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What is the Total Surface Area of a Frustum of a Cone?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Total Surface Area (TSA) of a frustum of a cone is the sum of the areas of its two circular bases (top and bottom) and its curved surface area. Imagine cutting the top off a cone; the remaining shape is a frustum, and TSA is the total area you would need to paint or cover on its entire outer surface.
Simple Example
Quick Example
Think of a bucket used to carry water or sand on a construction site. A bucket is a perfect example of a frustum of a cone. The Total Surface Area of the bucket would be the area of its bottom, the area of its top opening (if it had a lid), and the area of the curved metal or plastic part that forms its sides.
Worked Example
Step-by-Step
Let's find the TSA of a frustum with a top radius (r1) of 3 cm, a bottom radius (r2) of 7 cm, and a slant height (l) of 5 cm. Use pi = 22/7.
---Step 1: Calculate the area of the top circular base. Area_top = pi * r1^2 = (22/7) * 3 * 3 = (22/7) * 9 = 198/7 sq cm.
---Step 2: Calculate the area of the bottom circular base. Area_bottom = pi * r2^2 = (22/7) * 7 * 7 = (22/7) * 49 = 22 * 7 = 154 sq cm.
---Step 3: Calculate the Curved Surface Area (CSA) of the frustum. CSA = pi * (r1 + r2) * l = (22/7) * (3 + 7) * 5 = (22/7) * 10 * 5 = (22/7) * 50 = 1100/7 sq cm.
---Step 4: Add all the areas to find the Total Surface Area (TSA). TSA = Area_top + Area_bottom + CSA = (198/7) + 154 + (1100/7).
---Step 5: Combine the fractions: TSA = (198 + 1100)/7 + 154 = 1298/7 + 154.
---Step 6: Convert 154 to a fraction with denominator 7: 154 = 154 * 7 / 7 = 1078/7.
---Step 7: Add them: TSA = 1298/7 + 1078/7 = (1298 + 1078)/7 = 2376/7 sq cm.
---Step 8: Calculate the decimal value: 2376 / 7 approximately 339.43 sq cm.
Answer: The Total Surface Area of the frustum is approximately 339.43 sq cm.
Why It Matters
Understanding surface area is crucial in engineering for designing structures, in packaging for estimating material needs, and in physics for studying heat transfer. Civil engineers use this to calculate how much paint is needed for a conical water tank, and product designers use it to optimize packaging material for food items.
Common Mistakes
MISTAKE: Forgetting to add the areas of both the top and bottom circular bases. | CORRECTION: Remember that TSA includes the area of ALL surfaces, so both circles must be included along with the curved part.
MISTAKE: Confusing the height (h) with the slant height (l) in the formulas. | CORRECTION: The slant height (l) is used for the curved surface area, while height (h) is used to find the volume or slant height if it's not given.
MISTAKE: Using the radius of the original cone instead of the two radii of the frustum. | CORRECTION: A frustum has two different radii (r1 and r2). Always use these specific radii in the formulas for its bases and curved surface.
Practice Questions
Try It Yourself
QUESTION: A frustum has a top radius of 2 cm, a bottom radius of 6 cm, and a slant height of 8 cm. Calculate its Curved Surface Area (CSA) using pi = 3.14. | ANSWER: CSA = pi * (r1 + r2) * l = 3.14 * (2 + 6) * 8 = 3.14 * 8 * 8 = 3.14 * 64 = 200.96 sq cm.
QUESTION: Find the Total Surface Area (TSA) of a frustum if its top radius is 4 cm, bottom radius is 9 cm, and slant height is 10 cm. Use pi = 22/7. | ANSWER: Area_top = (22/7)*4*4 = 352/7. Area_bottom = (22/7)*9*9 = 1782/7. CSA = (22/7)*(4+9)*10 = (22/7)*13*10 = 2860/7. TSA = (352 + 1782 + 2860)/7 = 4994/7 approx 713.43 sq cm.
QUESTION: A frustum-shaped planter has a top diameter of 28 cm, a bottom diameter of 14 cm, and a height of 24 cm. First, find its slant height, then calculate its Total Surface Area (TSA) using pi = 22/7. (Hint: r1 = 14 cm, r2 = 7 cm. Slant height l = sqrt(h^2 + (r1-r2)^2)). | ANSWER: r1 = 14 cm, r2 = 7 cm. l = sqrt(24^2 + (14-7)^2) = sqrt(576 + 7^2) = sqrt(576 + 49) = sqrt(625) = 25 cm. Area_top = (22/7)*14*14 = 616 sq cm. Area_bottom = (22/7)*7*7 = 154 sq cm. CSA = (22/7)*(14+7)*25 = (22/7)*21*25 = 22*3*25 = 1650 sq cm. TSA = 616 + 154 + 1650 = 2420 sq cm.
MCQ
Quick Quiz
Which of the following formulas correctly represents the Total Surface Area (TSA) of a frustum of a cone?
pi * (r1 + r2) * l
pi * r1^2 + pi * r2^2
pi * (r1 + r2) * l + pi * r1^2 + pi * r2^2
pi * (r1 + r2) * h + pi * r1^2 + pi * r2^2
The Correct Answer Is:
C
Option C correctly adds the Curved Surface Area (pi * (r1 + r2) * l) to the areas of both the top and bottom circular bases (pi * r1^2 and pi * r2^2). Options A and B are only parts of the TSA, and option D incorrectly uses height (h) instead of slant height (l) for the curved surface.
Real World Connection
In the Real World
You see frustums everywhere in India! The shape of many traditional clay pots (matkas) and steel buckets, the base of a lamp shade, or even certain parts of rocket engines designed by ISRO, are frustums. Calculating their surface area helps in manufacturing, painting, or even understanding how much material is needed.
Key Vocabulary
Key Terms
Frustum: A part of a cone remaining when its top part is cut off by a plane parallel to the base. | Radius (r): The distance from the center of a circle to its edge. | Slant Height (l): The distance along the slanted surface from the top edge to the bottom edge. | Curved Surface Area (CSA): The area of the slanting side of the frustum, excluding the top and bottom circles. | Total Surface Area (TSA): The sum of the areas of all surfaces of a 3D object.
What's Next
What to Learn Next
Great job understanding frustums! Next, you can explore the 'Volume of a Frustum of a Cone'. Knowing how to find the surface area will help you visualize the shape better, which is key to understanding how much space it can hold.
