Area, Surface Area, and Volume: math practice, Grade 10

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Math practice Grade 10 : Area, Surface Area, and Volume — Zyro the alien explorer of Planète Maths

23 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

1 Area of a triangle ★★★

A triangular sail has a base of 12 cm and a height of 7 cm on a scale drawing. Find its area.

2 A trapezoid garden ★★★

A garden bed is a trapezoid with parallel sides 9 m and 15 m, and the distance between them is 6 m. What is its area?

3 A shoebox ★★★

A shoebox measures 12 in by 7 in by 5 in. Find its volume and its total surface area.

4 A soup can ★★★

A soup can has radius 4 cm and height 10 cm. What volume of soup can it hold? Give an exact value and a value rounded to the nearest hundredth.

5 A party hat ★★★

A cone-shaped party hat has base radius 9 cm and height 12 cm. Find the slant height, the volume, and the lateral area (rounded to the nearest hundredth).

6 A globe ★★★

A desk globe has radius 5 in. Find its volume and surface area, in terms of \(\pi\) and rounded to the nearest hundredth.

7 Name the cross section ★★★

Name the shape of each cross section.

A plane parallel to the bases of a cylinder.
A plane through the axis of a cylinder, perpendicular to its bases.
A plane through the apex of a cone and the center of its base.
A plane parallel to the base of a square pyramid.

8 Doubling the radius ★★★

True or false: if you double the radius of a sphere, its volume doubles. Justify with a calculation using \(r = 2\) and \(r = 4\), and say what happens to the surface area.

9 A patio with a curved end ★★★

A patio is a 20 ft by 12 ft rectangle with a semicircle attached to one 12 ft side, as shown.

Find the total area, to the nearest hundredth.

10 A hexagonal tile ★★★

A regular hexagon has side length 6 cm. Its apothem is \(3\sqrt{3}\) cm. Find its area, exact and rounded to the nearest hundredth.

11 A tent shaped like a prism ★★★

A camping tent is a triangular prism. The triangular end is a right triangle with legs 6 cm and 8 cm and hypotenuse 10 cm (this is a scale model), and the length is 15 cm.

Find the volume and the total surface area.

12 A glass pyramid ★★★

A museum entrance is a square pyramid with base side 10 m and height 12 m. Find its volume, its slant height, and its total surface area (all faces and the floor).

13 An ice cream cone ★★★

An empty waffle cone has base radius 6 cm and slant height 10 cm. Find its height, its volume, and the area of its lateral surface (nearest hundredth).

14 Slicing a cone ★★★

A cone has height 20 cm and base radius 8 cm. A plane parallel to the base cuts the cone 12 cm below the apex. Find the area of the cross section, to the nearest hundredth.

15 Slicing a sphere ★★★

A sphere has radius 13 cm. A plane cuts it at a distance of 5 cm from the center. Find the radius and the area of the circular section.

16 A water tank ★★★

A cylindrical tank has diameter 6 ft and is filled 5 ft deep. Using 1 ft³ ≈ 7.48 gallons, how many gallons does it hold, to the nearest gallon?

17 A steel rod ★★★

A solid steel cylinder has radius 2 cm and height 10 cm. Steel has density 7.85 g/cm³. Find its mass to the nearest tenth of a gram.

18 A hollow pipe ★★★

A pipe 20 in long has outer radius 5 in and inner radius 4 in. How much material is in the pipe wall? Give the exact value and a decimal rounded to the nearest hundredth.

19 A stack of coins ★★★

A straight stack of 40 coins, each 0.2 cm thick with radius 1.2 cm, is pushed into a leaning stack of the same height. Use Cavalieri’s principle to find the volume of the leaning stack, to the nearest hundredth.

20 An ice cream treat ★★★

A scoop of ice cream is a hemisphere of radius 3 cm resting on a cone of radius 3 cm and height 9 cm, as shown. Find the total volume that the cone and the scoop contain, to the nearest hundredth.

21 Scaling similar cones ★★★

Two similar cones have heights in the ratio 2 : 3. The smaller cone has volume 40 cm³ and surface area 50 cm². Find the volume and surface area of the larger cone.

22 A grain silo ★★★

A silo is a cylinder of radius 6 ft and height 20 ft topped by a hemisphere of the same radius. It is filled to 80% of its total volume with wheat of density 48 lb/ft³. Find the mass of the wheat to the nearest 10 pounds.