Frustum practice problems
A frustum (plural: frusta or frustums) is a geometric shape that results from cutting the top off a cone or pyramid with a plane parallel to its base. It is essentially a truncated cone or pyramid, consisting of two parallel bases (a larger base and a smaller top) connected by a curved or slanted surface.Key Features of a Frustum:
1. Bases: Two parallel polygonal or circular faces (one larger and one smaller).
2. Height (h): The perpendicular distance between the two bases.
3. Slant Height (l): The distance along the lateral (side) surface between the edges of the two bases (applies to cones and pyramids).
Types of Frustums:
1. Frustum of a Cone:
- The two bases are circles.
- The lateral surface is a curved surface.
2. Frustum of a Pyramid:
- The two bases are polygons (e.g., square, triangle).
- The lateral surface consists of trapezoidal faces.
Directions: Provide a careful solution to each problem, showing all steps in your work.
Number of problems found: 56
- A cone 4
A cone with a radius of 10 cm is divided into two parts by drawing a plane through the midpoint of its axis parallel to its base. Compare the volumes of the two parts. - A lamp
A lamp shade like that of a frustum has a height of 12 cm and an upper and lower diameter of 10 cm and 20 cm. What area of materials is required to cover the curved surface of the frustum? - Lamp shade
A lamp shade is in the form of a frustum of a cone with slant height 7 in., radii of bases 3 in. and 7 in. respectively. How much material is used in its construction if ¼ in. is allowed for the seam. - A frustum
A frustum of a pyramid consists of a square base of length 10 cm and a top square of length 7 cm. The height of the frustum is 6 cm. Calculate the surface area and volume. - The frustum A frustum of a pyramid is 4 cm at the top and 7 cm at the bottom square, and it's 6 cm high. Calculate the volume of the frustum.
- Slant height 3
The frustum of a right circular cone has the diameters of base 10 cm of top 6 cm and a height of 5 cm. Find the slant height. - Wooden bowls
Twenty wooden bowls in the shape of a truncated cone should be painted on the outside and inside with wood varnish. We need 0.1 l of paint to paint 200 cm². How many liters of paint do we have to buy if the bowls are 25 cm high, the bottom of the bowl has - Pit
The pit is 1.2 m deep and in the shape of a truncated pyramid with a rectangular base. Its length and width are the top 3 × 1.5 m and the bottom 1 m × 0.5 m. To paint one square meter of the pit, we use 0.8 l of green paint. How many liters of paint are n - Frustum of a cone
A reservoir contains 28.54 m³ of water when complete. The diameter of the upper base is 3.5 m, while the lower base is 2.5 m. Find the height if the reservoir is in the form of a frustum of a right circular cone. - Runcated pyramid teapot
The 35 cm high teapot has the shape of a truncated pyramid with the length of the edge of the lower square base a=50 cm and the edges of the rectangular base b: 20 cm and c: 30 cm. How many liters of water will fit in the teapot? - 2x cone
Circular cone height 36 cm was cut plane parallel with the base. The volume of these two small cones is the same. Calculate the height of the smaller cone. - Truncated cone
Calculate the height of the rotating truncated cone with volume V = 1471 cm³ and a base radii r1 = 6.1 cm and r2 = 7.9 cm. - Tetrahedron water level
A container shaped like a rotating cylinder with a base radius of 5 cm is filled with water. If a regular tetrahedron with an edge of 7 cm is immersed in it, how much will the water level in the container rise? - The surface
The surface of a truncated rotating cone with side s = 13 cm is S = 510π cm². Find the radii of the bases when their difference in lengths is 10cm. - Cone - bases
The volume of the cut cone is V = 38000π cm³. The radius of the lower base is 10 cm larger than the radius of the upper base. Determine the radius of the base if height v = 60 cm. - Flowerbed
The flowerbed has the shape of a truncated pyramid. The bottom edge of the base a = 10 m, and the upper base b = 9 m. The deviation angle between the edge and the base is alpha = 45°. What volume is needed to make this flowerbed? How many plants can be pl - Two vases
Michaela has two vases in her collection. The first vase has the shape of a cone with a base diameter of d = 20 cm; the second vase has the shape of a truncated cone with a lower base of d1 = 25 cm and a diameter of the upper base d2 = 15 cm. Which vase c - Similar frustums
The upper and lower radii of a frustum of a right circular cone are 8 cm and 32 cm, respectively. If the altitude of the frustum is 10 cm, how far from the bottom base must a cutting plane be made to form two similar frustums? - Cutting cone
A cone with a base radius of 10 cm and a height of 12 cm is given. At what height above the base should we divide it by a section parallel to the base so that the volumes of the two resulting bodies are the same? Express the result in cm. - Digging a pit
The pit has the shape of a regular quadrilateral truncated pyramid. The edges of the bases are 14m and 10m long. The sidewalls form an angle of 135° with a smaller base. Find how many m³ of soil were excavated when digging the pit.
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