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A square-based pyramid has a base side length of 6m and a vertical height of 4m. Calculate: a) The formula for the area of all triangular faces. b) The volume of the pyramid. c) The total surface area

Mathematics

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3 parts
a

LSA=2alLSA = 2al

b

48 m348 \text{ m}^3

c

96 m296 \text{ m}^2

Step-by-step solution

3 steps

यहाँ प्रश्न ५ को समाधान छ:

दिइएको जानकारी:
वर्ग आधार भएको पिरामिडको आधारको प्रत्येक भुजा (aa) = 6 मिटर
ठाडो उचाई (hh) = 4 मिटर

Part a

पिरामिडमा भएको सबै त्रिभुजाकार सतहहरूको क्षेत्रफल पत्ता लगाउने सूत्र लेख्नुहोस्।

पिरामिडको त्रिभुजाकार सतहहरूको क्षेत्रफल (Lateral Surface Area, LSA) पत्ता लगाउने सूत्र हो:

LSA=2alLSA = 2al

जहाँ aa आधारको भुजाको लम्बाइ र ll छड्के उचाई (slant height) हो।

LSA=2al\boxed{LSA = 2al}

Part b

48 m348 \text{ m}^3

उक्त पिरामिडको आयतन पत्ता लगाउनुहोस्।

पिरामिडको आयतन (VV) को सूत्र:

V=13×आधारको क्षेत्रफल×ठाडो उचाईV = \frac{1}{3} \times \text{आधारको क्षेत्रफल} \times \text{ठाडो उचाई}

आधार वर्ग भएकोले, आधारको क्षेत्रफल =a2= a^2

V=13a2hV = \frac{1}{3} a^2 h

  1. मानहरू सूत्रमा राख्नुहोस्।

    V=13×(6 m)2×4 mV = \frac{1}{3} \times (6 \text{ m})^2 \times 4 \text{ m} V=13×36 m2×4 mV = \frac{1}{3} \times 36 \text{ m}^2 \times 4 \text{ m} V=12 m2×4 mV = 12 \text{ m}^2 \times 4 \text{ m} V=48 m3V = 48 \text{ m}^3

    पिरामिडको आयतन 48 m3\boxed{48 \text{ m}^3} हो।

Part c

96 m296 \text{ m}^2

पिरामिडको पूरा सतहको क्षेत्रफल निकाल्नुहोस्।

पिरामिडको पूरा सतहको क्षेत्रफल (Total Surface Area, TSA) को सूत्र:

TSA=आधारको क्षेत्रफल+त्रिभुजाकार सतहहरूको क्षेत्रफलTSA = \text{आधारको क्षेत्रफल} + \text{त्रिभुजाकार सतहहरूको क्षेत्रफल} TSA=a2+2alTSA = a^2 + 2al

छड्के उचाई (ll) पत्ता लगाउनुहोस्।

l2=h2+(a2)2l^2 = h^2 + \left(\frac{a}{2}\right)^2

  1. छड्के उचाई (ll) पत्ता लगाउनुहोस्।

    l2=(4 m)2+(6 m2)2l^2 = (4 \text{ m})^2 + \left(\frac{6 \text{ m}}{2}\right)^2 l2=16 m2+(3 m)2l^2 = 16 \text{ m}^2 + (3 \text{ m})^2 l2=16 m2+9 m2l^2 = 16 \text{ m}^2 + 9 \text{ m}^2 l2=25 m2l^2 = 25 \text{ m}^2 l=25 m2l = \sqrt{25 \text{ m}^2} l=5 ml = 5 \text{ m}

  2. पूरा सतहको क्षेत्रफल पत्ता लगाउनुहोस्।

    TSA=a2+2alTSA = a^2 + 2al TSA=(6 m)2+2×6 m×5 mTSA = (6 \text{ m})^2 + 2 \times 6 \text{ m} \times 5 \text{ m} TSA=36 m2+60 m2TSA = 36 \text{ m}^2 + 60 \text{ m}^2 TSA=96 m2TSA = 96 \text{ m}^2

    पिरामिडको पूरा सतहको क्षेत्रफल 96 m2\boxed{96 \text{ m}^2} हो।

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