Mathematics
Grade 10
15 min
Surface area
Surface area
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1
Introduction & Learning Objectives
Learning Objectives
Define a solid of revolution and identify the 3D shape formed by rotating a line segment around an axis.
Use the distance formula to calculate the length of a line segment (slant height) in the coordinate plane.
Identify the correct radii of revolution based on a line segment's coordinates and the axis of revolution.
Calculate the lateral surface area of a cylinder, cone, or frustum generated by a line segment using a single, unified formula.
Distinguish between rotating around the x-axis versus the y-axis and its effect on the resulting surface area.
Solve multi-step problems involving the surface area of solids of revolution defined in the coordinate plane.
Ever wondered how a simple line on a graph can create a 3D object like a cone or a lampshade...
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Key Concepts & Vocabulary
TermDefinitionExample
Solid of RevolutionA 3D figure formed by rotating a 2D shape, such as a line segment, around a fixed line.Rotating a rectangle around one of its sides creates a cylinder. Rotating a right triangle around one of its legs creates a cone.
Axis of RevolutionThe fixed line in the coordinate plane around which a 2D shape is rotated. In this lesson, it will be either the x-axis or the y-axis.If we rotate the point (3, 4) around the x-axis, it traces a circle with a radius of 4.
Radius of Revolution (r)The perpendicular distance from a point on the line segment to the axis of revolution.For the point (5, 2), the radius of revolution around the x-axis is 2, and the radius of revolution around the y-axis is 5.
Slant Height (l)The length of the line segment being rotated. It is...
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Core Formulas
Distance Formula (for Slant Height)
l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
This formula calculates the slant height, `l`, which is the straight-line distance between two points (x₁, y₁) and (x₂, y₂). This is the first step in any problem.
General Formula for Lateral Surface Area of Revolution
A = \pi (r_1 + r_2) l
This powerful formula calculates the lateral surface area generated by rotating a line segment. `l` is the slant height, `r₁` is the radius of the first endpoint, and `r₂` is the radius of the second endpoint. This single formula works for cones (one radius is 0), cylinders (r₁ = r₂), and frustums.
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Challenging
Find the total lateral surface area of the solid formed by rotating the path from A(0,0) to B(3,4) and then from B(3,4) to C(5,4) around the x-axis.
A.20π
B.16π
C.52π
D.36π
Challenging
A frustum is formed by rotating a line segment around the x-axis. Its lateral surface area is 39π, its slant height is 3, and one endpoint of the segment is (2, 5). If the other endpoint (x, y) has y > 5, what are its coordinates?
A.(2, 8)
B.(5, 5)
C.(-1, 5)
D.(2, 13)
Challenging
A line segment connects P(2, 6) and Q(5, 10). Let A_y be the lateral surface area when rotated around the y-axis, and A_x be the area when rotated around the x-axis. What is the ratio A_y / A_x?
A.16/7
B.1/2
C.7/16
D.8/5
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