Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in cubic units, is _________
Show Hint
- \( \frac{\pi}{3} \)
- \( \frac{2\pi}{3} \)
- \( \frac{3\pi}{2} \)
- \( 3\pi \)
The Correct Option is A
Solution and Explanation
- The base radius \( r \) of the cone is half of the side of the square, so \( r = \frac{1}{2} \).
- The height \( h \) of the cone is the length of the other side of the triangle, which is also \( 1 \).
The formula for the volume of a cone is: \[ V = \frac{1}{3} \pi r^2 h. \] Substituting the values of \( r \) and \( h \): \[ V = \frac{1}{3} \pi \left( \frac{1}{2} \right)^2 \times 1 = \frac{1}{3} \pi \times \frac{1}{4} = \frac{\pi}{3}. \] Thus, the volume of the cone is \( \frac{\pi}{3} \) cubic units. Final Answer: \( \frac{\pi}{3} \)
Top Questions on Geometry
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
- In a $\triangle ABC$, points $D$ and $E$ are on the sides $BC$ and $AC$, respectively. $BE$ and $AD$ intersect at point $T$ such that $AD : AT = 4 : 3$, and $BE : BT = 5 : 4$. Point $F$ lies on $AC$ such that $DF$ is parallel to $BE$. Then, $BD : CD$ is:
- A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
Questions Asked in GATE EC exam
Courage : Bravery :: Yearning :
Select the most appropriate option to complete the analogy.The number of patients per shift (X) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is ₹1000(X - 0.2), what is the average amount (in ₹) she has earned per shift in the past 100 shifts?
- Water from a hand pump located near a landfill has 1 mg/L arsenic (oral carcinogenic potency factor = 1.75 (kg-day)/mg). A person who lives nearby drinks 2 L/day of water from this hand pump for 10 years. Assume a body weight of 70 kg and an average life duration of 70 years. The chances of this person getting an excess risk of cancer is ________ × 10-3 (rounded off to three decimal places).
- GATE ES - 2025
- Hydrology
- An industry releases three greenhouse gases (GHGs), CO2 (5 kg/day), CH4 (0.5 kg/day), and N2O (0.1 kg/day). The industry flares the CH4 before it is released to the atmosphere. The Global Warming Potential (GWP) are as follows: CO2 = 1, CH4 = 21, N2O = 310. The annual GWP of GHGs released from the industry is ________ kg CO2 equivalent (rounded off to the nearest integer).
- GATE ES - 2025
- Greenhouse Effect
- A solid waste of composition \( C_{60}H_{135}O_{50}N_5 \) is to be composted aerobically in a closed vessel mechanical composting facility. Given: all ammonia generated escapes the facility; air contains 23% of Oxygen by weight; 100% excess air requirement for the closed vessel composting facility. The atomic weights: C – 12, H – 1, O – 16, N – 14. The actual air required for composting is:
- GATE ES - 2025
- Waste Management