Three-dimensional geometry: solids, distance and angles：練習問題

Calculate the area of a triangle with a base of \(15\) cm and a perpendicular height of \(8\) cm.

A triangle has sides of length \(7\) cm, \(9\) cm, and \(12\) cm. Calculate the measure of the angle opposite the longest side, to one decimal place.

A storage container is in the shape of a frustum of a cone. The radius of the top circular base is \(4\) m, the radius of the bottom circular base is \(6\) m, and the perpendicular height is \(5\) m. Calculate the volume of the container to one decimal place.

A right-angled triangle has legs of length \(7\) cm and \(24\) cm. Calculate the length of the hypotenuse.

In triangle \(PQR\), \(PQ = 10\) cm, angle \(P = 45^\circ\) and angle \(Q = 75^\circ\). Calculate the length of side \(QR\) to one decimal place.

In triangle ABC, side \(a = 10\) cm, side \(b = 12\) cm, and angle \(A = 50^\circ\). Calculate the measure of angle \(B\) to one decimal place.

A cuboid \(ABCDEFGH\) has a base with dimensions \(AB = 8\) cm and \(BC = 6\) cm, and a vertical height \(AE = 5\) cm. Calculate the angle that the space diagonal \(AG\) makes with the horizontal base \(ABCD\).

A garden bed is shaped as a circular sector with radius \(r\) and central angle \(\theta\) radians. If the perimeter of the sector is fixed at \(20\) m, find the value of \(\theta\) that results in the maximum possible area for the garden.

A right-angled triangle ABC has its right angle at B. The length of side AB is 7 cm and the length of side BC is 24 cm.

a) Calculate the length of the hypotenuse AC.

b) Find the size of the angle \(\angle BAC\), giving your answer in degrees correct to one decimal place.

A ship leaves a port P and sails 25 km on a bearing of 035° to reach point A. From point A, the ship changes course and sails 40 km on a bearing of 160° to reach point B.

a) Calculate the distance from the port P to point B, giving your answer correct to two decimal places.

b) Find the bearing of B from the port P.