Exploring Surface Areas and Volumes of Common Shapes

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Questions and Answers

सही है कि दायरे के बेस के रूप में एक चक्र होने पर, दायरे का पृष्ठीय क्षेत्र क्या होगा?

2 पाई r(h) (correct)

2 पाई रूह

2 पाई r²h

2 पाई रू (चौड़ाई)

एक cuboid की कुल सतह क्षेत्रा कैसे निकाली जाती है?

समतल परिधि का आयतन

सभी तिर्यक समतल परिधियों का योग (correct)

समतल परिधि का क्षेत्रफल

कुल समतल परिधि का योग

कोने के आयतन की सही सूत्र क्या है?

(1/3) x p x r x h

(1/3) x पाई x r² x h (correct)

(1/3) x p x r²

(1/3) x p x r² x h

सभी मुख्य आकारों के सतह क्षेत्र की सही सूत्र सम्मिलित करने पर, प्रिज्म के लिए सही सूत्र क्या होगा?

(परिमाप मुक्त) + (2 x क्षेत्र) Signup and view all the answers

कोणीय चेकनाकार (Prism) के समतल पृष्ठ से संबंधित मिथक में से कौन सही है?

समतल परिधि + 2 x मुख्य पृष्ठीय क्षेत्र = पूरे Prism का क्षेत्र Signup and view all the answers

किस रूप में, Cylinder की पृष्ठीय क्षेत्र (Total Surface Area) सरलतम रूप से निकाली जा सकती है?

"TSA = LSA + (π d₁²) + (π d₂²) = 2 π rh + π (d₁²) + π (d₂²)" Signup and view all the answers

कौन सा सीधा उत्तर है जब हम किसी क्यूबॉइड के कुल पृष्ठ क्षेत्र को प्राप्त करने के लिए उपयोग करें?

2lw+2lh+2wh Signup and view all the answers

प्रिज्म की कुल पृष्ठ क्षेत्र की गणना के लिए किन दो कदमों का पालन करना चाहिए?

rac{B_1+B_2}{2} * n*h Signup and view all the answers

कोने की ऊर्ध्वकेंद्रित सतह की क्षेत्र निकालने के लिए हम किस सूत्र का उपयोग करते हैं?

rac{r}{s} Signup and view all the answers

किस प्रकार के प्रिज्म में हमें आंकलित सतहों का समर्पक पता लगाना महत्वपूर्ण है?

समानांतर प्रिज्म Signup and view all the answers

कुल पृष्ठ क्षेत्र (TSA) की व्याख्या में महत्वपूर्णता होने के मामले में, हमें Cone की पृष्ठ के क्षेत्र (surface area) की समीक्षा में नहीं **कहना* चाहिए, बल्कि कहना चाहिए?

' 'पृष्ठ-क्षेत्र ' Signup and view all the answers

Concave side (परिधि) की क्षेत्र (area) प्राप्त (known as) समीकरण (known as) है?

$ rac{rs}{ heta}$ Signup and view all the answers

Study Notes

Exploring Surfaces and Volumes of Common Shapes

In studying geometry, we encounter various shapes and their respective surface areas and volumes. As you delve into these concepts, let's examine some fundamental formulas related to common shapes like cylinders, cuboids, prisms, cones, and overall surface areas and volumes.

Cylinder Surface Area

A cylinder is defined by its base — which can be circular, square, rectangular, etc., and a height perpendicular to this plane. To find the total surface area, consider two surfaces formed from wrapping paper around the cylinder: one corresponding to the curved side (lateral surface) and another encompassing the entire top and bottom faces (curved + flat bases combined).

The formula to calculate the lateral surface area of a right cylinder with radius (r) and height (h):

[LSA = 2 \pi r(h)]

To obtain the total surface area ((TSA)), add the LSA to both base sections (one curved, one flat), using either radii or diameters depending on the shape of your base:

For a circle as the base: [TSA = LSA + (\pi d_1^2) + (\pi d_2^2) = 2 \pi rh + \pi (d_1^2) + \pi (d_2^2)]

For a rectangle (or square) as the base: [TSA = LSA + (\pi r^2)(2) = 2 \pi rh + 2 \pi r^2]

Cuboid Surface Area

Cuboids boast six distinct surfaces, each parallel to coordinate planes. These sides possess specific lengths based on the placement of x-, y-, z-coordinates. Here's how to find the cuboid's total surface area:

Add up all six individual face areas:[ TSA= lw+lh+hw ]

where (l), (w), (h) represent the length, width, and height respectively.

Prism Surface Area

Prisms consist of a polygon shaped base and vertical sides extending upward. They have two identical congruent bases joined by faces that connect these bases. Since there are countless ways to create prisms when considering varying base polygons, it's essential to determine what type of prism you're working with before calculating surface area.

To compute the total surface area of a general n-sided polygon base prism, follow these steps:

Calculate the area of both bases ((B_1), (B_2)).
Add together the product of (n) times the average of the base areas ((\frac{B_1+B_2}{2})) multiplied by the prism's height ((h)): [TSA=\left(\frac{B_1+B_2}{2} \right ) * n*h]

This expression will yield the same result regardless of whether you choose any base (B_{1}) or (B_{2}) since they share similar properties due to the symmetry present within prisms.

Cone Volume

Cones don't possess a surface area strictly speaking; however, mathematicians often refer to their surface area. A cone consists of a circular base, a vertex point above the center of that base, and a sloping side connecting them. It has four unique surfaces: two circles and a quadrilateral.

To calculate the area of the concave side (known as the lateral surface area), apply the following formula:

[ LSA = \pi rs,]

where (r) corresponds to the radius of the conical base and (s) denotes the slant height.

While technically not part of our surface area discussion, calculating a cone's volume remains relevant here because it provides insight into understanding solids and their spatial relationships. For cones, the volume can be derived using either the formula below or through Cavalieri's principle applied to their sliced disks:

[ V =\frac{1}{3}(\pi d^2)h]

where (d) represents the diameter of the base, and (h) refers to the height of the cone.

Summary

Within solid geometrical figures, such as those mentioned thus far—cylinders, cuboids, prisms, and cones—we discover the significance of surface area and volume calculations to understand structural behavior and characteristics. By exploring these relationships, we advance our knowledge of mathematical principles while strengthening analytic skills required for future scientific or engineering applications.

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Description

ज्यामिति की अध्ययन के दौरान हमें विभिन्न आकारों और उनके सतह क्षेत्रफल और आयतनों के सिद्धांतों से विचार करना पड़ता है। इस मौहरे में, हम साइलेंडर, क्यूबॉइड, प्रिज्म, और शंकु जैसे सामान्य आकारों के सतह क्षेत्र और आयतन सम्बंधित मौलिक सूत्रों की जांच करेंगे।