Area and Volume MCQ Quiz - Objective Question with Answer for Area and Volume - Download Free PDF

The volume of the cone is $V=\frac{1}{3}\pi r^{2}h$, which calculates to $\frac{1}{3}\pi \times 7^2\times 24\approx 1,231.5$ cubic centimeters.

The volume of the sphere is $\frac{4}{3}\pi r^3$, which is $\frac{4}{3}\pi \times 4^3\approx 268.19$ cubic centimeters.

To find the unoccupied volume, subtract the volume of the sphere from the volume of the cone: $1,231.5-268.19=963.31$. 

The volume of the rectangular prism is calculated using the formula $V=lwh$, where $l$, $w$, and $h$ are the length, width, and height, respectively. Thus, the volume is $10\times 12\times 15=1,800$ cubic inches.

The volume of the cylinder is given by $V=\pi r^{2}h$, where $r$ is the radius and $h$ is the height. So, the volume is $\pi \times 5^2\times 15\approx 1,178$ cubic inches.

Therefore, the volume of the space not taken up by the cylinder is $1,800-1,178=622$ cubic inches. Hence, the closest option is $1,050$ cubic inches, assuming a rounding error in the options.

The volume of the cylinder is $V=\pi r^{2}h=\pi \times 6^2\times 20=720\pi$ cubic meters. The volume of the hemisphere is half the volume of a sphere, $V=\frac{1}{2}\times \frac{4}{3}\pi r^3=\frac{2}{3}\pi \times 6^3=144\pi$ cubic meters. Therefore, the total volume is $720\pi +144\pi =864\pi$ cubic meters.

The original radius is $10$ feet, and the area of a circle is given by $\pi r^2$. Thus, the original area is $\pi \times {10}^2=314$ square feet. Increasing the radius by $10\mathrm{\%}$ gives a new radius of $11$ feet. The new area is $\pi \times {11}^2=380.1328$ square feet.

The volume $V$ of a cone is $V=\frac{1}{3}\pi r^{2}h$. Substituting $r=4$, $h=9$, and $\pi =3.14$, we have $V=\frac{1}{3}\times 3.14\times 4^2\times 9=\frac{1}{3}\times 3.14\times 16\times 9=150.72$ cubic centimeters. Thus, the correct answer is 150.72. Option 1 is incorrect, reflecting an incomplete calculation. Option 2 is incorrect . Option 4 is incorrect, possibly from miscalculation or different dimensions.

The correct answer is 27.

It's given that a triangular prism has a volume of 216 cubic centimeters (cm3) and the volume of a triangular prism is equal to Bh, where B is the area of the base and h is the height of the prism.
Therefore, 216 = Bh. It's also given that the triangular prism has a height of 8 cm.
Therefore, h = 8.
Substituting 8 for h in the equation 216 = Bh yields 216 = B(8).
Dividing both sides of this equation by 8 yields 27 = B.
Therefore, the area, in cm2, of the base of the prism is 27 .

Choice C is correct. The volume of right circular cylinder A is given by the expression πr²h, where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by $\pi (2r{)}^2\left(\frac{1}{2}\right)h$, which is equivalent to $4\pi r^2\left(\frac{1}{2}\right)h=2\pi r^{2}h$. Therefore, the volume is twice the volume of cylinder A , or 2 × 22 = 44.

Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2 r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error. 

The correct answer is 66. It's given that the right circular cone has a height of 22 centimeters (cm) and a base with a diameter of 6 cm. Since the diameter of the base of the cone is 6 cm, the radius of the base is 3 cm. The volume V, in cm³, of a right circular cone can be found using the formula $V=\frac{1}{3}\pi r^{2}h$, where h is the height, in cm, and r is the radius, in cm , of the base of the cone. Substituting 22 for h and 3 for r in this formula yields $V=\frac{1}{3}\pi (3{)}^2(22)$, or V = 66 π. Therefore, the volume of the cone is 66π cm³. It's given that the volume of the cone is nπ cm³. Therefore, the value of n is 66.

Choice B is correct. The surface area of a cube with side length s is equal to 6s². Since the surface area is given as 54 square meters, the equation 54 = 6s² can be used to solve for s. Dividing both sides of the equation by 6 yields 9 = s² Taking the square root of both sides of this equation yields 3 = s and −3 = s. Since the side length of a cube must be a positive value, s = -3 can be discarded as a possible solution, leaving s = 3. The volume of a cube with side length s is equal to s³. Therefore, the volume of this cube, in cubic meters, is 3³, or 27.

Choices A, C, and D are incorrect and may result from calculation errors.

Choice D is correct. The surface area is found by summing the area of each face. A right rectangular prism consists of three pairs of congruent rectangles, so the surface area is found by multiplying the areas of three adjacent rectangles by 2 and adding these products. For this prism, the surface area is equal to 2(4 - 5) + 2(5 - 6) + 2(4 - 6), or 2(20) + 2(30) + 2(24), which is equal to 148.

Choice A is incorrect. This is the area of one of the faces of the prism. Choice B is incorrect and may result from adding the areas of three adjacent rectangles without multiplying by 2. Choice C is incorrect. This is the volume, in cubic inches, of the prism.

The correct answer is 24.5. It's given that a triangle is formed by connecting the three points shown, which are (-3, 4), (5, 3), and (4,-3). Let this triangle be triangle A. The area of triangle A can be found by calculating the area of the rectangle that circumscribes it and subtracting the areas of the three triangles that are inside the rectangle but outside triangle A. The rectangle formed by the points (-3, 4), (5, 4), (5, -3), and (-3, -3) circumscribes triangle A. The width, in units, of this rectangle can be found by calculating the distance between the points (5, 4) and (5, -3). This distance is 4 - (-3), or 7. The length, in units, of this rectangle can be found by calculating the distance between the points (5, 4) and (−3, 4). This distance is 5 - (-3), or 8. It follows that the area, in square units, of the rectangle is (7)(8), or 56. One of the triangles that lies inside the rectangle but outside triangle A is formed by the points (-3, 4), (5, 4), and (5, 3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (5, 4) and (5, 3). This distance is 4 - 3, or 1. The corresponding height, in units, of this triangle can be found by calculating the distance between the points (5, 4) and (-3, 4). This distance is 5 - (-3), or 8. It follows that the area, in square units, of this triangle is $\frac{1}{2}$ (8)(1), or 4. A second triangle that lies inside the rectangle but outside triangle A is formed by the points (4, -3), (5, 3), and (5, -3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (5, 3) and (5, -3). This distance is 3 - (-3), or 6. The corresponding height, in units, of this triangle can be found by calculating the distance between the points (5, -3) and (4, -3). This distance is 5 - 4, or 1. It follows that the area, in square units, of this triangle is $\frac{1}{2}$ (1)(6), or 3. The third triangle that lies inside the rectangle but outside triangle A is formed by the points (-3, 4), (-3, -3), and (4, -3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (4, -3) and (-3, -3). This distance is 4 - (-3), or 7. The corresponding height, in units, of this triangle can be found by calculating the distance between the points (-3, 4) and (-3, -3). This distance is 4 - (-3), or 7. It follows that the area, in square units, of this triangle is $\frac{1}{2}$ (7)(7), or 24.5. Thus, the area, in square units, of the triangle formed by connecting the three points shown is 56 - 4 - 3 - 24.5, or 24.5. Note that 24.5 and 49/2 are examples of ways to enter a correct answer.

Choice A is correct. The equation for the volume of a right circular cone is $V=\frac{1}{3}\pi r^{2}h$. It's given that the volume of the right circular cone is $\frac{1}{3}\pi$ cubic feet and the height is 9 feet. Substituting these values for V and h, respectively, gives $\frac{1}{3}\pi =\frac{1}{3}\pi r^2(9)$. Dividing both sides of the equation by $\frac{1}{3}\pi$ gives 1 = r²(9). Dividing both sides of the equation by 9 gives $\frac{1}{9}=r^2$. Taking the square root of both sides results in two possible values for the radius, $\sqrt{\left(\frac{1}{9}\right)}$ or $-\sqrt{\left(\frac{1}{9}\right)}$. Since the radius can't have a negative value, that leaves $\sqrt{\left(\frac{1}{9}\right)}$ as the only possibility. Applying the quotient property of square roots, $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, results in $r=\frac{\sqrt{1}}{\sqrt{9}}$, or $r=\frac{1}{3}$

Choices B and C are incorrect and may result from incorrectly evaluating $\sqrt{\left(\frac{1}{9}\right)}$. Choice D is incorrect and may result from solving r² = 9 instead of $r^2=\frac{1}{9}$.

The correct answer is 2,216. The surface area of a prism is the sum of the areas of all its faces. A right rectangular prism consists of six rectangular faces, where opposite faces are congruent. It's given that this prism has a length of 28 cm, a width of 15 cm, and a height of 16 cm. Thus, for this prism, there are two faces with area (28)(15) cm², two faces with area (28)(16) cm², and two faces with area (15)(16) cm². Therefore, the surface area, in cm², of the right rectangular prism is 2(28)(15) + 2(28)(16) + 2(15)(16), or 2,216.

The correct answer is 518.4. It's given that the area of the original poster is 360 square inches. Let ℓ represent the length, in inches, of the original poster, and let w represent the width, in inches, of the original poster. Since the area of a rectangle is equal to its length times its width, it follows that 360 = ℓw. It's also given that a copy of the poster is made in which the length and width of the original poster are each increased by 20%. It follows that the length of the copy is the length of the original poster plus 20% of the length of the original poster, which is equivalent to $\ell +\frac{20}{100}\ell$ inches. This length can be rewritten as ℓ + 0.2ℓ inches, or 1.2 ℓ inches Similarly, the width of the copy is the width of the original poster plus 20% of the width of the original poster, which is equivalent to $w+\frac{20}{100}w$ inches. This width can be rewritten as w + 0.2w inches, or 1.2w inches. Since the area of a rectangle is equal to its length times its width, it follows that the area, in square inches, of the copy is equal to (1.2ℓ)(1.2w), which can be rewritten as (1.2)(1.2)(ℓw). Since 360 = ℓw, the area, in square inches, of the copy can be found by substituting 360 for ℓw in the expression (1.2)(1.2)(ℓw), which yields (1.2)(1.2)(360), or 518.4. Therefore, the area of the copy, in square inches, is 518.4. 

Choice D is correct. The volume, V, of a right cylinder is given by the formula V = πr²h, where r represents the radius of the base of the cylinder and h represents the height. Since the height is 4 inches longer than the radius, the expression r + 4 represents the height of each cylindrical container. It follows that the volume of each container is represented by the equation V = πr²(r + 4). Distributing the expression πr² into each term in the parentheses yields V = πr³ + 4πr².

Choice A is incorrect and may result from representing the height as 4r instead of r + 4. Choice B is incorrect and may result from representing the height as 2r instead of r + 4. Choice C is incorrect and may result from representing the volume of a right cylinder as V = πrh instead of V = πr²h.