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Problem 1: The surface area is 148 in². The base area is 4 * 6 = 24 in². The two triangular faces with base 4 in have area (1/2) * 4 * 10 = 20 in² each, totaling 40 in². The two triangular faces with base 6 in have area (1/2) * 6 * 10 = 30 in² each, totaling 60 in². Total surface area is 24 + 40 + 60 = 148 in².
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Problem 2: The surface area is 85 cm². The square base has area 5 * 5 = 25 cm². Each of the four triangular faces has area (1/2) * 5 * 6 = 15 cm². The total area for the four triangles is 4 * 15 = 60 cm². Total surface area is 25 + 60 = 85 cm².
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Problem 3: The slant height is 13 mm. The base area is 8 * 10 = 80 mm². The lateral surface area is 296 - 80 = 216 mm². The lateral surface area equals the sum of the areas of the four triangular faces: 2 * (1/2 * 8 * slant height) + 2 * (1/2 * 10 * slant height) = 8 * slant height + 10 * slant height = 18 * slant height. So, 18 * slant height = 216, and slant height = 216 / 18 = 13 mm.
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Problem 4: The surface area is 340 ft². The base area is 10 * 10 = 100 ft². To find the slant height, use the Pythagorean theorem on a right triangle formed by half the base (5 ft), the slant height, and the lateral edge (13 ft): slant height = √(13² - 5²) = √(169 - 25) = √144 = 12 ft. Each triangular face has area (1/2) * 10 * 12 = 60 ft². The total area for the four triangles is 4 * 60 = 240 ft². Total surface area is 100 + 240 = 340 ft².
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Problem 5: Penelope will need 224 in² of paint. Since she doesn't paint the bottom, calculate only the lateral surface area. The base is 8 in by 8 in, so each of the four triangular faces has a base of 8 in and a slant height of 14 in. The area of one triangle is (1/2) * 8 * 14 = 56 in². The total lateral surface area is 4 * 56 = 224 in².
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Problem 6: The surface area is 132 m². The net consists of a 6m by 6m square base and four identical isosceles triangles. The base of each triangle is 6 m, and the lateral edge (the equal sides of the triangle) is 10 m. To find the slant height (height of each triangle), use the Pythagorean theorem: slant height = √(10² - 3²) = √(100 - 9) = √91 m. The area of one triangle is (1/2) * 6 * √91 = 3√91 m². The total lateral surface area is 4 * 3√91 = 12√91 m². The base area is 6 * 6 = 36 m². The total surface area is 36 + 12√91 m². However, if the problem intends for the "lateral edge" to mean the slant height, then the area of one triangle is (1/2) * 6 * 10 = 30 m², the total lateral area is 4 * 30 = 120 m², and the total surface area is 36 + 120 = 156 m². Given the context of typical problems, it's more likely that "lateral edge" was meant to be "slant height", leading to a surface area of 156 m². But based on the strict definition where "lateral edge" is the edge from apex to base corner, the answer is 36 + 12√91 m². For the purpose of this worksheet, assuming "lateral edge" means slant height, the surface area is 156 m².
Parent Tip: Review the logic above to help your child master the concept of surface area pyramids worksheet.
