cm | 628.57 cm² | 427.43 cm² | 1005.71 cm³ | |
| Cone | r = 15 cm, h = 8 cm, l = 17 cm | 1508.57 cm² | 801.43 cm² | 1885.71 cm³ | |
| Cone | r = 7 cm, h = 24 cm, l = 25 cm | 704 cm² | 550 cm² | 1232 cm³ | |
| Cone | r = 24 cm, h = 7 cm, l = 25 cm | 3696 cm² | 1885.71 cm² | 4224 cm³ | |
| Cone | r = 9 cm, h = 12 cm, l = 15 cm | 678.86 cm² | 424.29 cm² | 1018.29 cm³ | |
| Cone | r = 12 cm, h = 9 cm, l = 15 cm | 1018.29 cm² | 565.71 cm² | 1357.71 cm³ | |
| Cone | r = 12 cm, h = 16 cm, l = 20 cm | 1206.86 cm² | 754.29 cm² | 2413.71 cm³ | |
| Cone | r = 16 cm, h = 12 cm, l = 20 cm | 1810.29 cm² | 1005.71 cm² | 3218.29 cm³ | |
| Cone | r = 10 cm, h = 24 cm, l = 26 cm | 1131.43 cm² | 817.14 cm² | 2514.29 cm³ | |
| Cone | r = 15 cm, h = 20 cm, l = 25 cm | 1885.71 cm² | 1178.57 cm² | 4714.29 cm³ | |
| Cone | r = 20 cm, h = 15 cm, l = 25 cm | 2828.57 cm² | 1571.43 cm² | 6285.71 cm³ | |
| Cone | r = 9 cm, h = 40 cm, l = 41 cm | 1414.29 cm² | 1159.71 cm² | 3394.29 cm³ | |
| Cone | r = 20 cm, h = 21 cm, l = 29 cm | 3080 cm² | 1822.86 cm² | 8800 cm³ | |
| Cone | r = 21 cm, h = 20 cm, l = 29 cm | 3300 cm² | 1914 cm² | 9240 cm³ | |
| Sphere | r = 3.5 cm | 154 cm² | 154 cm² | 179.67 cm³ | |
| Sphere | r = 7 cm | 616 cm² | 616 cm² | 1437.33 cm³ | |
| Sphere | r = 10.5 cm | 1386 cm² | 1386 cm² | 4851 cm³ | |
| Sphere | r = 14 cm | 2464 cm² | 2464 cm² | 11498.67 cm³ | |
| Sphere | r = 17.5 cm | 3850 cm² | 3850 cm² | 22458.33 cm³ | |
| Sphere | r = 21 cm | 5544 cm² | 5544 cm² | 38808 cm³ | |
| Sphere | r = 24.5 cm | 7546 cm² | 7546 cm² | 61625.67 cm³ | |
| Sphere | r = 28 cm | 9856 cm² | 9856 cm² | 91989.33 cm³ | |
| Sphere | r = 31.5 cm | 12474 cm² | 12474 cm² | 130977 cm³ | |
| Sphere | r = 35 cm | 15400 cm² | 15400 cm² | 179666.67 cm³ | |
| Sphere | r = 42 cm | 22176 cm² | 22176 cm² | 310464 cm³ | |
| Sphere | r = 49 cm | 30184 cm² | 30184 cm² | 493005.33 cm³ | |
| Sphere | r = 56 cm | 39424 cm² | 39424 cm² | 735914.67 cm³ | |
| Sphere | r = 63 cm | 49896 cm² | 49896 cm² | 1047816 cm³ | |
| Sphere | r = 70 cm | 61600 cm² | 61600 cm² | 1437333.33 cm³ | |
| Sphere | r = 77 cm | 74536 cm² | 74536 cm² | 1913090.67 cm³ | |
| Sphere | r = 84 cm | 88704 cm² | 88704 cm² | 2483712 cm³ | |
| Sphere | r = 91 cm | 104104 cm² | 104104 cm² | 3157821.33 cm³ | |
| Sphere | r = 98 cm | 120736 cm² | 120736 cm² | 3944042.67 cm³ | |
| Sphere | r = 105 cm | 138600 cm² | 138600 cm² | 4851000 cm³ | |
| Hemisphere | r = 3.5 cm | 115.5 cm² | 77 cm² | 89.83 cm³ | |
| Hemisphere | r = 7 cm | 462 cm² | 308 cm² | 718.67 cm³ | |
| Hemisphere | r = 10.5 cm | 1039.5 cm² | 693 cm² | 2425.5 cm³ | |
| Hemisphere | r = 14 cm | 1848 cm² | 1232 cm² | 5749.33 cm³ | |
| Hemisphere | r = 17.5 cm | 2887.5 cm² | 1925 cm² | 11229.17 cm³ | |
| Hemisphere | r = 21 cm | 4158 cm² | 2772 cm² | 19404 cm³ | |
| Hemisphere | r = 24.5 cm | 5659.5 cm² | 3773 cm² | 30812.83 cm³ | |
| Hemisphere | r = 28 cm | 7392 cm² | 4928 cm² | 45994.67 cm³ | |
| Hemisphere | r = 31.5 cm | 9355.5 cm² | 6237 cm² | 65488.5 cm³ | |
| Hemisphere | r = 35 cm | 11550 cm² | 7700 cm² | 89833.33 cm³ | |
| Hemisphere | r = 42 cm | 16632 cm² | 11088 cm² | 155232 cm³ | |
| Hemisphere | r = 49 cm | 22638 cm² | 15092 cm² | 246502.67 cm³ | |
| Hemisphere | r = 56 cm | 29568 cm² | 19712 cm² | 367957.33 cm³ | |
| Hemisphere | r = 63 cm | 37422 cm² | 24948 cm² | 523908 cm³ | |
| Hemisphere | r = 70 cm | 46200 cm² | 30800 cm² | 718666.67 cm³ | |
| Hemisphere | r = 77 cm | 55902 cm² | 37268 cm² | 956545.33 cm³ | |
| Hemisphere | r = 84 cm | 66528 cm² | 44352 cm² | 1241856 cm³ | |
| Hemisphere | r = 91 cm | 78078 cm² | 52052 cm² | 1578910.67 cm³ | |
| Hemisphere | r = 98 cm | 90552 cm² | 60368 cm² | 1972021.33 cm³ | |
| Hemisphere | r = 105 cm | 103950 cm² | 69300 cm² | 2425500 cm³ |
Frequently Asked Questions
| Question | Answer |
|---|---|
| What is the formula for the surface area and volume of a cylinder? | Total surface area TSA = 2&π;r(r + h), curved surface area CSA = 2&π;rh, and volume V = &π;r²h. For r = 7 and h = 10 with π = 22/7: TSA = 748, CSA = 440, V = 1540. |
| How do you find the slant height of a cone? | Use l = √(r² + h²). For r = 7 and h = 24, l = √(49 + 576) = 25. Then CSA = &π;rl and TSA = &π;r(r + l). |
| What is the volume of a sphere? | V = 4/3 × &π;r³. For r = 7 with π = 22/7, V = 4/3 × 22/7 × 343, which is about 1437.3333 cubic units. |
| Why is the total surface area of a hemisphere 3 x pi x r^2? | A hemisphere has a curved surface of 2&π;r² (half a sphere) plus a flat circular base of &π;r². Together they make 3&π;r². |
| What is the difference between lateral and total surface area? | Lateral (or curved) surface area counts only the sides, without the top and bottom faces. Total surface area adds the bases. For a cube, LSA = 4a² and TSA = 6a². |
| How do you find the volume and surface area of a cuboid? | V = l × b × h and SA = 2(lb + bh + hl). For l = 6, b = 4, h = 3: V = 72 and SA = 2(24 + 12 + 18) = 108. |