Slant Height of Cone
A cone is a three-dimensional solid with a circular base and a curved surface that tapers to a point called the apex (or vertex). The slant height of a cone is the distance from the apex to any point on the circumference of the base.
The slant height is different from the height (perpendicular distance from apex to centre of base). The relationship between slant height, height, and radius is derived using the Pythagorean Theorem.
The slant height is essential for calculating the curved surface area (CSA) of a cone. In NCERT Class 9, students learn to calculate slant height and use it in surface area problems.
What is Slant Height of Cone?
Definition: The slant height of a right circular cone is the distance from the apex to any point on the edge (circumference) of the circular base.
It is denoted by l.
l = √(r² + h²)
Where:
- l = slant height
- r = radius of the circular base
- h = perpendicular height of the cone (from apex to centre of base)
Slant Height of Cone Formula
Key Formulas Involving Slant Height:
1. Slant height from radius and height:
l = √(r² + h²)
2. Height from slant height and radius:
h = √(l² − r²)
3. Radius from slant height and height:
r = √(l² − h²)
4. Curved Surface Area (CSA) of cone:
- CSA = πrl
5. Total Surface Area (TSA) of cone:
- TSA = πr(r + l)
Derivation and Proof
Derivation of the Slant Height Formula:
Consider a right circular cone with:
- Apex at point V
- Centre of circular base at point O
- A point P on the circumference of the base
Step 1: The line VO is the height (h) — perpendicular from apex to base centre.
Step 2: The line OP is the radius (r) of the base.
Step 3: The line VP is the slant height (l).
Step 4: Since VO ⊥ the base, triangle VOP is a right-angled triangle with the right angle at O.
Step 5: By the Pythagorean Theorem:
- VP² = VO² + OP²
- l² = h² + r²
- l = √(r² + h²)
This formula applies to all right circular cones (where the apex is directly above the centre of the base).
Types and Properties
Problems Involving Slant Height:
1. Direct Calculation
- Given r and h, find l = √(r² + h²).
2. Finding Height from Slant Height
- Given l and r, find h = √(l² − r²).
3. Finding Radius from Slant Height
- Given l and h, find r = √(l² − h²).
4. Using Slant Height in CSA Problems
- CSA = πrl. Often you must first calculate l from r and h.
5. Word Problems
- Tents, ice cream cones, funnels — find the slant height from physical dimensions.
Solved Examples
Example 1: Example 1: Find slant height from radius and height
Problem: A cone has radius 6 cm and height 8 cm. Find the slant height.
Solution:
Given: r = 6 cm, h = 8 cm
Using the formula:
- l = √(r² + h²) = √(36 + 64) = √100 = 10 cm
Answer: The slant height is 10 cm.
Example 2: Example 2: Find height from slant height and radius
Problem: A cone has slant height 13 cm and radius 5 cm. Find the height.
Solution:
Given: l = 13 cm, r = 5 cm
Using the formula:
- h = √(l² − r²) = √(169 − 25) = √144 = 12 cm
Answer: The height is 12 cm.
Example 3: Example 3: Find radius from slant height and height
Problem: A cone has slant height 15 cm and height 12 cm. Find the radius.
Solution:
Given: l = 15 cm, h = 12 cm
- r = √(l² − h²) = √(225 − 144) = √81 = 9 cm
Answer: The radius is 9 cm.
Example 4: Example 4: CSA using slant height
Problem: Find the curved surface area of a cone with radius 7 cm and height 24 cm.
Solution:
Step 1: Find slant height
- l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm
Step 2: CSA = πrl
- = (22/7) × 7 × 25 = 22 × 25 = 550 sq cm
Answer: The CSA is 550 sq cm.
Example 5: Example 5: TSA of a cone
Problem: A cone has radius 3.5 cm and slant height 6 cm. Find the total surface area.
Solution:
Given: r = 3.5 cm, l = 6 cm
- TSA = πr(r + l) = (22/7) × 3.5 × (3.5 + 6)
- = (22/7) × 3.5 × 9.5 = 11 × 9.5 = 104.5 sq cm
Answer: The TSA is 104.5 sq cm.
Example 6: Example 6: Tent problem
Problem: A conical tent has a base diameter of 14 m and height 24 m. Find the slant height and the cost of canvas at Rs 50 per sq m.
Solution:
Given: d = 14 m, so r = 7 m; h = 24 m
Slant height:
- l = √(49 + 576) = √625 = 25 m
CSA (canvas needed):
- = πrl = (22/7) × 7 × 25 = 550 sq m
Cost = 550 × 50 = Rs 27,500
Answer: Slant height = 25 m; Cost = Rs 27,500.
Example 7: Example 7: Finding slant height from CSA
Problem: The CSA of a cone is 308 sq cm and its radius is 7 cm. Find the slant height.
Solution:
Given: CSA = 308 sq cm, r = 7 cm
- CSA = πrl
- 308 = (22/7) × 7 × l
- 308 = 22l
- l = 308/22 = 14 cm
Answer: The slant height is 14 cm.
Example 8: Example 8: Pythagorean triple in a cone
Problem: A cone has radius 9 cm and slant height 41 cm. Find the height and the volume.
Solution:
Height:
- h = √(41² − 9²) = √(1681 − 81) = √1600 = 40 cm
Volume:
- V = (1/3)πr²h = (1/3) × (22/7) × 81 × 40
- = (1/3) × (22/7) × 3240 = (22 × 3240)/21 = 71280/21 = 3394.3 cu cm
Answer: Height = 40 cm; Volume ≈ 3394.3 cu cm.
Real-World Applications
Applications:
- Tent and Canopy Design: Canvas requirement for conical tents depends on the CSA, which requires the slant height.
- Ice Cream Cones: Determining the amount of wafer material for cone-shaped containers.
- Funnels: Calculating the surface area of conical funnels in industrial applications.
- Architecture: Conical roofs, spires, and towers require slant height for material estimation.
- Party Caps and Decorations: Making conical paper caps requires knowing the slant height for the correct paper cut-out.
Key Points to Remember
- The slant height l is the distance from the apex to the edge of the base.
- l = √(r² + h²) is derived from the Pythagorean Theorem.
- The slant height is always greater than both the height and the radius.
- l ≠ h unless r = 0 (a degenerate cone).
- CSA of cone = πrl (requires slant height, not height).
- TSA of cone = πr(r + l).
- Given any two of r, h, l, the third can be calculated.
- Common Pythagorean triples used in problems: (3, 4, 5), (5, 12, 13), (6, 8, 10), (7, 24, 25), (8, 15, 17).
- This concept is covered in NCERT Class 9, Chapter 13 (Surface Areas and Volumes).
Practice Problems
- Find the slant height of a cone with radius 5 cm and height 12 cm.
- A cone has slant height 17 cm and height 15 cm. Find the radius and CSA.
- The CSA of a cone is 154 sq cm and its slant height is 7 cm. Find the radius.
- A conical tent has base radius 10.5 m and slant height 14 m. Find the canvas needed.
- A cone has height 21 cm and slant height 28 cm. Find the radius, CSA, and volume.
- The diameter of a cone is 28 cm and height is 45 cm. Find slant height and TSA.
Frequently Asked Questions
Q1. What is the slant height of a cone?
The slant height is the distance from the apex (tip) of the cone to any point on the circumference of the circular base. It is the length of the sloping side.
Q2. How is slant height different from height?
The height is the perpendicular distance from the apex to the centre of the base. The slant height is the distance from the apex to the edge of the base. The slant height is always greater than the height.
Q3. How do you calculate slant height?
Use l = √(r² + h²), where r is the base radius and h is the perpendicular height. This formula comes from the Pythagorean Theorem.
Q4. Why is slant height needed for CSA?
The curved surface of a cone, when unrolled, forms a sector of a circle with radius equal to the slant height. Hence CSA = πrl uses the slant height, not the perpendicular height.
Q5. Can the slant height be less than the height?
No. Since l² = r² + h² and r > 0, we always have l > h. The slant height is always the longest of the three measurements.
Q6. What is the slant height of a cone with equal radius and height?
If r = h, then l = √(r² + r²) = r√2. For example, if r = h = 7 cm, then l = 7√2 ≈ 9.9 cm.
Q7. How do you find slant height from CSA?
Rearrange CSA = πrl to get l = CSA / (πr). You need the radius to calculate the slant height from the CSA.
Q8. Is slant height in the NCERT Class 9 syllabus?
Yes. The slant height of a cone is taught in Chapter 13 (Surface Areas and Volumes) of NCERT Class 9 Mathematics.
