An open right circular cone has a base radius of 5cm and perpendicular height of 12cm. Calculate the surface area of the cone (take π=3.142)
The surface area of the cone is approximately 188.4 cm2.
OH NO! The robot tutor messed this one up.
You need the slant height, not the perpendicular height to find SA
SA of cone = π r l
= π(5)(13) , did you recognize the 5-12-13 right-angled triangle?
= appr 204.2 cm^2
I did not include the surface area of the base of the cone
To calculate the surface area of the cone, we need to find the area of the base and the lateral surface area.
1. Area of the base:
The base of the cone is a circle, so the area can be calculated using the formula: A = π * r^2, where π is approximately 3.142 and r is the radius.
A = 3.142 * 5^2
A = 3.142 * 25
A ≈ 78.55 cm^2
2. Lateral surface area:
The lateral surface area can be calculated using the formula: A = π * r * l, where r is the radius and l is the slant height of the cone.
To find the slant height, we can use the Pythagorean theorem:
l^2 = h^2 + r^2, where h is the height of the cone.
l^2 = 12^2 + 5^2
l^2 = 144 + 25
l^2 = 169
l = √169
l = 13 cm
Now we can calculate the lateral surface area:
A = 3.142 * 5 * 13
A = 3.142 * 65
A ≈ 204.21 cm^2
3. Total surface area:
The total surface area is the sum of the base area and the lateral surface area:
Total surface area = base area + lateral surface area
Total surface area = 78.55 + 204.21
Total surface area ≈ 282.76 cm^2
Therefore, the surface area of the cone is approximately 282.76 cm^2.
To calculate the surface area of a cone, we use the formula:
Surface Area = π × r × (r + l)
Where:
- π is a mathematical constant approximately equal to 3.142
- r is the radius of the base of the cone
- l is the slant height of the cone
To find the slant height (l), we can use the Pythagorean theorem in the right triangle formed by the height (h), the radius (r), and the slant height (l). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the height (h) of the cone is 12 cm and the radius (r) is 5 cm. We can use these values to calculate the slant height (l) as follows:
l² = h² + r²
l² = 12² + 5²
l² = 144 + 25
l² = 169
l = √169
l = 13 cm (approx)
Now that we have the radius (r = 5 cm) and the slant height (l = 13 cm), we can substitute these values into the formula for the surface area of the cone:
Surface Area = π × r × (r + l)
Surface Area = 3.142 × 5 × (5 + 13)
Surface Area = 3.142 × 5 × 18
Surface Area = 282.78 cm² (approx)
Therefore, the surface area of the cone is approximately 282.78 cm².