Question Type 5: Calculating the surface area of specific complex shapes made by combining two or more shapes Exercises

A solid is formed by joining the base of radius $5 \text{ cm}$ of a conical frustum (height $4 \text{ cm}$, radii $5 \text{ cm}$ and $3 \text{ cm}$) to the open face of a hemisphere of radius $5 \text{ cm}$. Calculate the total surface area of the composite solid.

An ice-cream shaped solid is made by attaching a hemisphere of radius 3 cm onto the base of a right circular cone with the same radius and slant height 10 cm. Find the total external surface area of the solid, excluding the circular area where they join. Give your answer in terms of $\pi$ and as a decimal to 1 d.p. (cm$^2$).

A solid is formed by joining a hemisphere of radius 3 cm to the smaller circular end of a frustum of a right circular cone. The frustum has radii $R=6$ cm and $r=3$ cm and slant height $s=10$ cm. The larger circular end of the frustum is a flat base.

Find the total external surface area of the solid, excluding the circular area where the hemisphere and frustum meet. Give your answer in terms of $\pi$ and as a decimal to 1 d.p. (cm$^2$).

A solid is formed by placing a right circular cone on top of a cylinder. The common radius is $5 \text{ cm}$. The cylinder has height $8 \text{ cm}$, and the cone has vertical height $12 \text{ cm}$. The base of the cylinder is a flat circular face.

Find the total external surface area of the solid, excluding the circular area where they join. Give your answer in terms of $\pi$ and as a decimal to $1$ d.p.

A solid comprises a hemisphere of radius $5 \text{ cm}$ attached to a right circular cone of base radius $5 \text{ cm}$ and height $4 \text{ cm}$. Calculate the total curved surface area of this composite solid.

A solid is made by attaching a half-cylinder (a semicircular cross-section) of radius 4 cm and length 12 cm along the entire top of a rectangular block measuring 12 cm by 8 cm by 5 cm. The flat face where the half-cylinder meets the block is not exposed.

Find the total external surface area of the solid. Give your answer in terms of $\pi$ and as a decimal to 1 d.p. (cm$^2$).

A conical cup of height $4 \text{ cm}$ and base radius $5 \text{ cm}$ is fitted with a hemispherical lid of the same radius. Calculate the total surface area exposed (excluding the rim of the cup).

A solid is formed by joining a hemisphere of radius 4 cm to the top of a right circular cylinder of radius 4 cm and height 10 cm. The base of the cylinder is a flat circular face. Find the total external surface area of the solid, excluding the circular area where the hemisphere and cylinder are joined. Give your answer in terms of $\pi$ and as a decimal to 1 d.p. (cm$^2$).

A capsule-shaped solid consists of a cylinder of radius 6 cm and height 20 cm with a hemisphere attached to each end. Find the total external surface area of the solid. Give your answer in terms of $\pi$ and as a decimal to 1 d.p. ($\text{cm}^2$).

Derive a general formula in terms of $r$ and $h$ for the total curved surface area of a hemisphere of radius $r$ attached to a right circular cone of base radius $r$ and height $h$, then evaluate it for $r=5 \text{ cm}$ and $h=4 \text{ cm}$.

Calculate the curved surface area of a hemisphere with radius $5 \text{ cm}$.

Determine the slant height of a right circular cone with base radius $5 \text{ cm}$ and height $4 \text{ cm}$, then find its curved surface area.