Find the surface area of a prism with a square base that is 5cm by 5cm and is 10cm tall.

Solution:

A prism with a square base will essentially have four slanting triangles converging at the vertex at the top. The height of the vertex from the square base is 10 cm. Therefore the surface area of the entire prism will comprise the four triangles and the base of the prism.

To find the area of the triangular face of the prism the slant height of the triangle has to be calculated. The area of the triangle is given as:

Area of the triangle = 1/2 × (base) × (height)

The height of the triangle here is the slant height. The slant height is calculated from the red color triangle in the image of the prism above. The red triangle is a right-angled triangle with a base = 5/2 = 2.5 cm (half of the length of the side of the square base)

Height = 10 cm

Since it is a right-angled triangle the Pythagoras theorem can be applied. The hypotenuse, in this case, is the slant height of the triangle which has to be found out to find the area of the triangle. Therefore,

(Slant Height)² = (10)² + (2.5)² = 100 + 6.25 = 106.25 cm²

Slant Height = √106.25 ≅ 10.3 cm

The area of the triangle = (1/2) × (5) × (10.3) = 25.75 cm²

Area of the square base = 5 × 5 = 25 cm²

Since it is a prism with a square base it has four triangles and one square base. The total surface area of the prism is calculated as follows:

The total surface area of the prism = 4 × (area of the triangle) + area of square

= [4 × (25.75) + (5 × 5)] cm²

= [103 + 25]cm²

= 128cm²

Find the surface area of a prism with a square base that is 5cm by 5cm and is 10cm tall.

Summary:

The surface area of the prism with a square base that is 5cm by 5cm and 10cm height is 128 cm².