Area of a Cone Calculator

Area of a Cone Example 1

Find the area of the closed cone below. Give your answer to 1 decimal place.

The height of the cone is 12 cm. The radius is 5 cm.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt {r^2 + h^2} \; ) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi \cdot (5) \cdot (5 + \sqrt {5^2 + 12^2} \; ) \]

If we simplify this, we get: \[ A = 5 \pi \cdot (5 + \sqrt {25 + 144}) = 5 \pi \cdot (5 + \sqrt {169} \;) \]

This means that: \[ A = 5 \pi \cdot (5 + 13) = 5 \pi \cdot (18) = 90 \pi \]

This gives us a final answer of: \[ A = 282.7 \;\ cm^2 \;\ to \;\ 1 \;\ decimal \;\ place \]

Area of a Cone Example 2

Find the surface area of the closed cone below to 1 decimal place.

The cone is lying on its side. The actual height of the cone is 8 inches. The radius is 3 ½ inches.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt {r^2 + h^2} \; ) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi \cdot (3 {1 \over 2}) \cdot (3 {1 \over 2} + \sqrt {(3 {1 \over 2})^2 + 8^2} \; ) \]

If we simplify this, we get: \[ A = 3 {1 \over 2} \pi \cdot (3 {1 \over 2} + \sqrt {12 {1 \over 4} + 64} \; ) = 3 {1 \over 2} \pi \cdot (3 {1 \over 2} + \sqrt {76 {1 \over 4}} \; ) \]

This means that: \[ A = 3 {1 \over 2} \pi \cdot (3 {1 \over 2} + 8.732...) = 3 {1 \over 2} \pi \cdot (12.232...) = 42.812... \pi \]

This gives us a final answer of: \[ A = 134.5 \;\ in^2 \;\ to \;\ 1 \;\ decimal \;\ place \]

Area of a Cone Example 3

Find the surface area of the closed cone below to 2 decimal places.

The height of the cone is 4.8 meters. The diameter of the cone is 4.6 meters.

To find the radius, we divide the diameter by 2. So the radius = 4.8 ÷ 2 = 2.4 meters.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt {r^2 + h^2} \; ) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi \cdot (2.4) \cdot (2.4 + \sqrt {(2.4)^2 + (4.6)^2} \; ) \]

If we simplify this, we get: \[ A = 2.4 \pi \cdot (2.4 + \sqrt {5.76 + 21.16} \; ) = 2.4 \pi \cdot (2.4 + \sqrt {26.92} \; ) \]

This means that: \[ A = 2.4 \pi \cdot (2.4 + 5.1884...) = 2.4 \pi \cdot (7.588...) = 18.212... \pi \]

This gives us a final answer of: \[ A = 57.22 \;\ m^2 \;\ to \;\ 2 \;\ decimal \;\ places \]

Area of a Cone Example 4

Find the surface area of the witch's black hat which is an open cone shown below. Give your answer to 1 decimal place.

The height of the cone is 60 cm. The diameter of the cone is 18 cm.

To find the radius, we divide the diameter by 2. So the radius = 18 ÷ 2 = 9 cm.

The formula for the area of an open cone is: \[ A = \pi r \sqrt {r^2 + h^2} \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi \cdot (9) \cdot \sqrt {9^2 + 60^2} \]

If we simplify this, we get: \[ A = 9 \pi \cdot \sqrt {81 + 3600} = 9 \pi \cdot \sqrt {3681} \]

This means that: \[ A = 9 \pi \cdot (60.671...) = 546.041... \pi \]

This gives us a final answer of: \[ A = 1715.4 \;\ cm^2 \;\ to \;\ 1 \;\ decimal \;\ place \]