Question:
A toy is in the form of a cone mounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.
A toy is in the form of a cone mounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.
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Recognize the standard Pythagorean triplet $(7, 24, 25)$ to instantly find the slant height $l = 25\text{ cm}$ without going through the step of calculating squares and square roots.
This saves significant calculation time in exam settings!
This saves significant calculation time in exam settings!
Updated On: Jul 7, 2026
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Solution and Explanation
Step 1: Understanding the Question:
This question is from "Surface Areas and Volumes".
We are given a toy that is a combination of two basic solids: a cone mounted on top of a hemisphere.
The common radius of both the hemisphere and the cone base is $r = 7\text{ cm}$.
The total height of the combined toy is $31\text{ cm}$.
We need to determine the total surface area of this toy, which consists of the outer curved surfaces of both the cone and the hemisphere.
Step 2: Key Formula or Approach:
1. Find the height of the cone ($h$) by subtracting the radius of the hemisphere ($r$) from the total height of the toy:
\[ h = \text{Total Height} - r \] 2. Compute the slant height ($l$) of the cone:
\[ l = \sqrt{r^2 + h^2} \] 3. Calculate the total surface area ($TSA$) of the toy, which is the sum of the curved surface area of the cone and the curved surface area of the hemisphere:
\[ TSA = \text{Curved Surface Area of Cone} + \text{Curved Surface Area of Hemisphere} \] \[ TSA = \pi r l + 2\pi r^2 = \pi r (l + 2r) \]
Step 3: Detailed Explanation:
• Identify the given dimensions:
- Radius $r = 7\text{ cm}$
- Total height of the toy $= 31\text{ cm}$
• Calculate the vertical height ($h$) of the cone:
\[ h = 31 - r = 31 - 7 = 24\text{ cm} \]
• Calculate the slant height ($l$) of the cone:
\[ l = \sqrt{r^2 + h^2} \] \[ l = \sqrt{7^2 + 24^2} \] \[ l = \sqrt{49 + 576} = \sqrt{625} \] \[ l = 25\text{ cm} \]
• Write down the expression for the Total Surface Area of the toy:
Note that the circular base of the cone is joined to the circular face of the hemisphere, so these flat faces are internal and not part of the external surface area.
\[ TSA = \pi r l + 2\pi r^2 = \pi r(l + 2r) \]
• Substitute the values ($\pi = \frac{22}{7}$, $r = 7$, and $l = 25$):
\[ TSA = \frac{22}{7} \times 7 \times (25 + 2(7)) \]
• Simplify the expression:
\[ TSA = 22 \times (25 + 14) \] \[ TSA = 22 \times 39 \]
• Perform the final multiplication:
\[ 22 \times 39 = 858\text{ cm}^2 \]
Step 4: Final Answer:
The total surface area of the toy is $858\text{ cm}^2$.
This question is from "Surface Areas and Volumes".
We are given a toy that is a combination of two basic solids: a cone mounted on top of a hemisphere.
The common radius of both the hemisphere and the cone base is $r = 7\text{ cm}$.
The total height of the combined toy is $31\text{ cm}$.
We need to determine the total surface area of this toy, which consists of the outer curved surfaces of both the cone and the hemisphere.
Step 2: Key Formula or Approach:
1. Find the height of the cone ($h$) by subtracting the radius of the hemisphere ($r$) from the total height of the toy:
\[ h = \text{Total Height} - r \] 2. Compute the slant height ($l$) of the cone:
\[ l = \sqrt{r^2 + h^2} \] 3. Calculate the total surface area ($TSA$) of the toy, which is the sum of the curved surface area of the cone and the curved surface area of the hemisphere:
\[ TSA = \text{Curved Surface Area of Cone} + \text{Curved Surface Area of Hemisphere} \] \[ TSA = \pi r l + 2\pi r^2 = \pi r (l + 2r) \]
Step 3: Detailed Explanation:
• Identify the given dimensions:
- Radius $r = 7\text{ cm}$
- Total height of the toy $= 31\text{ cm}$
• Calculate the vertical height ($h$) of the cone:
\[ h = 31 - r = 31 - 7 = 24\text{ cm} \]
• Calculate the slant height ($l$) of the cone:
\[ l = \sqrt{r^2 + h^2} \] \[ l = \sqrt{7^2 + 24^2} \] \[ l = \sqrt{49 + 576} = \sqrt{625} \] \[ l = 25\text{ cm} \]
• Write down the expression for the Total Surface Area of the toy:
Note that the circular base of the cone is joined to the circular face of the hemisphere, so these flat faces are internal and not part of the external surface area.
\[ TSA = \pi r l + 2\pi r^2 = \pi r(l + 2r) \]
• Substitute the values ($\pi = \frac{22}{7}$, $r = 7$, and $l = 25$):
\[ TSA = \frac{22}{7} \times 7 \times (25 + 2(7)) \]
• Simplify the expression:
\[ TSA = 22 \times (25 + 14) \] \[ TSA = 22 \times 39 \]
• Perform the final multiplication:
\[ 22 \times 39 = 858\text{ cm}^2 \]
Step 4: Final Answer:
The total surface area of the toy is $858\text{ cm}^2$.
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