Angles, polygons and properties of shapes：練習問題

A regular polygon has an exterior angle of \(45^\circ\).
How many sides does this polygon have?

The volume of a cylinder is \(180\pi\text{ cm}^3\). The radius of the cylinder is \(6\text{ cm}\).
Calculate the height of the cylinder.

The diagram shows a solid cylinder with a radius of \( r \text{ cm} \) and a height of \( 10 \text{ cm} \).
The total surface area of the cylinder is \( 192\pi \text{ cm}^2 \).
Find the value of \( r \).

Triangle ABC is a right-angled triangle where side \(AB = 3\text{ cm}\) and side \(BC = 4\text{ cm}\).
Calculate the length of the hypotenuse \(AC\).

In triangle \(ABC\), angle \(A = 42^\circ\) and angle \(B = 67^\circ\).
Calculate the size of the exterior angle at vertex \(C\).

A solid square-based pyramid has a base with side length 6 cm and a perpendicular height of 4 cm.
Calculate the total surface area of the pyramid, including the base.

The diagram shows a garden in the shape of a rectangle \(ABCD\) and a semicircle. The rectangle has a length \(AB = 12\text{ m}\) and a width \(BC = 8\text{ m}\). The semicircle is attached to the side \(BC\), where \(BC\) is the diameter of the semicircle.

a) Calculate the total area of the garden. Give your answer correct to 3 significant figures.
b) A decorative fence is to be placed around the entire perimeter of the garden (the boundary of the combined shape). Calculate the total length of the fence. Give your answer correct to 1 decimal place.

In the diagram, \(ABCD\) is a rectangle with \(AB = 10\text{ cm}\) and \(BC = 6\text{ cm}\).
A semicircular piece is removed from the rectangle, where \(BC\) is the diameter of the semicircle. This leaves a shape \(S\).

(a) Calculate the perimeter of the shape \(S\). Give your answer correct to 3 significant figures.

(b) Calculate the area of the shape \(S\). Give your answer correct to 3 significant figures.

(c) A mathematically similar shape \(T\) has an area of \(210.6\text{ cm}^2\). Work out the ratio of the perimeter of shape \(S\) to the perimeter of shape \(T\).