Advanced problem solving with further surface area and volume | Maths

Starter quiz

Match each measurement to the correct formula.

Volume of hemisphere

⇔

$\frac{2}{3} \pi r^3$ ✓

Volume of cone

⇔

$\frac{1}{3}\pi r^2 h$ ✓

Volume of cylinder

⇔

$\pi r^2 h$ ✓

Volume of pyramid

⇔

$\frac{1}{3}\times \text{area of base} \times \text {height}$ ✓

Volume of sphere

⇔

$\frac{4}{3} \pi r^3$ ✓
Select the algebraic expression for the volume of this cone.

$4 \pi x^2$

$12 \pi x^2$

$48\pi x$ ✓

$144\pi x$
The volume of this cone is the same as the volume of a sphere with a radius of 12 cm. The height, $x$ of the cone is ______ cm.

'48' ✓
Find the surface area of a cube with a volume of 512 cm².

150 cm²

216 cm²

294 cm²

384 cm² ✓

486 cm²
The height, $h$ , of a cone is three times the length of its radius. Find an algebraic expression for the volume of the cone in terms of $h$ .

$\frac {1}{27}\pi h^3$ ✓

$\frac {1}{9}\pi h^3$

$\frac {1}{3}\pi h^3$

$\pi h^3$

$3\pi h^3$
The radius of a cylinder is half of the height of the cylinder. The volume of the cylinder is 1024𝜋 cm³. The surface area of the cylinder, in terms of 𝜋, is ______ cm².

'384' ✓

Exit quiz

Match each measurement to the correct formula.

Surface area of a sphere

⇔

$2 \pi r^2$ ✓

Volume of cylinder

⇔

$\pi r^2 h$ ✓

Area of curved surface of cone

⇔

$\pi r l$ ✓

Volume of sphere

⇔

$\frac{4}{3}\pi r^3$ ✓

Curved surface area of cylinder

⇔

$2\pi r h$ ✓

Volume of cone

⇔

$\frac{1}{3}\pi r^2h$ ✓
The diagram shows a cuboid juice carton which is partially filled with juice. Laura changes the orientation of the carton. Which statement is true?

The height of the juice will be lower. ✓

The height of the juice will rise.

The height of the juice will remain the same.
Calculate the volume of this prism.

270 cm³ (3 s.f.)

468 cm³ (3 s.f.) ✓

540 cm³ (3 s.f.)

935 cm³ (3 s.f.)
The diagram shows a cuboid. The length of the base is 2 cm greater than the height. Find an expression for the volume of the cuboid.

$6x(x+2)$ ✓

$2x(x+6)$

$6x^2+2$

$6x^2+12$

$6x^2+12x$ ✓
The diagram shows a cuboid. The length of the base is 2 cm greater than the height. The volume of the cuboid is 90 cm³. The surface area of the cuboid is $x$ is ______ cm².

'126' ✓
The volume of the cone is 144𝜋 cm³. Its height is twice the length of its radius. The total surface area of the cone, in terms of 𝜋, is ______.

$144\pi$ cm²

$\left(36\sqrt{5}\right)\pi$ cm²

$\left(36+36\sqrt{5}\right)\pi$ cm² ✓

$\left(36+36\sqrt{2}\right)\pi$ cm²

Worksheet

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Lesson Details

Key learning points

The surface area of any solid can be calculated by a known method.
The volume of any solid can be calculated by a known method.
Writing an algebraic statement about surface area/volume can be done from a diagram.

Common misconception

When questions are in context, the length may be described as the depth or height and this can cause some pupils to struggle if they have learned a formula with a particular word.

Remind pupils that the volume of a 3D shape comes from the area of a cross section multiplied by the height or depth. They may need to evaluate a perpendicular length from other given information, using trigonometry or Pythagoras' theorem.

Keywords

Volume - The volume is the amount of space occupied by a closed 3D shape.
Surface area - The surface area is the total area of all the surfaces of a closed 3D shape. The surfaces include all faces and any curved surfaces.
Compound shape - A compound shape is a shape created using two or more basic shapes. A composite shape is an alternative for compound shape.