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Finding a missing radius in a composite solid

Finding a missing radius in a composite solid.

  • Split a compound solid into familiar shapes.
  • Use perpendicular height for volume and slant height for a cone’s curved area.
  • Add volumes, or subtract a removed part. Give cubic units.

Example 1

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.

Example 2

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.

Your turn

Question 1

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
Check answer
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.
Question 2

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
Check answer
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.
Question 3

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
Check answer
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.
Question 4

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
Check answer
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.
Question 5

A closed cylinder is topped with a hemisphere of the same radius. Its total exposed area minus the cylinder’s curved area is . Find the radius.

Cylinder with a hemisphere of the same radius5 cmDiagram not to scale
Check answer
The hemisphere and bottom circle contribute 3πr².Divide by 3π.Take the positive square root.
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