រូបធរណីមាត្រដែលមានវិមាត្របី

• a. ព្រីសត្រីកោណ ៖ \(F = 5, E = 9, V = 6\)

• b. ប្រលេពីប៉ែតកែង ៖ \(F = 6, E = 12, V = 8\)

• c. ពីរ៉ាមីតត្រីកោណ ៖ \(F = 4, E = 6, V = 4\)

• d. ពីរ៉ាមីតបាតការេ ៖ \(F = 5, E = 8, V = 5\)

ផ្ទៀងផ្ទាត់ \(V - E + F = 2\) ៖

• a. \(6 - 9 + 5 = 2\) (ពិត)

• b. \(8 - 12 + 6 = 2\) (ពិត)

• c. \(4 - 6 + 4 = 2\) (ពិត)

• d. \(5 - 8 + 5 = 2\) (ពិត)

• a. \(S_L = 4(6^2) = 144\text{ cm}^2\), \(S_T = 6(6^2) = 216\text{ cm}^2\), \(V = 6^3 = 216\text{ cm}^3\)

• b. \(S_L = 4(1.5^2) = 9\text{ m}^2\), \(S_T = 6(1.5^2) = 13.5\text{ m}^2\), \(V = 1.5^3 = 3.375\text{ m}^3\)

• a. \(S_L = 2(8+5)(12) = 312\text{ cm}^2\), \(S_T = 312 + 2(40) = 392\text{ cm}^2\), \(V = 8 \times 5 \times 12 = 480\text{ cm}^3\)

• b. \(S_L = 2(1.2+0.8)(2.5) = 10\text{ m}^2\), \(S_T = 10 + 2(0.96) = 11.92\text{ m}^2\), \(V = 1.2 \times 0.8 \times 2.5 = 2.4\text{ m}^3\)

\(P = 24\text{ cm}, S_B = \frac{1}{2}(6)(8) = 24\text{ cm}^2\) ៖ \(S_L = 24 \times 15 = 360\text{ cm}^2\)។ \(S_T = 360 + 2(24) = 408\text{ cm}^2\)។ \(V = 24 \times 15 = 360\text{ cm}^3\)។

\(S_B = \frac{6^2\sqrt{3}}{4} = 9\sqrt{3}\text{ cm}^2\), \(P = 18\text{ cm}\) ៖ \(S_T = 18(20) + 2(9\sqrt{3}) = 360 + 18\sqrt{3}\text{ cm}^2\)។ \(V = 9\sqrt{3} \times 20 = 180\sqrt{3}\text{ cm}^3\)។

\(S_B = \frac{1}{2}(12)(16) = 96\text{ cm}^2\)។ ជ្រុងបាត \(a = \sqrt{6^2+8^2} = 10\text{ cm} \implies P = 40\text{ cm}\)។ \(S_T = 40(10) + 2(96) = 592\text{ cm}^2\)។ \(V = 96 \times 10 = 960\text{ cm}^3\)។

\(S_B = \frac{(8+14) \times 6}{2} = 66\text{ cm}^2\)។ \(V = S_B \times h = 66 \times 14 = 924\text{ cm}^3\)។

\(S_L = 2\left(\frac{22}{7}\right)(7)(20) = 880\text{ cm}^2\)។ \(S_T = 880 + 2\left(\frac{22}{7}\right)(49) = 880 + 308 = 1\,188\text{ cm}^2\)។ \(V = \frac{22}{7}(49)(20) = 3\,080\text{ cm}^3\)។

\(2\pi r h = 120\pi \iff 2r(10) = 120 \implies r = 6\text{ cm}\)។ \(V = \pi(6^2)(10) = 360\pi\text{ cm}^3\)។

\(\pi(5^2)h = 500\pi \implies 25h = 500 \implies h = 20\text{ cm}\)។ \(S_T = 2\pi(5)(20+5) = 250\pi\text{ cm}^2\)។

• a. អាប៉ូតែម ៖ \(s = \sqrt{8^2 + 6^2} = 10\text{ cm}\)

• b. \(S_L = \frac{1}{2}(48)(10) = 240\text{ cm}^2\)។ \(S_T = 240 + 144 = 384\text{ cm}^2\)។ \(V = \frac{1}{3}(144)(8) = 384\text{ cm}^3\)។

\(h = \sqrt{13^2 - 5^2} = 12\text{ cm}\)។ \(S_B = 10^2 = 100\text{ cm}^2\)។ \(V = \frac{1}{3}(100)(12) = 400\text{ cm}^3\)។

\(\frac{1}{3}a^2(8) = 384 \implies a^2 = 144 \implies a = 12\text{ cm}\)។ \(s = \sqrt{8^2 + 6^2} = 10\text{ cm}\)។ \(S_T = \frac{1}{2}(48)(10) + 144 = 384\text{ cm}^2\)។

• a. \(s = \sqrt{5^2 + 12^2} = 13\text{ cm}\)

• b. \(S_L = \pi(5)(13) = 65\pi\text{ cm}^2\)។ \(S_T = 65\pi + 25\pi = 90\pi\text{ cm}^2\)។ \(V = \frac{1}{3}\pi(25)(12) = 100\pi\text{ cm}^3\)។

\(h = \sqrt{25^2 - 7^2} = 24\text{ cm}\)។ \(V = \frac{1}{3}\pi(7^2)(24) = 392\pi\text{ cm}^3\)។

• a. \(V = \frac{1}{3}S_B h \implies 18\pi = \frac{1}{3}(6\pi)h \implies h = 9\text{ cm}\)

• b. \(S_B = \pi r^2 = 6\pi \implies r = \sqrt{6}\text{ cm}\)។ \(S_L = \pi r s \implies 16\pi = \pi\sqrt{6}s \implies s = \frac{16}{\sqrt{6}} = \frac{8\sqrt{6}}{3}\text{ cm}\)

\(V = \frac{1}{3}\pi r^2(15) = 5\pi r^2 = 420\pi \implies r^2 = 84\)។ \(S_B = 84\pi\text{ cm}^2\)។ \(S_T = S_L + S_B = 216\pi + 84\pi = 300\pi\text{ cm}^2\)។

\(S_B = \pi r^2 = 160 \implies r = \sqrt{\frac{160}{3.14}} \approx 7.14\text{ cm}\)។ \(S_T = \pi r s + S_B = 3.14(7.14)(12) + 160 \approx 269 + 160 = 429\text{ cm}^2\)។

• a. \(V_{\text{កោន}} = \frac{1}{3}\left(\frac{22}{7}\right)(18^2)(49) = 16\,632\text{ cm}^3\)។ \(V_{\text{ស៊ីឡាំង}} = \frac{22}{7}(18^2)(198) = 201\,569\text{ cm}^3\)។ \(V_{\text{សរុប}} \approx 218\,201\text{ cm}^3\)។

• b. \(\rho = \frac{m}{V} = \frac{2145}{218201} \approx 0.0098\text{ g/cm}^3\)។

• a. \(V = \frac{1}{3}(8 \times 6)(12) = 192\text{ cm}^3\)

• b. \(OA = \sqrt{4^2+3^2} = 5\text{ cm} \implies SA = \sqrt{12^2+5^2} = 13\text{ cm}\)

\(V_{\text{ទឹក}} = \pi(4^2)(15) \approx 3.14(240) = 753.6\text{ cm}^3\)។ \(V_{\text{ផើង}} = 10 \times 8 \times 12 = 960\text{ cm}^3\)។ ដោយ \(753.6 < 960\) នាំឱ្យទឹកមិនហៀរចេញទេ។

\(V_1 = 40 \times 30 \times 20 = 24\,000\text{ cm}^3\)។ \(V_2 = \frac{1}{2}\pi(15^2)(40) \approx 14\,137\text{ cm}^3\)។ \(V = 24\,000 + 14\,137 = 38\,137\text{ cm}^3\)។

\(V_{\text{ជញ្ជាំង}} = 25 \times 20 = 500\text{ m}^3\)។ \(h_{\text{ពីរ៉ាមីត}} = 24 - 20 = 4\text{ m}\)។ \(V_{\text{ដំបូល}} = \frac{1}{3}(25)(4) = \frac{100}{3} \approx 33.33\text{ m}^3\)។ \(V = 533.33\text{ m}^3\)។

\(r = 3\text{ m}\)។ \(h_{\text{កោន}} = \sqrt{5^2-3^2} = 4\text{ m}\)។ \(V = \pi(3^2)(50) + \frac{1}{3}\pi(3^2)(4) = 450\pi + 12\pi = 462\pi\text{ m}^3\)។

ផលធៀបកម្ពស់ \(k = \frac{6}{12} = \frac{1}{2} \implies\) ផលធៀបមាឌ \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)។ មាឌនៅសល់គឺ \(1 - \frac{1}{8} = \frac{7}{8}\) នៃមាឌកោនធំ ៖ \(V = \frac{7}{8}\left(\frac{1}{3}\pi(16)(12)\right) = 56\pi\text{ cm}^3\)។

• a. មាឌប្រេង ៖ \(150 \times 60\% = 90\text{ m}^3\)

• b. មាឌនៅសល់ ៖ \(150 - 90 = 60\text{ m}^3\)

កន្លះអង្កត់ទ្រូងបាតគឺ \(5\text{ dm}\)។ \(h = \sqrt{13^2 - 5^2} = 12\text{ dm}\)។ ផ្ទៃបាតការេ ៖ \(S_B = \frac{1}{2}(10^2) = 50\text{ dm}^2\)។ \(V = \frac{1}{3}(50)(12) = 200\text{ dm}^3\)។

\(S_B = \frac{(1+2) \times 25}{2} = 37.5\text{ m}^2\)។ \(V = 37.5 \times 10 = 375\text{ m}^3 = 375\,000\text{ L}\)។

\(V = \pi(10^2 - 8^2)(100) = 3\,600\pi \approx 11\,309.7\text{ cm}^3\)។ ម៉ាស ៖ \(m = 11\,309.7 \times 7.8 \approx 88\,216\text{ g} \approx 88.2\text{ kg}\)។

\(V_{\text{គូប}} = 20^3 = 8\,000\text{ cm}^3\)។ \(V_{\text{ស៊ីឡាំង}} = \pi(5^2)(20) = 500\pi \approx 1\,570.8\text{ cm}^3\)។ \(V = 8\,000 - 1\,570.8 = 6\,429.2\text{ cm}^3\)។

\(V = 6^3 = 216\text{ cm}^3\)។ \(\pi(3^2)h = 216 \implies h = \frac{216}{9\pi} = \frac{24}{\pi} \approx 7.64\text{ cm}\)។

\(V_{\text{ថ្ម}} = \pi(5^2)(2) = 50\pi \approx 157.08\text{ cm}^3\)។

\(V = \frac{1}{3}(3.14)(4^2)(9) = 150.72\text{ cm}^3 = 150.72\text{ mL}\)។

\(S_B = \frac{6^2\sqrt{3}}{4} = 9\sqrt{3}\text{ cm}^2\)។ \(V = \frac{1}{3}(9\sqrt{3})(10) = 30\sqrt{3}\text{ cm}^3\)។

\(S_T = 4 \times \frac{a^2\sqrt{3}}{4} = 6^2\sqrt{3} = 36\sqrt{3}\text{ cm}^2\)។

\(V = \frac{6^3\sqrt{2}}{12} = \frac{216\sqrt{2}}{12} = 18\sqrt{2}\text{ cm}^3\)។

\(2\pi(6)(h+6) = 168\pi \implies 12(h+6) = 168 \implies h = 8\text{ cm}\)។ \(V = \pi(6^2)(8) = 288\pi\text{ cm}^3\)។

\(\pi(5)s = 65\pi \implies s = 13\text{ cm}\)។ \(h = \sqrt{13^2 - 5^2} = 12\text{ cm}\)។ \(V = \frac{1}{3}\pi(25)(12) = 100\pi\text{ cm}^3\)។

\((a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca)\) ៖ \(18^2 = 12^2 + S_T \implies S_T = 324 - 144 = 180\text{ cm}^2\)។

ប្រវែងទ្រនុងដំបូល ៖ \(\sqrt{3^2 + 4^2} = 5\text{ m}\)។ ផ្ទៃដំបូលសងខាង ៖ \(2(5 \times 15) = 150\text{ m}^2\)។

\(V = \frac{1}{2}(\pi \times 1^2 \times 4) = 2\pi\text{ m}^3 \approx 6.28\text{ m}^3 = 6\,280\text{ L}\)។

\(V = 4^3 + \frac{1}{3}(16)(3) = 64 + 16 = 80\text{ m}^3\)។

\(\frac{V_2}{V_1} = k^3 = 27 \implies k = 3\) (កើនឡើង ៣ ដង)។

\(\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\)។

\(\frac{V_1}{V_2} = \frac{h_1}{h_2} = \frac{3}{4}\)។

\(r = \frac{a\sqrt{2}}{2}, h = a \implies V_{\text{ស៊ីឡាំង}} = \pi\left(\frac{a^2}{2}\right)a = \frac{\pi}{2}a^3\)។ \(\frac{V_{\text{ស៊ីឡាំង}}}{V_{\text{គូប}}} = \frac{\pi}{2} \approx 1.57\)។

\(a\sqrt{2} = 2R \implies a = R\sqrt{2}\)។ \(V_{\text{គូប}} = (R\sqrt{2})^3 = 2\sqrt{2}R^3\)។ \(\frac{V_{\text{គូប}}}{V_{\text{ស៊ីឡាំង}}} = \frac{2\sqrt{2}R^3}{2\pi R^3} = \frac{\sqrt{2}}{\pi}\)។

\(V = \frac{h}{3}(S_1 + S_2 + \sqrt{S_1 S_2}) = \frac{6}{3}(36 + 100 + 60) = 2(196) = 392\text{ cm}^3\)។

\(S' = k^2 S = 4(200) = 800\text{ cm}^2\)។ \(V' = k^3 V = 8(250) = 2\,000\text{ cm}^3\)។