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Math  /  Geometry

QuestionFind the surface area of a rectangular pyramid with side length 13 cm width 7 cm and 10 cm .
Answer Attempt 2 out of 2 453 cm2453 \mathrm{~cm}^{2} 291 cm2291 \mathrm{~cm}^{2} 125 cm2125 \mathrm{~cm}^{2} 264 cm2264 \mathrm{~cm}^{2}

Studdy Solution

STEP 1

1. The base of the pyramid is a rectangle with dimensions 13 cm×7 cm 13 \, \text{cm} \times 7 \, \text{cm} .
2. The slant height of the triangular faces is 10 cm 10 \, \text{cm} .
3. The pyramid has four triangular faces and one rectangular base.

STEP 2

1. Calculate the area of the rectangular base.
2. Calculate the area of the triangular faces.
3. Sum the areas to find the total surface area.

STEP 3

Calculate the area of the rectangular base:
Base Area=Length×Width \text{Base Area} = \text{Length} \times \text{Width} =13 cm×7 cm = 13 \, \text{cm} \times 7 \, \text{cm} =91 cm2 = 91 \, \text{cm}^2

STEP 4

Calculate the area of the triangular faces. There are two pairs of triangular faces with different bases.
For the triangles with base 13 cm 13 \, \text{cm} :
Area of one triangle=12×Base×Slant Height \text{Area of one triangle} = \frac{1}{2} \times \text{Base} \times \text{Slant Height} =12×13 cm×10 cm = \frac{1}{2} \times 13 \, \text{cm} \times 10 \, \text{cm} =65 cm2 = 65 \, \text{cm}^2
Since there are two such triangles:
Total area for these triangles=2×65 cm2=130 cm2 \text{Total area for these triangles} = 2 \times 65 \, \text{cm}^2 = 130 \, \text{cm}^2
For the triangles with base 7 cm 7 \, \text{cm} :
Area of one triangle=12×7 cm×10 cm \text{Area of one triangle} = \frac{1}{2} \times 7 \, \text{cm} \times 10 \, \text{cm} =35 cm2 = 35 \, \text{cm}^2
Since there are two such triangles:
Total area for these triangles=2×35 cm2=70 cm2 \text{Total area for these triangles} = 2 \times 35 \, \text{cm}^2 = 70 \, \text{cm}^2

STEP 5

Sum the areas to find the total surface area:
Total Surface Area=Base Area+Total Area of Triangles \text{Total Surface Area} = \text{Base Area} + \text{Total Area of Triangles} =91 cm2+130 cm2+70 cm2 = 91 \, \text{cm}^2 + 130 \, \text{cm}^2 + 70 \, \text{cm}^2 =291 cm2 = 291 \, \text{cm}^2
The surface area of the rectangular pyramid is:
291 cm2 \boxed{291 \, \text{cm}^2}

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