Key takeaway: This grade 7 area surface area lesson teaches students to split a complicated figure into simple shapes and to measure prisms in three ways. To start, area covers a flat region, surface area covers every face of a solid, and volume counts the space inside. Seventh graders also use the formulas V = Bh and SA = 2B + Ph, which come from standard 7.G.B.6.
First, you cut a composite figure into rectangles and triangles. Next, you find the base area of a prism. Finally, you combine that base with the height to get volume or with the perimeter to get surface area.
To find the area of a composite figure, split it into rectangles and triangles and add the pieces. The volume of a right prism is \(V=Bh\), the base area times the height. Surface area is the sum of the areas of all faces. Grade 7 students apply these to prisms and composite solids.
Grade 7 Area Surface Area Lesson: key facts
- Area of a triangle: \(A=\frac{1}{2}bh\).
- Volume of a right prism: \(V=Bh\).
- Surface area of a right prism: \(SA=2B+Ph\).
- Area uses square units, and volume uses cubic units.
- Break composite figures and solids into simple parts, then add.
How do you find the area of a composite figure?
A composite figure is built from simple shapes such as rectangles and triangles. To find its area, break it into pieces you know, find each area, and add. You can also draw a larger rectangle and subtract the missing piece.
| Shape | Area formula |
|---|---|
| Rectangle | \(A=\ell w\) |
| Triangle | \(A=\frac{1}{2}bh\) |
| Parallelogram | \(A=bh\) |
| Trapezoid | \(A=\frac{1}{2}(b_1+b_2)h\) |
In Figure 1, the area is \(10\times 4+6\times 3=40+18=58\text{ ft}^2\).
How do you find the volume of a prism?
\(V=Bh\), where \(B\) is the area of one base and \(h\) is the height of the prism. Volume is measured in cubic units.
For the prism in Figure 2, \(B=6\times 4=24\) and \(V=24\times 3=72\text{ in}^3\).
How do you find the surface area of a prism?
Surface area is the total area of all faces. Draw the net, find the area of each face, and add. For a right prism, \(SA=2B+Ph\), where \(P\) is the perimeter of the base.
For the prism in Figure 3, \(SA=2(24)+2(18)+2(12)=48+36+24=108\text{ in}^2\).
How do you solve a prism problem step by step?
- Sketch or label the figure, including the base and the height.
- Find the area of the base (split it into simple shapes if needed).
- For volume, multiply the base area by the height. For surface area, add the areas of every face.
- Write the answer with square or cubic units.
Worked example. Find the volume and surface area of the triangular prism with right-triangle bases (legs \(3\text{ cm}\) and \(4\text{ cm}\), hypotenuse \(5\text{ cm}\)) and length \(10\text{ cm}\).
Base area: \(B=\frac{1}{2}\times 3\times 4=6\), so \(V=6\times 10=60\text{ cm}^3\). Surface area: the two bases give \(12\), and the rectangles give \((3+4+5)\times 10=120\), so \(SA=132\text{ cm}^2\).
For a composite solid, find the volume of each part and add them.
Area, Surface Area and Volume of Prisms: frequently asked questions
What does B mean in V = Bh?
The letter \(B\) stands for the area of the base of the prism, not a side length. For a rectangular prism, \(B=\ell w\). For a triangular prism, \(B=\frac{1}{2}bh\) using the triangle’s base and height.
Does V = Bh work for a triangular prism?
Yes. It works for every right prism. Find the area of the triangular base first, then multiply by the length of the prism, which is the distance between the two triangular bases.
What is the difference between area and surface area?
Area measures a flat 2D region, such as one face. Surface area measures the total area of all the faces of a 3D solid. Both use square units.
How do you find the volume of a composite solid?
Break the solid into rectangular prisms or other right prisms, find the volume of each part, and add them. You can also find the volume of a larger box and subtract any missing piece.
Why is volume measured in cubic units?
Volume counts how many unit cubes fit inside a solid. A unit cube has edges of 1 unit, so the count is in cubic units, such as \(\text{in}^3\) or \(\text{cm}^3\).
Why split a composite figure before finding its area?
First, sketch dashed lines that cut the figure into pieces you already know. For example, an L-shaped patio becomes two rectangles, so you add their areas. A banner becomes a rectangle plus a triangle, because the triangle uses one half times base times height. However, if a piece is cut out, such as a 5 by 4 foot pool inside a 20 by 12 foot lawn, subtract it instead. In that case 240 minus 20 leaves 220 square feet.
Which formula gives the volume of a prism?
First work out the base area, and then multiply by the height. For instance, a triangular prism with right-triangle legs of 7 and 6 inches has a base area of 21 square inches. Because the prism is 8 inches long, the volume is 168 cubic inches.
What does a grade 7 area surface area lesson say about surface area?
Simply add the areas of all faces. For a prism, the two bases give 2B, and then the perimeter of the base times the height gives the sides. As an example, a cube with 7-foot edges has six faces of 49, so the surface area is 294 square feet. Its volume, in contrast, is 343 cubic feet.
What mixes up the units?
Area and surface area use square units, but volume uses cubic units. In addition, many students forget that a base is the face the prism stands on, not always the bottom of a picture.
Try these steps after the lesson
Pick a cereal box and a wedge of cheese from the kitchen and sketch their nets. Measure the edges, then compute both surface area and volume on paper. Once those sketches feel natural, move to the worksheet and begin with its trapezoid and L-shape items.
See every chapter on the Grade 7 math page.
