Topic 3.1: Volumes and Surface Areas of 3D Shapes Questions

Point P has coordinates (2, 5) and point Q has coordinates (8, 13). What is the distance PQ?

Point A has coordinates (−3, 4) and point B has coordinates (5, −2). What are the coordinates of the midpoint M of [AB]?

Two towns are at (0, 0) and (5, 12) on a coordinate grid where 1 unit = 1 km. What is the distance between the towns?

A cylinder has radius 5 cm and height 8 cm. What is its curved surface area?

A cone has radius r and slant height l. What is its curved surface area?

Point P has coordinates (−3, 2) and point Q has coordinates (15, −8). Point M is the midpoint of [PQ]. (a) Find the coordinates of M. (b) Find the distance PQ.

Two cafes are at A(1, 3) and B(7, 7) on a coordinate grid where 1 unit = 1 km. Find the exact distance AB.

A sphere has radius 3 cm. Which expression gives its volume?

To find the slant height of a cone with vertical height h and base radius r, which formula is used?

A closed cylinder has radius 4 cm and height 10 cm. (a) Calculate the total surface area of the cylinder. (b) Calculate the volume of the cylinder.

A vertical pole stands at the origin O. The top F of the pole is at (0, 0, 5.8). A rope runs from F to the ground at A(3.2, 4.5, 0). All units are in metres. (a) Find the length of the rope AF. (b) Find the angle the rope makes with the ground.

Three vertices of a triangle are at P(0, 0), Q(6, 0) and R(3, 4). (a) Find the lengths PQ, QR and PR. (b) State whether the triangle is isosceles. Justify your answer.

The midpoint of a line segment is M(5, 2). One endpoint is A(3, −1). Find the coordinates of the other endpoint B.

Show that the point M(3, 0) is equidistant from A(−1, 3) and B(7, 3).

A map shows a park with corners at P(0, 0), Q(12, 0), R(12, 9) and S(0, 9). A path runs directly from P to R.

A pyramid has a square base of side 6 cm and vertical height 4 cm. What is the volume?

In 3D, a point F has coordinates (0, 0, 5.8) and a point A has coordinates (3.2, 4.5, 0). Which expression gives the length AF?

If the midpoint of [PQ] is M(4, 3) and P is at (1, 1), what are the coordinates of Q?

A scoop of ice cream is a sphere with radius 3.4 cm. The ice cream sits in a cone with slant height 11 cm and base radius 3 cm. The outside of the cone is covered with chocolate. (a) Calculate the volume of the sphere of ice cream. (b) Calculate the curved surface area of the cone covered with chocolate.

The front view of a doghouse consists of a square with an isosceles triangle on top. The doghouse is 1.35 m high and 0.9 m wide. The top of the rectangular roof surfaces is to be painted. Find the total area of the two rectangular roof surfaces to be painted.