For numbers 13-20 a.squares b. cylinder c. rectangular prism d.net e. surface area f.plane figure g.cube h.triangular prism ___ 13 The sum of all areas of lateral faces and bases of a solid figure. ___ 14 The flat pattern that you can flod to form a solid figure ___ 15 The sum of the areas of its six faces is the surface area of a _____ of each face is equal to the product of length and width ___ 16 The sum of the areas of the two circular bases and rectangular lateral area is the surface area of the _____ ___ 17 To measure surface area, we use ______ units. ___ 18 To determine the surface area of a rectangular prism, you must identify what __________ is represented by every face of it. ___ 19 It has six square face ___ 20 It has rectangular lateral faces and two bases which are triangles.
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Find the surface area of the prism shown below. 12 in. 6.76 in. 26 in. 8 in. 6.6 in. 18 in. Round your answer to the nearest square inch. Surface area = square inches the tolerance is +/-2% Find the surface area of the following prism. 20 16 24 44 Surface area = square units exact number, no tolerance Find the surface area of the following cylinder. 18 18 Enter the exact answer. Use "pi" or symbol π to enter the π. Surface area = square units Find the surface area of the following ball whose circumference is as shown. 16π Enter the answer in terms of π. The surface area of the following ball = square units Find the surface area of the cylindrical juice can with r = 5.7 cm, h = 17.4 cm to the nearest square centimeter. The surface area of the cylindrical is: cm² the tolerance is +/-2% A square 8 centimeters on a side is rolled up to form the lateral surface of a right circular cylinder. What is the surface area of the cylinder, including the top and the bottom? Enter the exact answer. Surface area = cm²
Jerelyn N.
17. A rectangular solid whose dimensions are 4cm, 8cm, and 12cm is inscribed in a sphere. What is the diagonal of the rectangular solid? a) 7.49 cm b) 8.94 cm c) 14.42 cm d) 14.97 cm 18. In question 17, what is the surface area of the sphere? a) 176.24 b) 251.09 c) 704.03 d) 1,405.25 19. The base of a right prism is an equilateral triangle whose edge is 10cm each and its lateral edge is 80cm. Find its volume. a) 2,486.60 b) 3,464.10 c) 4,000.00 d) 8,000.00 20. Determine the total area of the prism in cm². a) 2,486.60 b) 3,464.10 c) 4,000.00 d) 8,000.00 21. The surface area of a sphere inscribed in a regular tetrahedron is 144 cm². What is the radius of the sphere? a) 12 cm b) 6 cm c) 5 cm 22. What is the altitude of the tetrahedron? a) 20 cm b) 24 cm c) 28 cm d) 48 cm 23. If a sphere is inscribed in a cube of side length 6 cm, what is the volume of the sphere? a) 36 cm³ b) 3 cm³ c) 288 cm³ d) 972 cm³ 24. A prismatoid has an upper base of 90 cm², a lower base of 2100 cm², an altitude of 100 cm, and a midsection of 120 cm². What is its volume? a) 7000 cm³ b) 11500 cm³ c) 13000 cm³ d) 17500 cm³ 25. Find the midsection of a frustum of a square pyramid if its lower base is 20 cm² on each side, its upper base is 12 cm² on each side, and the altitude of the frustum is 14 cm. a) 144 cm² b) 240 cm² c) 256 cm² d) 400 cm²
Sri K.
Heron's formula: Approximately 2000 years ago, Heron of Alexandria derived a formula for the area of a triangle in terms of the lengths of the sides. A more modern derivation of Heron's formula is indicated in the steps that follow. (a) Use the expression for $\sin A$ in Exercise $56(b)$ to show that $\sin ^{2} A=\frac{(a-b+c)(a+b-c)(b+c-a)(b+c+a)}{4 b^{2} c^{2}}$ Hint: Use difference-of-squares factoring repeatedly. (b) Let $s$ denote one-half of the perimeter of $\triangle A B C$. That is, let $s=\frac{1}{2}(a+b+c) .$ Using this notation (which is due to Euler), verify that (i) $a+b+c=2 s$ (ii) $-a+b+c=2(s-a)$ (iii) $a-b+c=2(s-b)$ (iv) $a+b-c=2(s-c)$ Then, using this notation and the result in part (a), show that $$ \sin A=\frac{2 \sqrt{s}(s-a)(s-b)(s-c)}{b c} $$ Note: since $\sin A$ is positive (Why?), the positive root is appropriate here. (c) Use the result in part (b) and the formula area $\triangle A B C=\frac{1}{2} b c \sin A$ to conclude that $$\text { area } \triangle A B C=\sqrt{s(s-a)(s-b)(s-c)} $$ This is Heron's formula. For historical background and a purely geometric proof, see An Introduction to the History of Mathematics, 6 th ed., by Howard Eves (Philadelphia: Saunders College Publishing, 1990 ), pp. 178 and 194
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