Soal-Soal Matematika/Volume bangun ruang

Rumus volume bangun ruang:

1. kubus: r³
2. balok: plt
3. prisma
   1. segitiga (tergantung jenisnya): luas alas x t
   2. segi-n:
   3. persegi panjang: sama dengan balok
4. limas:
   1. persegi: ${\frac {l_{a}\times t}{3}}$
   2. segitiga (tergantung jenisnya): ${\frac {l_{a}\times t}{3}}$
   3. persegi panjang: ${\frac {\pi \cdot r^{2}\times t}{3}}$
5. tabung:
   1. biasa: πr²t
   2. berongga: π(R-r)²t
6. kerucut:
   1. biasa: ${\frac {\pi \times r^{2}\times t}{3}}$
   2. terpancung: ${\frac {\pi \times (r^{2}+r\times R+R^{2})\times t}{3}}$
7. bola:
   1. biasa: ${\frac {4\times \pi \times r^{3}}{3}}$
   2. terpancung: ${\frac {\pi \times (r^{2}+R^{2}+t^{2})\times t}{6}}$

1. Tentukan volume sebuah kubus yang memiliki rusuknya 6 m!

Jawaban

$${\begin{aligned}V&=r^{3}\\&=6m\times 6m\times 6m\\&=216m^{3}\\\end{aligned}}$$

2. Tangki air berbentuk balok memiliki panjang 3 m, lebar 80 cm, dan tingginya 60 cm. Tentukan volume tangki dalam satuan liter!

Jawaban

$${\begin{aligned}V&=p\times l\times t\\&=3m\times 80cm\times 60cm\\&=300cm\times 80cm\times 60cm\\&=24.000cm^{2}\times 60cm\\&=1.440.000cm^{3}\\&=1.440l\\\end{aligned}}$$

3. Tentukan volume tabung dengan diameter 14 cm dan tinggi 21 cm!

Jawaban

$${\begin{aligned}d&=14cm{\text{ maka }}r=7cm\\V&=\pi \times r^{2}\times t\\&={\frac {22}{7}}\times 7cm\times 7cm\times 21cm\\&=3.234cm^{3}\\\end{aligned}}$$

4. Tentukan volume kerucut yang berjari-jari 18 cm dan tinggi 10 cm!

Jawaban

$${\begin{aligned}V&={\frac {1}{3}}\times \pi \times r^{2}\times t\\&={\frac {1}{3}}\times 3.14\times 18cm\times 18cm\times 10cm\\&=3,391.2cm^{3}\\\end{aligned}}$$

5. Tentukan volume prisma segitiga sisi dengan panjang sisi alas 10 cm dan tingginya 15 cm!

Jawaban

$${\begin{aligned}t1&={\sqrt {10^{2}-5^{2}}}cm\\&={\sqrt {5{\sqrt {3}}}}cm\\&=8.66cm\\V&={\frac {1}{2}}\times a\times t1\times t2\\&={\frac {1}{2}}\times 10cm\times 8.66cm\times 15cm\\&={\frac {1}{2}}\times 1,299cm^{3}\\&=649.5cm^{3}\\atau\\V&={\frac {1}{2}}\times a\times a\times sin\alpha \times t\\&=\times {\frac {1}{2}}\times 10cm\times 10cm\times sin60\times 15cm\\&={\frac {1}{2}}\times 1,299cm^{3}\\&=649.5cm^{3}\\\end{aligned}}$$

6. Tentukan volume limas persegi dengan panjang sisi alas 10 cm dan tingginya 12 cm!

Jawaban

$${\begin{aligned}V&={\frac {1}{3}}\times s^{2}\times t\\&={\frac {1}{3}}\times 10cm\times 10cm\times 12cm\\&=4cm\times 100cm^{2}\\&=400cm^{3}\\\end{aligned}}$$

7. Tentukan volume bola yang berdiameter 40 cm! Jari-jari = 20cm

Jawaban

$${\begin{aligned}d&=40cm{\text{ maka }}r=20cm\\V&={\frac {4}{3}}\times \pi \times r^{3}\\&={\frac {4}{3}}\times 3.14\times 20cm\times 20cm\times 20cm\\&={\frac {4}{3}}\times 25.120cm^{3}\\&=33,493.34cm^{3}\end{aligned}}$$

8. Tentukan volume kerucut terpancung jika diameter alasnya 10 dm, diameter atas 4 dm, dan tinggi 4 dm!

Jawaban

$${\begin{aligned}d1&=10dm{\text{ maka }}r1=5dm\\d2&=4dm{\text{ maka }}r2=2dm\\V&={\frac {1}{3}}\times \pi \times (r1^{2}+r1\times r2+r2^{2})\times t\\&={\frac {1}{3}}\times 3.14\times (5dm\times 5dm+5dm\times 2dm+2dm\times 2dm)\times 4dm\\&=4.18dm\times (25+10+4)dm^{2}\\&=4.18dm\times 39dm^{2}\\&=163.02dm^{3}\\\end{aligned}}$$

9. Tentukan volume bola terpancung jika jari-jari alas 10 dm, jari-jari atas 7 dm dan tinggi 14 dm!

Jawaban

$${\begin{aligned}V&={\frac {1}{6}}\times \pi \times (r1^{2}+r2^{2}+t^{2})\times t\\&={\frac {1}{6}}\times {\frac {22}{7}}\times (10\times 10+7\times 7+14\times 14)dm^{2}\times 14dm\\&=7.33dm\times (100+49+196)dm^{2}\\&=7.33dm\times 345dm^{2}\\&=2,528.85dm^{3}\\\end{aligned}}$$

10. Berapa volume pipa berongga dengan diameter masing-masing 10 mm dan 18 mm serta tinggi 3.5 cm! (dalam mm persegi)

Jawaban

$${\begin{aligned}d1&=10mm{\text{ maka }}r1=5mm\\d2&=18mm{\text{ maka }}r2=9mm\\t&=3.5cm=35mm\\V&=\pi \times (r2^{2}-r1^{2})\times t\\&={\frac {22}{7}}\times (9\times 9-5\times 5)mm^{2}\times 35mm\\&=110mm\times (81-25)mm^{2}\\&=110mm\times 56mm^{2}\\&=6,160mm^{3}\\\end{aligned}}$$