Question: A right circular cone has a base radius of $ 5x $ and a slant height of $ 13x $. What is the total surface area of the cone in terms of $ x $?
["Title: How to Calculate the Total Surface Area of a Right Circular Cone with Given Dimensions", "When studying geometry, one common question involves finding the total surface area of a right circular cone when the base radius and slant height are known. This article explains step-by-step how to calculate the total surface area of a cone with a base radius of $ 5x $ and a slant height of $ 13x $, in terms of $ x $.", "---", "Understanding the Geometry of the Cone", "A right circular cone consists of three key components:\n- Base radius $ r = 5x $\n- Slant height $ l = 13x $ — the distance from the apex to any point on the edge of the circular base\n- Height $ h $ — the vertical distance from the base center to the apex", "The total surface area (TSA) of a cone is the sum of the lateral (curved) surface area and the area of the circular base:", "$$\n\ ext{TSA} = \ ext{Lateral Surface Area} + \ ext{Base Area}\n$$", "$$\n\ ext{TSA} = \pi r l + \pi r^2\n$$", "---", "Step 1: Find the height $ h $ using the Pythagorean Theorem\nIn a right circular cone, the radius $ r $, height $ h $, and slant height $ l $ form a right triangle. Therefore:", "$$\nl^2 = r^2 + h^2\n$$", "Substitute $ l = 13x $ and $ r = 5x $:", "$$\n(13x)^2 = (5x)^2 + h^2\n$$\n$$\n169x^2 = 25x^2 + h^2\n$$\n$$\nh^2 = 169x^2 - 25x^2 = 144x^2\n$$\n$$\nh = \sqrt{144x^2} = 12x\n$$", "---", "Step 2: Compute the lateral surface area\n$$\n\ ext{Lateral Surface Area} = \pi r l = \pi (5x)(13x) = 65\pi x^2\n$$", "---", "Step 3: Compute the base area\n$$\n\ ext{Base Area} = \pi r^2 = \pi (5x)^2 = 25\pi x^2\n$$", "---", "Step 4: Add to find total surface area\n$$\n\ ext{TSA} = 65\pi x^2 + 25\pi x^2 = 90\pi x^2\n$$", "---", "Conclusion", "The total surface area of a right circular cone with base radius $ 5x $ and slant height $ 13x $ is:", "$$\n\boxed{90\pi x^2}\n$$", "This formula is essential for solving geometry problems efficiently and is commonly required in standardized tests and engineering applications.", "---", "Keywords:\nright circular cone surface area, total surface area cone, cone surface area calculation, cone dimensions formula, geometry problem solution", "Meta Description:\nLearn how to calculate the total surface area of a right circular cone with base radius $ 5x $ and slant height $ 13x $. Follow step-by-step formulas and verify using the Pythagorean Theorem."]
