Cylinders, cones, and spheres are three important 3D shapes that you’ll encounter in geometry. Each has unique properties related to their volume and surface area, and they’re all around us in the real world—like in cans, ice cream cones, and basketballs!
What is a Cylinder?
A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. Think of a can or a stack of coins.
Properties
- Faces: 2 flat circular faces (top and bottom), 1 curved face
- Edges: 2 circular edges where the curved surface meets the flat faces
- Vertices: No vertices, as the top and bottom are circles, not points.
- The top and bottom faces are circles of the same size.
- The height (h) is the distance between the two circular bases.
- The radius (r) is the distance from the center to the edge of the circular base.
Volume of a Cylinder
The volume of a cylinder is the amount of space it occupies. We can think of a cylinder as a stack of circles (its circular base) that extend upwards to form a solid shape. Hence, the volume of a cylinder depends on two things:
- The area of the circular base (how big each circular slice is)
- The height of the cylinder (how tall the cylinder is, or how many of those circular slices are stacked up)
Step-by-Step Explanation
Area of the Base (πr²): The base of a cylinder is a circle. To calculate the area of this circle, we use the formula for the area of a circle.
Height (h): The height (h) of the cylinder tells us how many of these circular areas are stacked on top of one another. Each “layer” of the cylinder is a flat circle, and we are stacking up h layers.
Multiplying the Area of the base (πr²) by the Height (h): To get the total volume of the cylinder, we multiply the area of the base (πr²) by the height (h). This gives us the total amount of space inside the cylinder.
Formula
Volume of a Cylinder = πr²h
Example: Find the volume of a cylinder with a radius of 3 cm and a height of 14 cm.
Volume = πr²h = πx3x3x14 = 126π = 126x$\frac{22}{7}$ = 18 cm³
Surface Area of a Cylinder
The surface area of a cylinder is the total area of its outer surface, which consists of two circular bases and a curved surface that wraps around the sides. To calculate the surface area, we need to consider each part of the cylinder separately and then combine them.
A cylinder has:
- Two circular bases (top and bottom).
- One curved surface (the side that wraps around the cylinder, also called the lateral surface).
Step-by-Step Explanation
Area of the Circular Bases: Each cylinder has two identical circular bases (top and bottom), and the area of each circular base is calculated using the formula for the area of a circle:
Area of one base = πr²
Since there are two circular bases, the total area of the two bases is:
Total area of both bases = 2πr²
Area of the Curved Surface (Lateral Surface): Now, let’s calculate the area of the curved surface that wraps around the sides of the cylinder. If you were to cut the curved surface vertically and lay it flat, you would get a rectangle.
- The height (h) of this rectangle is the same as the height of the cylinder.
- The width of the rectangle is equal to the circumference of the circular base, which is the distance around the edge of the circle. The circumference of a circle is given by
So, the area of the curved surface (lateral surface) is the height of the cylinder multiplied by the circumference of the base (which gives the area of the rectangle):
Combining Both Parts: To find the total surface area of the cylinder, we add the area of the two circular bases and the area of the curved surface:
Formula
= 2πr² + 2πrh
Example: Calculate the Surface Area of a Cylinder with radius 4 cm and height 10 cm.
= 2πr (h+r) = 2x$\frac{22}{7}$x4x(10+4) = 2x$\frac{22}{7}$x4x14 = 352 cm²
What is a Cone?
A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. Think of a can or a stack of coins.
Properties
- Faces: 2 flat circular faces (top and bottom), 1 curved face
- Edges: 2 circular edges where the curved surface meets the flat faces
- Vertices: No vertices, as the top and bottom are circles, not points.
- The top and bottom faces are circles of the same size.
- The height (h) is the distance between the two circular bases.
- The radius (r) is the distance from the center to the edge of the circular base.
Volume of a Cone
The volume of a cone represents the amount of space it occupies. A cone is a 3D shape with a circular base that tapers smoothly up to a single point called the apex. To derive the volume of a cone, we can think of it as part of a cylinder. If you imagine a cone inside a cylinder with the same base and height, the cone occupies one-third of the cylinder’s volume. This is because of the way the cone tapers, reducing the space it occupies compared to the full cylinder.
Volume of a Cylinder = πr²h
Since the cone only occupies a third of this volume, the volume of a cone is
Volume of Cone = $\frac{1}{3}$xVolume of Cylinder
Formula
Volume of a Cone = $\frac{1}{3}$πr²h
Example: Let’s calculate the volume of a cone with a radius (r) of 7 cm and a height (l) of 15 cm.
Volume of a Cone = $\frac{1}{3}$πr²h = $\frac{1}{3}$x$\frac{22}{7}$x7x7x15 = 770 cm³
Surface Area of a Cone
The surface area of a cone is the total area that covers the outer surface of the cone. A cone has two components to its surface:
- The circular base at the bottom.
- The curved surface (lateral surface) that wraps around the cone from the base to the apex.
To calculate the total surface area of a cone, we need to find the area of both the base and the curved surface, then add them together.
Step-by-Step Explanation
Area of the Circular Base: The base of the cone is a circle, and the area of a circle is given by the formula:
Area of base = πr
This gives the area of the flat circular base at the bottom of the cone.
Area of the Curved Surface (Lateral Surface): The curved surface of the cone is like a “side” that wraps around from the circular base to the apex. If you were to cut the cone along its slant height and flatten it out, you would get a sector of a larger circle.
The slant height (l) is the length of the side of the cone from the edge of the base to the apex.
The curved surface area (also called the lateral surface area) of a cone is given by the formula: Curved surface area = πrl
This formula can be understood as the area of the sector of the larger circle that forms when you unroll the curved surface. The radius of this sector is the slant height (l), and the arc length of the sector is the circumference of the circular base.
Combining Both Parts: To get the total surface area of the cone, we add the area of the circular base and the area of the curved surface.
Formula
Surface Area of a Cone =
Example: Calculate the surface area of a cone with radius 7 cm and slant height 10 cm.
Surface Area of a Cone =
What is a Sphere?
A sphere is a perfectly round 3D shape, like a ball or the Earth. Every point on its surface is the same distance from the center.
Properties
- Faces: 1 continuous curved face
- Edges: No edges, since the surface is completely smooth
- Vertices: No vertices, as there are no points or corners.
- A sphere has no edges, corners, or faces.
- The radius (r) is the distance from the center to any point on the surface.
Volume of a Sphere
The volume of a sphere measures the amount of space inside the sphere, which is a perfectly symmetrical 3D object where every point on its surface is equidistant from the center.
Step-by-Step Explanation
- The Radius (r³): The volume of a sphere is directly proportional to the cube of its radius. A larger sphere has more space inside it. Cubing the radius accounts for the three dimensions (length, width, and depth) of the sphere.
- Multiplying by : The constant is used because it adjusts the formula to account for the spherical shape. comes from the fact that the base of the sphere’s circular slices is a circle, and 4 is a geometric constant related to how spheres behave in 3D space.
- Dividing by 3: Dividing by 3 adjusts for the 3D nature of the shape, just like in the volume formula for a cone, which is also divided by 3. This division accounts for the way the sphere’s volume “spreads out” compared to more straightforward shapes like cubes.
Formula
Volume of a Sphere = $\frac{4}{3}$
Example: Calculate the volume of a sphere with radius 6 cm.
Volume of a Sphere = $\frac{4}{3}$
Surface Area of a Sphere
The surface area of a sphere is the total area of its curved outer surface, which is completely smooth and symmetrical. Imagine a basketball. If you could “unwrap” the entire outer surface of the ball and lay it flat, it would cover an area equal to four times the area of a flat circle with the same radius as the basketball.
Formula
Surface Area of a Sphere = 4
Example: Let’s calculate the volume of a sphere with a radius (r) of 5 cm.
Surface Area = 4
FAQs on Cylinder, Cone and Sphere
- No, by definition, a cylinder has circular bases. If a solid has a square base, it is not a cylinder.
- The height (or altitude) of a cylinder is the perpendicular distance between its two circular bases.
- The lateral surface area of a cone (excluding the base) is: Lateral Surface Area=πrl where r is the radius of the base and l is the slant height.
- The height is the perpendicular distance from the centre of the base to the apex, while the slant height is the diagonal distance from any point on the edge of the base to the apex.
- A right circular cone is a cone where the apex is directly above the centre of the circular base, and the height forms a perpendicular with the base.
- A sphere is a fully round 3D shape where every point on its surface is equidistant from the centre.
- A hemisphere is half of a sphere, divided along its diameter. It has one flat circular face (the cut surface) and one curved surface.
- A hemisphere is half of a sphere, formed by cutting a sphere along its diameter. It has one flat circular face and one curved surface.
- The volume of a hemisphere is half of the volume of a sphere:
Volume = 2/3 πr³
- The surface area of a hemisphere is the sum of the curved surface and the area of the circular base: Surface Area=3πr²
- If the base area and height are the same, a cone can have one-third of the volume of a cylinder. For example, a cone with the same base radius and height as a cylinder has a volume of ⅓ of the cylinder’s volume.