Can You Find The Volume Of Pyramids And Cones?

1. What is the volume of a cone with a height of 25 mm and with a base having diameter 30 mm?

785 mm^3

5888 mm^3

6830 mm^3

23 550 mm^3

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this case, the base has a diameter of 30 mm, so the radius is half of that, which is 15 mm. The height is given as 25 mm. Plugging these values into the formula, we get V = (1/3)π(15^2)(25) = 5888 mm^3.

Explanation

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this case, the base has a diameter of 30 mm, so the radius is half of that, which is 15 mm. The height is given as 25 mm. Plugging these values into the formula, we get V = (1/3)π(15^2)(25) = 5888 mm^3.

2. A paperweight in the shape of a square pyramid is filled with water. The paperweight has a side length of 7 cm and a height of 7.5 cm. How much water can the paperweight hold?

30.6 cm^3

35 cm^3

48.1 cm^3

122.5 cm^3

The paperweight is in the shape of a square pyramid, which means it has a square base. The side length of the square base is given as 7 cm. The volume of a pyramid is calculated by multiplying the area of the base by the height and then dividing by 3. The area of a square is calculated by multiplying the side length by itself. In this case, the area of the base is 7 cm * 7 cm = 49 cm^2. The height of the paperweight is given as 7.5 cm. Plugging these values into the formula, we get (49 cm^2 * 7.5 cm) / 3 = 122.5 cm^3. Therefore, the paperweight can hold 122.5 cm^3 of water.

Explanation

The paperweight is in the shape of a square pyramid, which means it has a square base. The side length of the square base is given as 7 cm. The volume of a pyramid is calculated by multiplying the area of the base by the height and then dividing by 3. The area of a square is calculated by multiplying the side length by itself. In this case, the area of the base is 7 cm * 7 cm = 49 cm^2. The height of the paperweight is given as 7.5 cm. Plugging these values into the formula, we get (49 cm^2 * 7.5 cm) / 3 = 122.5 cm^3. Therefore, the paperweight can hold 122.5 cm^3 of water.

3. Chocolate candy is shaped like a regular hexagonal pyramid with base edges 1.2cm and a height of 1.5cm. The candy is filled with caramel. How much caramel will fit in the candy if the distance from the center of the base to the midpoint of each side is 1.04cm?

0.72 cm^3

1.87 cm^3

2.25 cm^3

3.74 cm^3

The volume of a regular hexagonal pyramid can be calculated using the formula V = (1/3) * A * h, where A is the area of the base and h is the height of the pyramid. In this case, the base is a regular hexagon, so its area can be calculated using the formula A = (3√3/2) * s^2, where s is the length of one side of the hexagon. The length of one side of the hexagon can be calculated using the distance from the center of the base to the midpoint of each side, which is given as 1.04 cm. Therefore, s = 2 * 1.04 cm = 2.08 cm. Plugging these values into the formulas, we get A ≈ 10.392 cm^2 and V ≈ 1.87 cm^3.

Explanation

The volume of a regular hexagonal pyramid can be calculated using the formula V = (1/3) * A * h, where A is the area of the base and h is the height of the pyramid. In this case, the base is a regular hexagon, so its area can be calculated using the formula A = (3√3/2) * s^2, where s is the length of one side of the hexagon. The length of one side of the hexagon can be calculated using the distance from the center of the base to the midpoint of each side, which is given as 1.04 cm. Therefore, s = 2 * 1.04 cm = 2.08 cm. Plugging these values into the formulas, we get A ≈ 10.392 cm^2 and V ≈ 1.87 cm^3.

4. A paper cup at a water fountain in the shape of a cone has a height of 9cm and a diameter of 7cm. How much water can the cup hold?

115.4 cm^3

124.4 cm^3

131.9 cm^3

461.6 cm^3

The volume of a cone can be calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. In this case, the radius is half of the diameter, so it is 3.5 cm. Plugging in the values, we get V = (1/3)π(3.5)^2(9) = 115.4 cm^3. Therefore, the cup can hold 115.4 cm^3 of water.

Explanation

The volume of a cone can be calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. In this case, the radius is half of the diameter, so it is 3.5 cm. Plugging in the values, we get V = (1/3)π(3.5)^2(9) = 115.4 cm^3. Therefore, the cup can hold 115.4 cm^3 of water.

5. A tepee stands 8.5 m tall and 8 m in diameter. What is the volume of the tepee?

71.2 m^2

142.3 m^2

157.4 m^2

569.4 m^2

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Explanation

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6. What is the volume of a pyramid with a slant height 25 cm and a base that is an equilateral triangle with a perimeter of 120 cm?

5000 cm^3

8000 cm^3

13 333 cm^3

15 467 cm^3

The volume of a pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base of the pyramid is an equilateral triangle with a perimeter of 120 cm, which means each side is 40 cm. The height of the pyramid can be found using the Pythagorean theorem, where the slant height is the hypotenuse and half the base length is one of the legs. Solving for the height, we get h = sqrt(25^2 - 20^2) = 15 cm. The base area of an equilateral triangle can be calculated using the formula (sqrt(3)/4) * side^2, which gives us a base area of (sqrt(3)/4) * 40^2 = 400sqrt(3) cm^2. Plugging in these values into the volume formula, we get V = (1/3) * 400sqrt(3) * 15 = 8000 cm^3.

Explanation

The volume of a pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base of the pyramid is an equilateral triangle with a perimeter of 120 cm, which means each side is 40 cm. The height of the pyramid can be found using the Pythagorean theorem, where the slant height is the hypotenuse and half the base length is one of the legs. Solving for the height, we get h = sqrt(25^2 - 20^2) = 15 cm. The base area of an equilateral triangle can be calculated using the formula (sqrt(3)/4) * side^2, which gives us a base area of (sqrt(3)/4) * 40^2 = 400sqrt(3) cm^2. Plugging in these values into the volume formula, we get V = (1/3) * 400sqrt(3) * 15 = 8000 cm^3.

7. A square pyramid has a base with side length 24 cm and has a slant height of 15 cm. What is the volume of the pyramid?

432 cm^3

720 cm^3

1728 cm^3

2880 cm^3

The volume of a square pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base area is equal to the side length squared (24 cm * 24 cm) and the height is equal to the slant height (15 cm). Plugging these values into the formula, we get V = (1/3) * 24 cm * 24 cm * 15 cm = 1728 cm^3. Therefore, the correct answer is 1728 cm^3.

Explanation

The volume of a square pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base area is equal to the side length squared (24 cm * 24 cm) and the height is equal to the slant height (15 cm). Plugging these values into the formula, we get V = (1/3) * 24 cm * 24 cm * 15 cm = 1728 cm^3. Therefore, the correct answer is 1728 cm^3.

8. A cone has a base with a diameter of 12 cm and a slant height of 10 cm. What is the volume of the cone?

251.2 cm^3

301.4 cm^3

376.8 cm^3

1205.8 cm^3

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this question, the diameter of the base is given as 12 cm, so the radius would be half of that, which is 6 cm. The slant height of the cone is given as 10 cm, which is the height of the cone. Plugging these values into the formula, we get V = (1/3)π(6^2)(10) = 301.4 cm^3.

Explanation

The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this question, the diameter of the base is given as 12 cm, so the radius would be half of that, which is 6 cm. The slant height of the cone is given as 10 cm, which is the height of the cone. Plugging these values into the formula, we get V = (1/3)π(6^2)(10) = 301.4 cm^3.

9. Stephanie made a cone out of paper. The cone has the same measurements for the height and diameter. What is the radius of the cone if the volume measures 16.75 cm^3

2 cm

3 cm

4 cm

5 cm

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the volume is given as 16.75 cm^3 and the height and diameter are the same. Since the height and diameter are the same, the height can be used as the diameter. We can rearrange the formula to solve for the radius: r = √((3V)/(πh)). Plugging in the given values, we get r = √((3*16.75)/(π*2)). Simplifying this expression gives r ≈ 2 cm. Therefore, the radius of the cone is 2 cm.

Explanation

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the volume is given as 16.75 cm^3 and the height and diameter are the same. Since the height and diameter are the same, the height can be used as the diameter. We can rearrange the formula to solve for the radius: r = √((3V)/(πh)). Plugging in the given values, we get r = √((3*16.75)/(π*2)). Simplifying this expression gives r ≈ 2 cm. Therefore, the radius of the cone is 2 cm.

10. A cone has a diameter of 4 m and a volume of 23 m^2. What is the slant height of the cone?

5 m

5.5 m

5.75 m

5.9 m

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. Since the diameter is 4 m, the radius is 2 m. We can rearrange the formula to solve for h: h = (3V)/(πr^2). Plugging in the given volume of 23 m^2 and radius of 2 m, we get h = (3 * 23) / (π * 2^2) = 69 / (4π) ≈ 5.9 m. Therefore, the slant height of the cone is 5.9 m.

Explanation

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. Since the diameter is 4 m, the radius is 2 m. We can rearrange the formula to solve for h: h = (3V)/(πr^2). Plugging in the given volume of 23 m^2 and radius of 2 m, we get h = (3 * 23) / (π * 2^2) = 69 / (4π) ≈ 5.9 m. Therefore, the slant height of the cone is 5.9 m.