What is the surface area of a regular pyramid with a height 10 base 9 by 9?

Answer 1

Since clearly the question was copied out of an assignment with no thought whatsoever, I'm going to have to guess at what shape the base is and which dimensions have been given.

As the base is described as 9 [units] by 9 [units], I'll guess that it has a square base.

Next, the height of the pyramid is given as 10 [units]. With no picture, I'm going to have to guess where the apex is above the base - the position of the apex will affect the total surface area; I'll workout two such scenarios:

• Right square based pyramid:

As the apex is located over the centre of the base, all four triangular sides are equal. The height of each triangle can be worked out using Pythagoras: by joining the apex to the centre of the base, and then the centre of the base to the centre of one of one of the sides of the square base and thence from here to the apex. Third side, the hypotenuse of the triangle just described, is the height of the triangular side used to find its area:

height_triangle = √(102 + (9/2)2) = 1/2 x √(202 + 92) = 1/2 x √481

→ total surface area = area_base + 4 x area_triangular_sides

= 9 x 9 + 4 x 1/2 x 9 x 1/2 x √481

= 9 x 9 + 9 x √481

≈ 278.39 sq units

• Oblique square based pyramid with the apex directly over one of the corners of the base:

In this case, there are two sets of identical triangles - the two next to the corner over which the apex is situated, and the other two triangles.

For the two next to the corner, their total area is: 2 x 1/2 x 9 x 10 sq units

For the other two triangles, their height is the hypotenuse of the first two triangles, so their total area is: 2 x 1/2 x 9 x √(102 + 92) sq units

→ total surface area = area_base + area_sides_next_to_corner_under_apex + area_other_two_sides

= 9 x 9 + 2 x 1/2 x 9 x 10 + 2 x 1/2 x 9 x √(102 + 92)

= 9 x 9 + 9 x 10 + 9 x √181

≈ 292.08 sq units

The original answer (below) assumes that it is a right square based pyramid with the a slant height (ie height of each triangular side) of 10 [units].

You will need to use the area of 4 triangles, and 1 square to find the surface area of a regular pyramid. The answer to the surface area of a regular pyramid is 261 sq. units.